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CONTENTS
DIFFERENTIAL CALCULUS
CHAPTER I
COLLECTION OF FORMULAS
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1. | Formulas from Algebra, Trigonometry, and Analytic Geometry ................................................................................................................................................................................................................................................................................................................................................................................................ | 1
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2. | Greek alphabet ................................................................................................................................................................................................................................................................................................................................................................................................ | 3
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3. | Rules for signs in the four quadrants ................................................................................................................................................................................................................................................................................................................................................................................................ | 3
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4. | Natural values of the trigonometric functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 4
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5. | Tables of logarithms ................................................................................................................................................................................................................................................................................................................................................................................................ | 5
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6. | Variables and constants ................................................................................................................................................................................................................................................................................................................................................................................................ | 6
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7. | Interval of a variable ................................................................................................................................................................................................................................................................................................................................................................................................ | 6
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8. | Continuous variation ................................................................................................................................................................................................................................................................................................................................................................................................ | 6
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9. | Functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 7
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10. | Independent and dependent variables ................................................................................................................................................................................................................................................................................................................................................................................................ | 7
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11. | Notation of functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 8
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12. | Values of the independent variable for which a function is defined ................................................................................................................................................................................................................................................................................................................................................................................................ | 8
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13. | Limit of a variable ................................................................................................................................................................................................................................................................................................................................................................................................ | 11
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14. | Division by zero excluded ................................................................................................................................................................................................................................................................................................................................................................................................ | 12
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15. | Infinitesimals ................................................................................................................................................................................................................................................................................................................................................................................................ | 13
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16. | The concept of infinity ( ) ................................................................................................................................................................................................................................................................................................................................................................................................ | 13
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17. | Limiting value of a function ................................................................................................................................................................................................................................................................................................................................................................................................ | 14
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18. | Continuous and discontinuous functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 14
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19. | Continuity and discontinuity of functions illustrated by their graphs ................................................................................................................................................................................................................................................................................................................................................................................................ | 16
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20. | Fundamental theorems on limits ................................................................................................................................................................................................................................................................................................................................................................................................ | 18
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21. | Special limiting values ................................................................................................................................................................................................................................................................................................................................................................................................ | 20
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22. | The limit of as  ................................................................................................................................................................................................................................................................................................................................................................................................ | 21
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23. | The number  ................................................................................................................................................................................................................................................................................................................................................................................................ | 22
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24. | Expressions assuming the form  ................................................................................................................................................................................................................................................................................................................................................................................................ | 23
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25. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 25
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26. | Increments ................................................................................................................................................................................................................................................................................................................................................................................................ | 25
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27. | Comparison of increments ................................................................................................................................................................................................................................................................................................................................................................................................ | 26
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28. | Derivative of a function of one variable ................................................................................................................................................................................................................................................................................................................................................................................................ | 27
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29. | Symbols for derivatives ................................................................................................................................................................................................................................................................................................................................................................................................ | 28
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30. | Differentiable functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 29
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31. | General rule for differentiation ................................................................................................................................................................................................................................................................................................................................................................................................ | 29
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32. | Applications of the derivative to Geometry ................................................................................................................................................................................................................................................................................................................................................................................................ | 31
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CHAPTER V
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS
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33. | Importance of General Rule ................................................................................................................................................................................................................................................................................................................................................................................................ | 34
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34. | Differentiation of a constant ................................................................................................................................................................................................................................................................................................................................................................................................ | 36
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35. | Differentiation of a variable with respect to itself ................................................................................................................................................................................................................................................................................................................................................................................................ | 37
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36. | Differentiation of a sum ................................................................................................................................................................................................................................................................................................................................................................................................ | 37
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37. | Differentiation of the product of a constant and a function ................................................................................................................................................................................................................................................................................................................................................................................................ | 37
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38. | Differentiation of the product of two functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 38
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39. | Differentiation of the product of any finite number of functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 38
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40. | Differentiation of a function with a constant exponent ................................................................................................................................................................................................................................................................................................................................................................................................ | 39
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41. | Differentiation of a quotient ................................................................................................................................................................................................................................................................................................................................................................................................ | 40
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42. | Differentiation of a function of a function ................................................................................................................................................................................................................................................................................................................................................................................................ | 44
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43. | Differentiation of inverse functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 45
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44. | Differentiation of a logarithm ................................................................................................................................................................................................................................................................................................................................................................................................ | 46
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45. | Differentiation of the simple exponential function ................................................................................................................................................................................................................................................................................................................................................................................................ | 48
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46. | Differentiation of the general exponential function ................................................................................................................................................................................................................................................................................................................................................................................................ | 49
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47. | Logarithmic differentiation ................................................................................................................................................................................................................................................................................................................................................................................................ | 50
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48. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 54
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49. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 55
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50. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 56
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51. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 56
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52. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 56
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53. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 57
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54. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 57
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55. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 61
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56. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 62
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57. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 62
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58. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 63
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59. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 63
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60. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 64
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61. | Differentiation of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 65
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62. | Implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 69
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63. | Differentiation of implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 69
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64. | Direction of a curve ................................................................................................................................................................................................................................................................................................................................................................................................ | 73
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65. | Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 76
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66. | Parametric equations of a curve ................................................................................................................................................................................................................................................................................................................................................................................................ | 79
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67. | Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point ................................................................................................................................................................................................................................................................................................................................................................................................ | 83
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68. | Lengths of polar subtangent and polar subnormal ................................................................................................................................................................................................................................................................................................................................................................................................ | 86
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69. | Solution of equations having multiple roots ................................................................................................................................................................................................................................................................................................................................................................................................ | 88
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70. | Applications of the derivative in mechanics. Velocity ................................................................................................................................................................................................................................................................................................................................................................................................ | 90
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71. | Component velocities ................................................................................................................................................................................................................................................................................................................................................................................................ | 91
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72. | Acceleration ................................................................................................................................................................................................................................................................................................................................................................................................ | 92
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73. | Component accelerations ................................................................................................................................................................................................................................................................................................................................................................................................ | 93
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74. | Definition of successive derivatives ................................................................................................................................................................................................................................................................................................................................................................................................ | 97
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75. | Notation ................................................................................................................................................................................................................................................................................................................................................................................................ | 97
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76. | The derivative ................................................................................................................................................................................................................................................................................................................................................................................................ | 98
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77. | Leibnitz's formula for the derivative of a product ................................................................................................................................................................................................................................................................................................................................................................................................ | 98
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78. | Successive differentiation of implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 100
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CHAPTER VIII
MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING
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79. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 103
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80. | Increasing and decreasing functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 106
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81. | Tests for determining when a function is increasing and when decreasing ................................................................................................................................................................................................................................................................................................................................................................................................ | 108
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82. | Maximum and minimum values of a function ................................................................................................................................................................................................................................................................................................................................................................................................ | 109
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83. | First method for examining a function for maximum and minimum values ................................................................................................................................................................................................................................................................................................................................................................................................ | 111
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84. | Second method for examining a function for maximum and minimum values ................................................................................................................................................................................................................................................................................................................................................................................................ | 112
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85. | Definition of points of inflection and rule for finding points of inflection ................................................................................................................................................................................................................................................................................................................................................................................................ | 125
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86. | Curve tracing ................................................................................................................................................................................................................................................................................................................................................................................................ | 128
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87. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 131
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88. | Definitions ................................................................................................................................................................................................................................................................................................................................................................................................ | 131
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89. | Infinitesimals ................................................................................................................................................................................................................................................................................................................................................................................................ | 132
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90. | Derivative of the arc in rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 134
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91. | Derivative of the arc in polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 135
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92. | Formulas for finding the differentials of functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 137
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93. | Successive differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 139
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94. | The derivative considered as the ratio of two rates ................................................................................................................................................................................................................................................................................................................................................................................................ | 141
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95. | Interchange of dependent and independent variables ................................................................................................................................................................................................................................................................................................................................................................................................ | 148
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96. | Change of the dependent variable ................................................................................................................................................................................................................................................................................................................................................................................................ | 149
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97. | Change of the independent variable ................................................................................................................................................................................................................................................................................................................................................................................................ | 150
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98. | Simultaneous change of both independent and dependent variables ................................................................................................................................................................................................................................................................................................................................................................................................ | 152
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99. | Curvature ................................................................................................................................................................................................................................................................................................................................................................................................ | 155
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100. | Curvature of a circle ................................................................................................................................................................................................................................................................................................................................................................................................ | 155
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101. | Curvature at a point ................................................................................................................................................................................................................................................................................................................................................................................................ | 156
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102. | Formulas for curvature ................................................................................................................................................................................................................................................................................................................................................................................................ | 159
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103. | Radius of curvature ................................................................................................................................................................................................................................................................................................................................................................................................ | 159
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104. | Circle of curvature ................................................................................................................................................................................................................................................................................................................................................................................................ | 161
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105. | Rolle's Theorem ................................................................................................................................................................................................................................................................................................................................................................................................ | 164
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106. | The Theorem of Mean Value ................................................................................................................................................................................................................................................................................................................................................................................................ | 165
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107. | The Extended Theorem of Mean Value ................................................................................................................................................................................................................................................................................................................................................................................................ | 166
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108. | Maxima and minima treated analytically ................................................................................................................................................................................................................................................................................................................................................................................................ | 167
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109. | Indeterminate forms ................................................................................................................................................................................................................................................................................................................................................................................................ | 170
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110. | Evaluation of a function taking on an indeterminate form ................................................................................................................................................................................................................................................................................................................................................................................................ | 170
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111. | Evaluation of the indeterminate form  ................................................................................................................................................................................................................................................................................................................................................................................................ | 171
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112. | Evaluation of the indeterminate form  ................................................................................................................................................................................................................................................................................................................................................................................................ | 174
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113. | Evaluation of the indeterminate form  ................................................................................................................................................................................................................................................................................................................................................................................................ | 174
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114. | Evaluation of the indeterminate form  ................................................................................................................................................................................................................................................................................................................................................................................................ | 175
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115. | Evaluation of the indeterminate forms  ................................................................................................................................................................................................................................................................................................................................................................................................ | 176
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116. | Circle of curvature. Center of curvature ................................................................................................................................................................................................................................................................................................................................................................................................ | 178
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117. | Second method for finding center of curvature ................................................................................................................................................................................................................................................................................................................................................................................................ | 180
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118. | Center of curvature the limiting position of the intersection of normals at neighboring points ................................................................................................................................................................................................................................................................................................................................................................................................ | 181
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119. | Evolutes ................................................................................................................................................................................................................................................................................................................................................................................................ | 182
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120. | Properties of the evolute ................................................................................................................................................................................................................................................................................................................................................................................................ | 186
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121. | Involutes and their mechanical construction ................................................................................................................................................................................................................................................................................................................................................................................................ | 187
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122. | Continuous functions of two or more independent variables ................................................................................................................................................................................................................................................................................................................................................................................................ | 190
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123. | Partial derivatives ................................................................................................................................................................................................................................................................................................................................................................................................ | 191
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124. | Partial derivatives interpreted geometrically ................................................................................................................................................................................................................................................................................................................................................................................................ | 192
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125. | Total derivatives ................................................................................................................................................................................................................................................................................................................................................................................................ | 194
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126. | Total differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 197
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127. | Differentiation of implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................ | 198
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128. | Successive partial derivatives ................................................................................................................................................................................................................................................................................................................................................................................................ | 202
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129. | Order of differentiation immaterial ................................................................................................................................................................................................................................................................................................................................................................................................ | 203
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130. | Family of curves. Variable parameter ................................................................................................................................................................................................................................................................................................................................................................................................ | 205
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131. | Envelope of a family of curves depending on one parameter ................................................................................................................................................................................................................................................................................................................................................................................................ | 205
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132. | The evolute of a given curve considered as the envelope of its normals ................................................................................................................................................................................................................................................................................................................................................................................................ | 208
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133. | Two parameters connected by one equation of condition ................................................................................................................................................................................................................................................................................................................................................................................................ | 209
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134. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 212
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135. | Infinite series ................................................................................................................................................................................................................................................................................................................................................................................................ | 213
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136. | Existence of a limit ................................................................................................................................................................................................................................................................................................................................................................................................ | 215
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137. | Fundamental test for convergence ................................................................................................................................................................................................................................................................................................................................................................................................ | 216
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138. | Comparison test for convergence ................................................................................................................................................................................................................................................................................................................................................................................................ | 217
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139. | Cauchy's ratio test for convergence ................................................................................................................................................................................................................................................................................................................................................................................................ | 218
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140. | Alternating series ................................................................................................................................................................................................................................................................................................................................................................................................ | 220
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141. | Absolute convergence ................................................................................................................................................................................................................................................................................................................................................................................................ | 220
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142. | Power series ................................................................................................................................................................................................................................................................................................................................................................................................ | 223
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143. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 227
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144. | Taylor's Theorem and Taylor's Series ................................................................................................................................................................................................................................................................................................................................................................................................ | 228
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145. | Maclaurin's Theorem and Maclaurin's Series ................................................................................................................................................................................................................................................................................................................................................................................................ | 230
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146. | Computation by series ................................................................................................................................................................................................................................................................................................................................................................................................ | 234
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147. | Approximate formulas derived from series. Interpolation ................................................................................................................................................................................................................................................................................................................................................................................................ | 237
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148. | Taylor's Theorem for functions of two or more variables ................................................................................................................................................................................................................................................................................................................................................................................................ | 240
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149. | Maxima and minima of functions of two independent variables ................................................................................................................................................................................................................................................................................................................................................................................................ | 243
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150. | Rectilinear asymptotes ................................................................................................................................................................................................................................................................................................................................................................................................ | 249
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151. | Asymptotes found by method of limiting intercepts ................................................................................................................................................................................................................................................................................................................................................................................................ | 249
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152. | Method of determining asymptotes to algebraic curves ................................................................................................................................................................................................................................................................................................................................................................................................ | 250
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153. | Asymptotes in polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 254
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154. | Singular points ................................................................................................................................................................................................................................................................................................................................................................................................ | 255
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155. | Determination of the tangent to an algebraic curve at a given point by inspection ................................................................................................................................................................................................................................................................................................................................................................................................ | 255
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156. | Nodes ................................................................................................................................................................................................................................................................................................................................................................................................ | 258
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157. | Cusps ................................................................................................................................................................................................................................................................................................................................................................................................ | 259
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158. | Conjugate or isolated points ................................................................................................................................................................................................................................................................................................................................................................................................ | 260
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159. | Transcendental singularities ................................................................................................................................................................................................................................................................................................................................................................................................ | 260
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160. | Tangent line and normal plane to a skew curve whose equations are given in parametric form ................................................................................................................................................................................................................................................................................................................................................................................................ | 262
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161. | Tangent plane to a surface ................................................................................................................................................................................................................................................................................................................................................................................................ | 264
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162. | Normal line to a surface ................................................................................................................................................................................................................................................................................................................................................................................................ | 266
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163. | Another form of the equations of the tangent line to a skew curve ................................................................................................................................................................................................................................................................................................................................................................................................ | 268
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164. | Another form of the equation of the normal plane to a skew curve ................................................................................................................................................................................................................................................................................................................................................................................................ | 269
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CHAPTER XXI
CURVES FOR REFERENCE
INTEGRAL CALCULUS
CHAPTER XXII
INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS
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165. | Integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 279
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166. | Constant of integration. Indefinite integral ................................................................................................................................................................................................................................................................................................................................................................................................ | 281
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167. | Rules for integrating standard elementary forms ................................................................................................................................................................................................................................................................................................................................................................................................ | 282
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168. | Trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 298
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169. | Integration of expressions containing or by a trigonometric substitution ................................................................................................................................................................................................................................................................................................................................................................................................ | 304
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170. | Determination of the constant of integration by means of initial conditions ................................................................................................................................................................................................................................................................................................................................................................................................ | 307
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171. | Geometrical signification of the constant of integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 307
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172. | Physical signification of the constant of integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 309
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173. | Differential of an area ................................................................................................................................................................................................................................................................................................................................................................................................ | 314
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174. | The definite integral ................................................................................................................................................................................................................................................................................................................................................................................................ | 314
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175. | Calculation of a definite integral ................................................................................................................................................................................................................................................................................................................................................................................................ | 316
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176. | Calculation of areas ................................................................................................................................................................................................................................................................................................................................................................................................ | 318
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177. | Geometrical representation of an integral ................................................................................................................................................................................................................................................................................................................................................................................................ | 319
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178. | Mean value of  ................................................................................................................................................................................................................................................................................................................................................................................................ | 320
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179. | Interchange of limits ................................................................................................................................................................................................................................................................................................................................................................................................ | 320
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180. | Decomposition of the interval ................................................................................................................................................................................................................................................................................................................................................................................................ | 321
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181. | The definite integral a function of its limits ................................................................................................................................................................................................................................................................................................................................................................................................ | 321
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182. | Infinite limits ................................................................................................................................................................................................................................................................................................................................................................................................ | 321
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183. | When is discontinuous ................................................................................................................................................................................................................................................................................................................................................................................................ | 322
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184. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 325
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185. | Case I ................................................................................................................................................................................................................................................................................................................................................................................................ | 325
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186. | Case II ................................................................................................................................................................................................................................................................................................................................................................................................ | 327
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187. | Case III ................................................................................................................................................................................................................................................................................................................................................................................................ | 329
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188. | Case IV ................................................................................................................................................................................................................................................................................................................................................................................................ | 331
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CHAPTER XXVI
INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION
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189. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 335
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190. | Differentials containing fractional powers of only ................................................................................................................................................................................................................................................................................................................................................................................................ | 335
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191. | Differentials containing fractional powers of only ................................................................................................................................................................................................................................................................................................................................................................................................ | 336
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192. | Change in limits corresponding to change in variable ................................................................................................................................................................................................................................................................................................................................................................................................ | 336
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193. | Differentials containing no radical except  ................................................................................................................................................................................................................................................................................................................................................................................................ | 338
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194. | Differentials containing no radical except  ................................................................................................................................................................................................................................................................................................................................................................................................ | 338
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195. | Binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 340
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196. | Conditions of integrability of binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 341
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197. | Transformation of trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 343
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198. | Miscellaneous substitutions ................................................................................................................................................................................................................................................................................................................................................................................................ | 345
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199. | Formula for integration by parts ................................................................................................................................................................................................................................................................................................................................................................................................ | 347
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200. | Reduction formulas for binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 350
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201. | Reduction formulas for trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................ | 356
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202. | To find and  ................................................................................................................................................................................................................................................................................................................................................................................................ | 359
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203. | Introduction ................................................................................................................................................................................................................................................................................................................................................................................................ | 361
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204. | The fundamental theorem of Integral Calculus ................................................................................................................................................................................................................................................................................................................................................................................................ | 361
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205. | Analytical proof of the Fundamental Theorem ................................................................................................................................................................................................................................................................................................................................................................................................ | 364
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206. | Areas of plane curves. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 365
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207. | Area when curve is given in parametric form ................................................................................................................................................................................................................................................................................................................................................................................................ | 368
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208. | Areas of plane curves. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 370
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209. | Length of a curve ................................................................................................................................................................................................................................................................................................................................................................................................ | 372
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210. | Lengths of plane curves. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 373
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211. | Lengths of plane curves. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 375
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212. | Volumes of solids of revolution ................................................................................................................................................................................................................................................................................................................................................................................................ | 377
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213. | Areas of surfaces of revolution ................................................................................................................................................................................................................................................................................................................................................................................................ | 381
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214. | Miscellaneous applications ................................................................................................................................................................................................................................................................................................................................................................................................ | 385
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215. | Successive integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 393
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216. | Partial integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 395
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217. | Definite double integral. Geometric interpretation ................................................................................................................................................................................................................................................................................................................................................................................................ | 396
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218. | Value of a definite double integral over a region ................................................................................................................................................................................................................................................................................................................................................................................................ | 400
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219. | Plane area as a definite double integral. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 402
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220. | Plane area as a definite double integral. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 406
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221. | Moment of area ................................................................................................................................................................................................................................................................................................................................................................................................ | 408
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222. | Center of area ................................................................................................................................................................................................................................................................................................................................................................................................ | 408
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223. | Moment of inertia. Plane areas ................................................................................................................................................................................................................................................................................................................................................................................................ | 410
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224. | Polar moment of inertia. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 410
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225. | Polar moment of inertia. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................ | 411
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226. | General method for finding the areas of surfaces ................................................................................................................................................................................................................................................................................................................................................................................................ | 413
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227. | Volumes found by triple integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 417
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228. | Differential equations. Order and degree ................................................................................................................................................................................................................................................................................................................................................................................................ | 421
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229. | Solutions of differential equations ................................................................................................................................................................................................................................................................................................................................................................................................ | 422
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230. | Verifications of solutions ................................................................................................................................................................................................................................................................................................................................................................................................ | 423
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231. | Differential equations of the first order and of the first degree ................................................................................................................................................................................................................................................................................................................................................................................................ | 424
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232. | Differential equations of the order and of the first degree ................................................................................................................................................................................................................................................................................................................................................................................................ | 432
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CHAPTER XXXI
INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS
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233. | Mechanical integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 443
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234. | Integral curves ................................................................................................................................................................................................................................................................................................................................................................................................ | 443
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235. | The integraph ................................................................................................................................................................................................................................................................................................................................................................................................ | 445
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236. | Polar planimeter ................................................................................................................................................................................................................................................................................................................................................................................................ | 446
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237. | Area swept over by a line ................................................................................................................................................................................................................................................................................................................................................................................................ | 446
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238. | Approximate integration ................................................................................................................................................................................................................................................................................................................................................................................................ | 448
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239. | Trapezoidal rule ................................................................................................................................................................................................................................................................................................................................................................................................ | 448
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240. | Simpson's rule (parabolic rule) ................................................................................................................................................................................................................................................................................................................................................................................................ | 449
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241. | Integrals for reference ................................................................................................................................................................................................................................................................................................................................................................................................ | 451
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................................................................................................................................................................................................................................................................................................................................................................................................ | 461
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