Index:Elements of the Differential and Integral Calculus - Granville - Revised.djvu

Title	Elements of the Differential and Integral Calculus
Author	William Anthony Granville
Year	1911
Source	djvu
Progress	To be proofread

Pages   (key to Page Status)   

Cover — — — — — — frontispiece Title  iv  v  vi  vii  viii  ix  x  xi  xii  xiii  xiv  xv — — Image 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468

1. CONTENTS

    

   DIFFERENTIAL CALCULUS

    

   CHAPTER I
   COLLECTION OF FORMULAS

2. SECTION

   PAGE

3. 1.

   Formulas from Algebra, Trigonometry, and Analytic Geometry ................................................................................................................................................................................................................................................................................................................................................................................................

   1

4. 2.

   Greek alphabet ................................................................................................................................................................................................................................................................................................................................................................................................

   3

5. 3.

   Rules for signs in the four quadrants ................................................................................................................................................................................................................................................................................................................................................................................................

   3

6. 4.

   Natural values of the trigonometric functions ................................................................................................................................................................................................................................................................................................................................................................................................

   4

7. 5.

   Tables of logarithms ................................................................................................................................................................................................................................................................................................................................................................................................

   5

8.  

   CHAPTER II
   VARIABLES AND FUNCTIONS

9. 6.

   Variables and constants ................................................................................................................................................................................................................................................................................................................................................................................................

   6

10. 7.

    Interval of a variable ................................................................................................................................................................................................................................................................................................................................................................................................

    6

11. 8.

    Continuous variation ................................................................................................................................................................................................................................................................................................................................................................................................

    6

12. 9.

    Functions ................................................................................................................................................................................................................................................................................................................................................................................................

    7

13. 10.

    Independent and dependent variables ................................................................................................................................................................................................................................................................................................................................................................................................

    7

14. 11.

    Notation of functions ................................................................................................................................................................................................................................................................................................................................................................................................

    8

15. 12.

    Values of the independent variable for which a function is defined ................................................................................................................................................................................................................................................................................................................................................................................................

    8

16.  

    CHAPTER III
    THEORY OF LIMITS

17. 13.

    Limit of a variable ................................................................................................................................................................................................................................................................................................................................................................................................

    11

18. 14.

    Division by zero excluded ................................................................................................................................................................................................................................................................................................................................................................................................

    12

19. 15.

    Infinitesimals ................................................................................................................................................................................................................................................................................................................................................................................................

    13

20. 16.

    The concept of infinity ($\scriptstyle {\infty }$) ................................................................................................................................................................................................................................................................................................................................................................................................

    13

21. 17.

    Limiting value of a function ................................................................................................................................................................................................................................................................................................................................................................................................

    14

22. 18.

    Continuous and discontinuous functions ................................................................................................................................................................................................................................................................................................................................................................................................

    14

23. 19.

    Continuity and discontinuity of functions illustrated by their graphs ................................................................................................................................................................................................................................................................................................................................................................................................

    16

24. 20.

    Fundamental theorems on limits ................................................................................................................................................................................................................................................................................................................................................................................................

    18

25. 21.

    Special limiting values ................................................................................................................................................................................................................................................................................................................................................................................................

    20

26. 22.

    The limit of $\scriptstyle {\frac {\sin x}{x}}$ as $\scriptstyle {x\doteq 0}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    21

27. 23.

    The number $\scriptstyle {e}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    22

28. 24.

    Expressions assuming the form $\scriptstyle {\frac {\infty }{\infty }}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    23

29.  
30. CHAPTER IV
    

    DIFFERENTIATION

31. 25.

    Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

    25

32. 26.

    Increments ................................................................................................................................................................................................................................................................................................................................................................................................

    25

33. 27.

    Comparison of increments ................................................................................................................................................................................................................................................................................................................................................................................................

    26

34. 28.

    Derivative of a function of one variable ................................................................................................................................................................................................................................................................................................................................................................................................

    27

35. 29.

    Symbols for derivatives ................................................................................................................................................................................................................................................................................................................................................................................................

    28

36. 30.

    Differentiable functions ................................................................................................................................................................................................................................................................................................................................................................................................

    29

37. 31.

    General rule for differentiation ................................................................................................................................................................................................................................................................................................................................................................................................

    29

38. 32.

    Applications of the derivative to Geometry ................................................................................................................................................................................................................................................................................................................................................................................................

    31

39.  

    CHAPTER V
    RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS

40. 33.

    Importance of General Rule ................................................................................................................................................................................................................................................................................................................................................................................................

    34

41. 34.

    Differentiation of a constant ................................................................................................................................................................................................................................................................................................................................................................................................

    36

42. 35.

    Differentiation of a variable with respect to itself ................................................................................................................................................................................................................................................................................................................................................................................................

    37

43. 36.

    Differentiation of a sum ................................................................................................................................................................................................................................................................................................................................................................................................

    37

44. 37.

    Differentiation of the product of a constant and a function ................................................................................................................................................................................................................................................................................................................................................................................................

    37

45. 38.

    Differentiation of the product of two functions ................................................................................................................................................................................................................................................................................................................................................................................................

    38

46. 39.

    Differentiation of the product of any finite number of functions ................................................................................................................................................................................................................................................................................................................................................................................................

    38

47. 40.

    Differentiation of a function with a constant exponent ................................................................................................................................................................................................................................................................................................................................................................................................

    39

48. 41.

    Differentiation of a quotient ................................................................................................................................................................................................................................................................................................................................................................................................

    40

49. 42.

    Differentiation of a function of a function ................................................................................................................................................................................................................................................................................................................................................................................................

    44

50. 43.

    Differentiation of inverse functions ................................................................................................................................................................................................................................................................................................................................................................................................

    45

51. 44.

    Differentiation of a logarithm ................................................................................................................................................................................................................................................................................................................................................................................................

    46

52. 45.

    Differentiation of the simple exponential function ................................................................................................................................................................................................................................................................................................................................................................................................

    48

53. 46.

    Differentiation of the general exponential function ................................................................................................................................................................................................................................................................................................................................................................................................

    49

54. 47.

    Logarithmic differentiation ................................................................................................................................................................................................................................................................................................................................................................................................

    50

55. 48.

    Differentiation of $\scriptstyle {\sin v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    54

56. 49.

    Differentiation of $\scriptstyle {\cos v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    55

57. 50.

    Differentiation of $\scriptstyle {\tan v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    56

58. 51.

    Differentiation of $\scriptstyle {\cot v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    56

59. 52.

    Differentiation of $\scriptstyle {\sec v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    56

60. 53.

    Differentiation of $\scriptstyle {\csc v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    57

61. 54.

    Differentiation of $\scriptstyle {\operatorname {vers} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    57

62. 55.

    Differentiation of $\scriptstyle {\operatorname {arc~sin} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    61

63. 56.

    Differentiation of $\scriptstyle {\operatorname {arc~cos} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    62

64. 57.

    Differentiation of $\scriptstyle {\operatorname {arc~tan} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    62

65. 58.

    Differentiation of $\scriptstyle {\operatorname {arc~cot} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    63

66. 59.

    Differentiation of $\scriptstyle {\operatorname {arc~sec} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    63

67. 60.

    Differentiation of $\scriptstyle {\operatorname {arc~csc} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    64

68. 61.

    Differentiation of $\scriptstyle {\operatorname {arc~vers} v}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    65

69. 62.

    Implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................

    69

70. 63.

    Differentiation of implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................

    69

71.  

    CHAPTER VI
    SIMPLE APPLICATIONS OF THE DERIVATIVE

72. 64.

    Direction of a curve ................................................................................................................................................................................................................................................................................................................................................................................................

    73

73. 65.

    Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

    76

74. 66.

    Parametric equations of a curve ................................................................................................................................................................................................................................................................................................................................................................................................

    79

75. 67.

    Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point ................................................................................................................................................................................................................................................................................................................................................................................................

    83

76. 68.

    Lengths of polar subtangent and polar subnormal ................................................................................................................................................................................................................................................................................................................................................................................................

    86

77. 69.

    Solution of equations having multiple roots ................................................................................................................................................................................................................................................................................................................................................................................................

    88

78. 70.

    Applications of the derivative in mechanics. Velocity ................................................................................................................................................................................................................................................................................................................................................................................................

    90

79. 71.

    Component velocities ................................................................................................................................................................................................................................................................................................................................................................................................

    91

80. 72.

    Acceleration ................................................................................................................................................................................................................................................................................................................................................................................................

    92

81. 73.

    Component accelerations ................................................................................................................................................................................................................................................................................................................................................................................................

    93

82.  

    CHAPTER VII
    SUCCESSIVE DIFFERENTIATION

83. 74.

    Definition of successive derivatives ................................................................................................................................................................................................................................................................................................................................................................................................

    97

84. 75.

    Notation ................................................................................................................................................................................................................................................................................................................................................................................................

    97

85. 76.

    The $\scriptstyle {n{\text{th}}}$ derivative ................................................................................................................................................................................................................................................................................................................................................................................................

    98

86. 77.

    Leibnitz's formula for the $\scriptstyle {n{\text{th}}}$ derivative of a product ................................................................................................................................................................................................................................................................................................................................................................................................

    98

87. 78.

    Successive differentiation of implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................

    100

88.  

    CHAPTER VIII
    MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

89. 79.

    Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

    103

90. 80.

    Increasing and decreasing functions ................................................................................................................................................................................................................................................................................................................................................................................................

    106

91. 81.

    Tests for determining when a function is increasing and when decreasing ................................................................................................................................................................................................................................................................................................................................................................................................

    108

92. 82.

    Maximum and minimum values of a function ................................................................................................................................................................................................................................................................................................................................................................................................

    109

93. 83.

    First method for examining a function for maximum and minimum values ................................................................................................................................................................................................................................................................................................................................................................................................

    111

94. 84.

    Second method for examining a function for maximum and minimum values ................................................................................................................................................................................................................................................................................................................................................................................................

    112

95. 85.

    Definition of points of inflection and rule for finding points of inflection ................................................................................................................................................................................................................................................................................................................................................................................................

    125

96. 86.

    Curve tracing ................................................................................................................................................................................................................................................................................................................................................................................................

    128

97.  

    CHAPTER IX
    DIFFERENTIALS

98. 87.

    Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

    131

99. 88.

    Definitions ................................................................................................................................................................................................................................................................................................................................................................................................

    131

100. 89.

     Infinitesimals ................................................................................................................................................................................................................................................................................................................................................................................................

     132

101. 90.

     Derivative of the arc in rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     134

102. 91.

     Derivative of the arc in polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     135

103. 92.

     Formulas for finding the differentials of functions ................................................................................................................................................................................................................................................................................................................................................................................................

     137

104. 93.

     Successive differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     139

105.  

     CHAPTER X
     RATES

106. 94.

     The derivative considered as the ratio of two rates ................................................................................................................................................................................................................................................................................................................................................................................................

     141

107.  

     CHAPTER XI
     CHANGE OF VARIABLE

108. 95.

     Interchange of dependent and independent variables ................................................................................................................................................................................................................................................................................................................................................................................................

     148

109. 96.

     Change of the dependent variable ................................................................................................................................................................................................................................................................................................................................................................................................

     149

110. 97.

     Change of the independent variable ................................................................................................................................................................................................................................................................................................................................................................................................

     150

111. 98.

     Simultaneous change of both independent and dependent variables ................................................................................................................................................................................................................................................................................................................................................................................................

     152

112.  

     CHAPTER XII
     CURVATURE. RADIUS OF CURVATURE

113. 99.

     Curvature ................................................................................................................................................................................................................................................................................................................................................................................................

     155

114. 100.

     Curvature of a circle ................................................................................................................................................................................................................................................................................................................................................................................................

     155

115. 101.

     Curvature at a point ................................................................................................................................................................................................................................................................................................................................................................................................

     156

116. 102.

     Formulas for curvature ................................................................................................................................................................................................................................................................................................................................................................................................

     159

117. 103.

     Radius of curvature ................................................................................................................................................................................................................................................................................................................................................................................................

     159

118. 104.

     Circle of curvature ................................................................................................................................................................................................................................................................................................................................................................................................

     161

119.  

     CHAPTER XIII
     THEOREM OF MEAN VALUE. INDETERMINATE FORMS

120. 105.

     Rolle's Theorem ................................................................................................................................................................................................................................................................................................................................................................................................

     164

121. 106.

     The Theorem of Mean Value ................................................................................................................................................................................................................................................................................................................................................................................................

     165

122. 107.

     The Extended Theorem of Mean Value ................................................................................................................................................................................................................................................................................................................................................................................................

     166

123. 108.

     Maxima and minima treated analytically ................................................................................................................................................................................................................................................................................................................................................................................................

     167

124. 109.

     Indeterminate forms ................................................................................................................................................................................................................................................................................................................................................................................................

     170

125. 110.

     Evaluation of a function taking on an indeterminate form ................................................................................................................................................................................................................................................................................................................................................................................................

     170

126. 111.

     Evaluation of the indeterminate form $\scriptstyle {\frac {0}{0}}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     171

127. 112.

     Evaluation of the indeterminate form $\scriptstyle {\frac {\infty }{\infty }}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     174

128. 113.

     Evaluation of the indeterminate form $\scriptstyle {0\cdot \infty }$ ................................................................................................................................................................................................................................................................................................................................................................................................

     174

129. 114.

     Evaluation of the indeterminate form $\scriptstyle {\infty -\infty }$ ................................................................................................................................................................................................................................................................................................................................................................................................

     175

130. 115.

     Evaluation of the indeterminate forms $\scriptstyle {0^{0},~1^{\infty },~\infty ^{0}}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     176

131.  

     CHAPTER XIV
     CIRCLE OF CURVATURE. CENTER OF CURVATURE

132. 116.

     Circle of curvature. Center of curvature ................................................................................................................................................................................................................................................................................................................................................................................................

     178

133. 117.

     Second method for finding center of curvature ................................................................................................................................................................................................................................................................................................................................................................................................

     180

134. 118.

     Center of curvature the limiting position of the intersection of normals at neighboring points ................................................................................................................................................................................................................................................................................................................................................................................................

     181

135. 119.

     Evolutes ................................................................................................................................................................................................................................................................................................................................................................................................

     182

136. 120.

     Properties of the evolute ................................................................................................................................................................................................................................................................................................................................................................................................

     186

137. 121.

     Involutes and their mechanical construction ................................................................................................................................................................................................................................................................................................................................................................................................

     187

138.  

     CHAPTER XV
     PARTIAL DIFFERENTIATION

139. 122.

     Continuous functions of two or more independent variables ................................................................................................................................................................................................................................................................................................................................................................................................

     190

140. 123.

     Partial derivatives ................................................................................................................................................................................................................................................................................................................................................................................................

     191

141. 124.

     Partial derivatives interpreted geometrically ................................................................................................................................................................................................................................................................................................................................................................................................

     192

142. 125.

     Total derivatives ................................................................................................................................................................................................................................................................................................................................................................................................

     194

143. 126.

     Total differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     197

144. 127.

     Differentiation of implicit functions ................................................................................................................................................................................................................................................................................................................................................................................................

     198

145. 128.

     Successive partial derivatives ................................................................................................................................................................................................................................................................................................................................................................................................

     202

146. 129.

     Order of differentiation immaterial ................................................................................................................................................................................................................................................................................................................................................................................................

     203

147.  

     CHAPTER XVI
     ENVELOPES

148. 130.

     Family of curves. Variable parameter ................................................................................................................................................................................................................................................................................................................................................................................................

     205

149. 131.

     Envelope of a family of curves depending on one parameter ................................................................................................................................................................................................................................................................................................................................................................................................

     205

150. 132.

     The evolute of a given curve considered as the envelope of its normals ................................................................................................................................................................................................................................................................................................................................................................................................

     208

151. 133.

     Two parameters connected by one equation of condition ................................................................................................................................................................................................................................................................................................................................................................................................

     209

152.  

     CHAPTER XVII
     SERIES

153. 134.

     Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

     212

154. 135.

     Infinite series ................................................................................................................................................................................................................................................................................................................................................................................................

     213

155. 136.

     Existence of a limit ................................................................................................................................................................................................................................................................................................................................................................................................

     215

156. 137.

     Fundamental test for convergence ................................................................................................................................................................................................................................................................................................................................................................................................

     216

157. 138.

     Comparison test for convergence ................................................................................................................................................................................................................................................................................................................................................................................................

     217

158. 139.

     Cauchy's ratio test for convergence ................................................................................................................................................................................................................................................................................................................................................................................................

     218

159. 140.

     Alternating series ................................................................................................................................................................................................................................................................................................................................................................................................

     220

160. 141.

     Absolute convergence ................................................................................................................................................................................................................................................................................................................................................................................................

     220

161. 142.

     Power series ................................................................................................................................................................................................................................................................................................................................................................................................

     223

162.  

     CHAPTER XVIII
     EXPANSION OF FUNCTIONS

163. 143.

     Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

     227

164. 144.

     Taylor's Theorem and Taylor's Series ................................................................................................................................................................................................................................................................................................................................................................................................

     228

165. 145.

     Maclaurin's Theorem and Maclaurin's Series ................................................................................................................................................................................................................................................................................................................................................................................................

     230

166. 146.

     Computation by series ................................................................................................................................................................................................................................................................................................................................................................................................

     234

167. 147.

     Approximate formulas derived from series. Interpolation ................................................................................................................................................................................................................................................................................................................................................................................................

     237

168. 148.

     Taylor's Theorem for functions of two or more variables ................................................................................................................................................................................................................................................................................................................................................................................................

     240

169. 149.

     Maxima and minima of functions of two independent variables ................................................................................................................................................................................................................................................................................................................................................................................................

     243

170.  

     CHAPTER XIX
     ASYMPTOTES. SINGULAR POINTS

171. 150.

     Rectilinear asymptotes ................................................................................................................................................................................................................................................................................................................................................................................................

     249

172. 151.

     Asymptotes found by method of limiting intercepts ................................................................................................................................................................................................................................................................................................................................................................................................

     249

173. 152.

     Method of determining asymptotes to algebraic curves ................................................................................................................................................................................................................................................................................................................................................................................................

     250

174. 153.

     Asymptotes in polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     254

175. 154.

     Singular points ................................................................................................................................................................................................................................................................................................................................................................................................

     255

176. 155.

     Determination of the tangent to an algebraic curve at a given point by inspection ................................................................................................................................................................................................................................................................................................................................................................................................

     255

177. 156.

     Nodes ................................................................................................................................................................................................................................................................................................................................................................................................

     258

178. 157.

     Cusps ................................................................................................................................................................................................................................................................................................................................................................................................

     259

179. 158.

     Conjugate or isolated points ................................................................................................................................................................................................................................................................................................................................................................................................

     260

180. 159.

     Transcendental singularities ................................................................................................................................................................................................................................................................................................................................................................................................

     260

181.  

     CHAPTER XX
     APPLICATIONS TO GEOMETRY OF SPACE

182. 160.

     Tangent line and normal plane to a skew curve whose equations are given in parametric form ................................................................................................................................................................................................................................................................................................................................................................................................

     262

183. 161.

     Tangent plane to a surface ................................................................................................................................................................................................................................................................................................................................................................................................

     264

184. 162.

     Normal line to a surface ................................................................................................................................................................................................................................................................................................................................................................................................

     266

185. 163.

     Another form of the equations of the tangent line to a skew curve ................................................................................................................................................................................................................................................................................................................................................................................................

     268

186. 164.

     Another form of the equation of the normal plane to a skew curve ................................................................................................................................................................................................................................................................................................................................................................................................

     269

187.  

     CHAPTER XXI
     CURVES FOR REFERENCE

      

     ---

      

     INTEGRAL CALCULUS

      

     CHAPTER XXII
     INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

188. 165.

     Integration ................................................................................................................................................................................................................................................................................................................................................................................................

     279

189. 166.

     Constant of integration. Indefinite integral ................................................................................................................................................................................................................................................................................................................................................................................................

     281

190. 167.

     Rules for integrating standard elementary forms ................................................................................................................................................................................................................................................................................................................................................................................................

     282

191. 168.

     Trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     298

192. 169.

     Integration of expressions containing $\scriptstyle {\sqrt {a^{2}-x^{2}}}$ or $\scriptstyle {\sqrt {x^{2}\pm a^{2}}}$ by a trigonometric substitution ................................................................................................................................................................................................................................................................................................................................................................................................

     304

193.  

     CHAPTER XXIII
     CONSTANT OF INTEGRATION

194. 170.

     Determination of the constant of integration by means of initial conditions ................................................................................................................................................................................................................................................................................................................................................................................................

     307

195. 171.

     Geometrical signification of the constant of integration ................................................................................................................................................................................................................................................................................................................................................................................................

     307

196. 172.

     Physical signification of the constant of integration ................................................................................................................................................................................................................................................................................................................................................................................................

     309

197.  

     CHAPTER XXIV
     THE DEFINITE INTEGRAL

198. 173.

     Differential of an area ................................................................................................................................................................................................................................................................................................................................................................................................

     314

199. 174.

     The definite integral ................................................................................................................................................................................................................................................................................................................................................................................................

     314

200. 175.

     Calculation of a definite integral ................................................................................................................................................................................................................................................................................................................................................................................................

     316

201. 176.

     Calculation of areas ................................................................................................................................................................................................................................................................................................................................................................................................

     318

202. 177.

     Geometrical representation of an integral ................................................................................................................................................................................................................................................................................................................................................................................................

     319

203. 178.

     Mean value of $\scriptstyle {\phi (x)}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     320

204. 179.

     Interchange of limits ................................................................................................................................................................................................................................................................................................................................................................................................

     320

205. 180.

     Decomposition of the interval ................................................................................................................................................................................................................................................................................................................................................................................................

     321

206. 181.

     The definite integral a function of its limits ................................................................................................................................................................................................................................................................................................................................................................................................

     321

207. 182.

     Infinite limits ................................................................................................................................................................................................................................................................................................................................................................................................

     321

208. 183.

     When $\scriptstyle {y=\phi (x)}$ is discontinuous ................................................................................................................................................................................................................................................................................................................................................................................................

     322

209.  

     CHAPTER XXV
     INTEGRATION OF RATIONAL FRACTIONS

210. 184.

     Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

     325

211. 185.

     Case I ................................................................................................................................................................................................................................................................................................................................................................................................

     325

212. 186.

     Case II ................................................................................................................................................................................................................................................................................................................................................................................................

     327

213. 187.

     Case III ................................................................................................................................................................................................................................................................................................................................................................................................

     329

214. 188.

     Case IV ................................................................................................................................................................................................................................................................................................................................................................................................

     331

215.  

     CHAPTER XXVI
     INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

216. 189.

     Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

     335

217. 190.

     Differentials containing fractional powers of $\scriptstyle {x}$ only ................................................................................................................................................................................................................................................................................................................................................................................................

     335

218. 191.

     Differentials containing fractional powers of $\scriptstyle {a+bx}$ only ................................................................................................................................................................................................................................................................................................................................................................................................

     336

219. 192.

     Change in limits corresponding to change in variable ................................................................................................................................................................................................................................................................................................................................................................................................

     336

220. 193.

     Differentials containing no radical except $\scriptstyle {\sqrt {a+bx+x^{2}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     338

221. 194.

     Differentials containing no radical except $\scriptstyle {\sqrt {a+bx-x^{2}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     338

222. 195.

     Binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     340

223. 196.

     Conditions of integrability of binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     341

224. 197.

     Transformation of trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     343

225. 198.

     Miscellaneous substitutions ................................................................................................................................................................................................................................................................................................................................................................................................

     345

226.  

     CHAPTER XXVII
     INTEGRATION BY PARTS. REDUCTION FORMULAS

227. 199.

     Formula for integration by parts ................................................................................................................................................................................................................................................................................................................................................................................................

     347

228. 200.

     Reduction formulas for binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     350

229. 201.

     Reduction formulas for trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................

     356

230. 202.

     To find $\scriptstyle {\int e^{ax}\sin {nx}dx}$ and $\scriptstyle {\int e^{ax}\cos {nx}dx}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     359

231.  

     CHAPTER XXVIII
     INTEGRATION A PROCESS OF SUMMATION

232. 203.

     Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

     361

233. 204.

     The fundamental theorem of Integral Calculus ................................................................................................................................................................................................................................................................................................................................................................................................

     361

234. 205.

     Analytical proof of the Fundamental Theorem ................................................................................................................................................................................................................................................................................................................................................................................................

     364

235. 206.

     Areas of plane curves. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     365

236. 207.

     Area when curve is given in parametric form ................................................................................................................................................................................................................................................................................................................................................................................................

     368

237. 208.

     Areas of plane curves. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     370

238. 209.

     Length of a curve ................................................................................................................................................................................................................................................................................................................................................................................................

     372

239. 210.

     Lengths of plane curves. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     373

240. 211.

     Lengths of plane curves. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     375

241. 212.

     Volumes of solids of revolution ................................................................................................................................................................................................................................................................................................................................................................................................

     377

242. 213.

     Areas of surfaces of revolution ................................................................................................................................................................................................................................................................................................................................................................................................

     381

243. 214.

     Miscellaneous applications ................................................................................................................................................................................................................................................................................................................................................................................................

     385

244.  

     CHAPTER XXIX
     SUCCESSIVE AND PARTIAL INTEGRATION

245. 215.

     Successive integration ................................................................................................................................................................................................................................................................................................................................................................................................

     393

246. 216.

     Partial integration ................................................................................................................................................................................................................................................................................................................................................................................................

     395

247. 217.

     Definite double integral. Geometric interpretation ................................................................................................................................................................................................................................................................................................................................................................................................

     396

248. 218.

     Value of a definite double integral over a region ................................................................................................................................................................................................................................................................................................................................................................................................

     400

249. 219.

     Plane area as a definite double integral. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     402

250. 220.

     Plane area as a definite double integral. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     406

251. 221.

     Moment of area ................................................................................................................................................................................................................................................................................................................................................................................................

     408

252. 222.

     Center of area ................................................................................................................................................................................................................................................................................................................................................................................................

     408

253. 223.

     Moment of inertia. Plane areas ................................................................................................................................................................................................................................................................................................................................................................................................

     410

254. 224.

     Polar moment of inertia. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     410

255. 225.

     Polar moment of inertia. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

     411

256. 226.

     General method for finding the areas of surfaces ................................................................................................................................................................................................................................................................................................................................................................................................

     413

257. 227.

     Volumes found by triple integration ................................................................................................................................................................................................................................................................................................................................................................................................

     417

258.  

     CHAPTER XXX
     ORDINARY DIFFERENTIAL EQUATIONS

259. 228.

     Differential equations. Order and degree ................................................................................................................................................................................................................................................................................................................................................................................................

     421

260. 229.

     Solutions of differential equations ................................................................................................................................................................................................................................................................................................................................................................................................

     422

261. 230.

     Verifications of solutions ................................................................................................................................................................................................................................................................................................................................................................................................

     423

262. 231.

     Differential equations of the first order and of the first degree ................................................................................................................................................................................................................................................................................................................................................................................................

     424

263. 232.

     Differential equations of the $\scriptstyle {n{\text{th}}}$ order and of the first degree ................................................................................................................................................................................................................................................................................................................................................................................................

     432

264.  

     CHAPTER XXXI
     INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

265. 233.

     Mechanical integration ................................................................................................................................................................................................................................................................................................................................................................................................

     443

266. 234.

     Integral curves ................................................................................................................................................................................................................................................................................................................................................................................................

     443

267. 235.

     The integraph ................................................................................................................................................................................................................................................................................................................................................................................................

     445

268. 236.

     Polar planimeter ................................................................................................................................................................................................................................................................................................................................................................................................

     446

269. 237.

     Area swept over by a line ................................................................................................................................................................................................................................................................................................................................................................................................

     446

270. 238.

     Approximate integration ................................................................................................................................................................................................................................................................................................................................................................................................

     448

271. 239.

     Trapezoidal rule ................................................................................................................................................................................................................................................................................................................................................................................................

     448

272. 240.

     Simpson's rule (parabolic rule) ................................................................................................................................................................................................................................................................................................................................................................................................

     449

273. 241.

     Integrals for reference ................................................................................................................................................................................................................................................................................................................................................................................................

     451

274.  

275. INDEX ................................................................................................................................................................................................................................................................................................................................................................................................

     461