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

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
Transclusion Index not transcluded or unreviewed

Pages   (key to Page Status)   

Cover frontispiece Title  iv  v  vi  vii  viii  ix  x  xi  xii  xiii  xiv  xv Image 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
  1. CONTENTS

    DIFFERENTIAL CALCULUS

    CHAPTER I
    COLLECTION OF FORMULAS

  2. SECTIONPAGE
  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 ()
    ................................................................................................................................................................................................................................................................................................................................................................................................
    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 as
    ................................................................................................................................................................................................................................................................................................................................................................................................
    21
  27. 23.
    The number
    ................................................................................................................................................................................................................................................................................................................................................................................................
    22
  28. 24.
    Expressions assuming the form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    23
  29. CHAPTER IV

    DIFFERENTIATION

  30. 25.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    25
  31. 26.
    Increments
    ................................................................................................................................................................................................................................................................................................................................................................................................
    25
  32. 27.
    Comparison of increments
    ................................................................................................................................................................................................................................................................................................................................................................................................
    26
  33. 28.
    Derivative of a function of one variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    27
  34. 29.
    Symbols for derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    28
  35. 30.
    Differentiable functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    29
  36. 31.
    General rule for differentiation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    29
  37. 32.
    Applications of the derivative to Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    31
  38. CHAPTER V
    RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORMS

  39. 33.
    Importance of General Rule
    ................................................................................................................................................................................................................................................................................................................................................................................................
    34
  40. 34.
    Differentiation of a constant
    ................................................................................................................................................................................................................................................................................................................................................................................................
    36
  41. 35.
    Differentiation of a variable with respect to itself
    ................................................................................................................................................................................................................................................................................................................................................................................................
    37
  42. 36.
    Differentiation of a sum
    ................................................................................................................................................................................................................................................................................................................................................................................................
    37
  43. 37.
    Differentiation of the product of a constant and a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    37
  44. 38.
    Differentiation of the product of two functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    38
  45. 39.
    Differentiation of the product of any finite number of functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    38
  46. 40.
    Differentiation of a function with a constant exponent
    ................................................................................................................................................................................................................................................................................................................................................................................................
    39
  47. 41.
    Differentiation of a quotient
    ................................................................................................................................................................................................................................................................................................................................................................................................
    40
  48. 42.
    Differentiation of a function of a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    44
  49. 43.
    Differentiation of inverse functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    45
  50. 44.
    Differentiation of a logarithm
    ................................................................................................................................................................................................................................................................................................................................................................................................
    46
  51. 45.
    Differentiation of the simple exponential function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    48
  52. 46.
    Differentiation of the general exponential function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    49
  53. 47.
    Logarithmic differentiation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    50
  54. 48.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    54
  55. 49.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    55
  56. 50.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  57. 51.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  58. 52.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  59. 53.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    57
  60. 54.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    57
  61. 55.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    61
  62. 56.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    62
  63. 57.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    62
  64. 58.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    63
  65. 59.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    63
  66. 60.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    64
  67. 61.
    Differentiation of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    65
  68. 62.
    Implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    69
  69. 63.
    Differentiation of implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    69
  70. CHAPTER VI
    SIMPLE APPLICATIONS OF THE DERIVATIVE

  71. 64.
    Direction of a curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    73
  72. 65.
    Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    76
  73. 66.
    Parametric equations of a curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    79
  74. 67.
    Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point
    ................................................................................................................................................................................................................................................................................................................................................................................................
    83
  75. 68.
    Lengths of polar subtangent and polar subnormal
    ................................................................................................................................................................................................................................................................................................................................................................................................
    86
  76. 69.
    Solution of equations having multiple roots
    ................................................................................................................................................................................................................................................................................................................................................................................................
    88
  77. 70.
    Applications of the derivative in mechanics. Velocity
    ................................................................................................................................................................................................................................................................................................................................................................................................
    90
  78. 71.
    Component velocities
    ................................................................................................................................................................................................................................................................................................................................................................................................
    91
  79. 72.
    Acceleration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    92
  80. 73.
    Component accelerations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    93
  81. CHAPTER VII
    SUCCESSIVE DIFFERENTIATION

  82. 74.
    Definition of successive derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    97
  83. 75.
    Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    97
  84. 76.
    The derivative
    ................................................................................................................................................................................................................................................................................................................................................................................................
    98
  85. 77.
    Leibnitz's formula for the derivative of a product
    ................................................................................................................................................................................................................................................................................................................................................................................................
    98
  86. 78.
    Successive differentiation of implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    100
  87. CHAPTER VIII
    MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

  88. 79.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    103
  89. 80.
    Increasing and decreasing functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    106
  90. 81.
    Tests for determining when a function is increasing and when decreasing
    ................................................................................................................................................................................................................................................................................................................................................................................................
    108
  91. 82.
    Maximum and minimum values of a function
    ................................................................................................................................................................................................................................................................................................................................................................................................
    109
  92. 83.
    First method for examining a function for maximum and minimum values
    ................................................................................................................................................................................................................................................................................................................................................................................................
    111
  93. 84.
    Second method for examining a function for maximum and minimum values
    ................................................................................................................................................................................................................................................................................................................................................................................................
    112
  94. 85.
    Definition of points of inflection and rule for finding points of inflection
    ................................................................................................................................................................................................................................................................................................................................................................................................
    125
  95. 86.
    Curve tracing
    ................................................................................................................................................................................................................................................................................................................................................................................................
    128
  96. CHAPTER IX
    DIFFERENTIALS

  97. 87.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    131
  98. 88.
    Definitions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    131
  99. 89.
    Infinitesimals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    132
  100. 90.
    Derivative of the arc in rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    134
  101. 91.
    Derivative of the arc in polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    135
  102. 92.
    Formulas for finding the differentials of functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    137
  103. 93.
    Successive differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    139
  104. CHAPTER X
    RATES

  105. 94.
    The derivative considered as the ratio of two rates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    141
  106. CHAPTER XI
    CHANGE OF VARIABLE

  107. 95.
    Interchange of dependent and independent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    148
  108. 96.
    Change of the dependent variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    149
  109. 97.
    Change of the independent variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    150
  110. 98.
    Simultaneous change of both independent and dependent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    152
  111. CHAPTER XII
    CURVATURE. RADIUS OF CURVATURE

  112. 99.
    Curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    155
  113. 100.
    Curvature of a circle
    ................................................................................................................................................................................................................................................................................................................................................................................................
    155
  114. 101.
    Curvature at a point
    ................................................................................................................................................................................................................................................................................................................................................................................................
    156
  115. 102.
    Formulas for curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    159
  116. 103.
    Radius of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    159
  117. 104.
    Circle of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    161
  118. CHAPTER XIII
    THEOREM OF MEAN VALUE. INDETERMINATE FORMS

  119. 105.
    Rolle's Theorem
    ................................................................................................................................................................................................................................................................................................................................................................................................
    164
  120. 106.
    The Theorem of Mean Value
    ................................................................................................................................................................................................................................................................................................................................................................................................
    165
  121. 107.
    The Extended Theorem of Mean Value
    ................................................................................................................................................................................................................................................................................................................................................................................................
    166
  122. 108.
    Maxima and minima treated analytically
    ................................................................................................................................................................................................................................................................................................................................................................................................
    167
  123. 109.
    Indeterminate forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    170
  124. 110.
    Evaluation of a function taking on an indeterminate form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    170
  125. 111.
    Evaluation of the indeterminate form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    171
  126. 112.
    Evaluation of the indeterminate form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    174
  127. 113.
    Evaluation of the indeterminate form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    174
  128. 114.
    Evaluation of the indeterminate form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    175
  129. 115.
    Evaluation of the indeterminate forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    176
  130. CHAPTER XIV
    CIRCLE OF CURVATURE. CENTER OF CURVATURE

  131. 116.
    Circle of curvature. Center of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    178
  132. 117.
    Second method for finding center of curvature
    ................................................................................................................................................................................................................................................................................................................................................................................................
    180
  133. 118.
    Center of curvature the limiting position of the intersection of normals at neighboring points
    ................................................................................................................................................................................................................................................................................................................................................................................................
    181
  134. 119.
    Evolutes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    182
  135. 120.
    Properties of the evolute
    ................................................................................................................................................................................................................................................................................................................................................................................................
    186
  136. 121.
    Involutes and their mechanical construction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    187
  137. CHAPTER XV
    PARTIAL DIFFERENTIATION

  138. 122.
    Continuous functions of two or more independent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    190
  139. 123.
    Partial derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    191
  140. 124.
    Partial derivatives interpreted geometrically
    ................................................................................................................................................................................................................................................................................................................................................................................................
    192
  141. 125.
    Total derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    194
  142. 126.
    Total differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    197
  143. 127.
    Differentiation of implicit functions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    198
  144. 128.
    Successive partial derivatives
    ................................................................................................................................................................................................................................................................................................................................................................................................
    202
  145. 129.
    Order of differentiation immaterial
    ................................................................................................................................................................................................................................................................................................................................................................................................
    203
  146. CHAPTER XVI
    ENVELOPES

  147. 130.
    Family of curves. Variable parameter
    ................................................................................................................................................................................................................................................................................................................................................................................................
    205
  148. 131.
    Envelope of a family of curves depending on one parameter
    ................................................................................................................................................................................................................................................................................................................................................................................................
    205
  149. 132.
    The evolute of a given curve considered as the envelope of its normals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    208
  150. 133.
    Two parameters connected by one equation of condition
    ................................................................................................................................................................................................................................................................................................................................................................................................
    209
  151. CHAPTER XVII
    SERIES

  152. 134.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    212
  153. 135.
    Infinite series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    213
  154. 136.
    Existence of a limit
    ................................................................................................................................................................................................................................................................................................................................................................................................
    215
  155. 137.
    Fundamental test for convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    216
  156. 138.
    Comparison test for convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    217
  157. 139.
    Cauchy's ratio test for convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    218
  158. 140.
    Alternating series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    220
  159. 141.
    Absolute convergence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    220
  160. 142.
    Power series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    223
  161. CHAPTER XVIII
    EXPANSION OF FUNCTIONS

  162. 143.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    227
  163. 144.
    Taylor's Theorem and Taylor's Series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    228
  164. 145.
    Maclaurin's Theorem and Maclaurin's Series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    230
  165. 146.
    Computation by series
    ................................................................................................................................................................................................................................................................................................................................................................................................
    234
  166. 147.
    Approximate formulas derived from series. Interpolation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    237
  167. 148.
    Taylor's Theorem for functions of two or more variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    240
  168. 149.
    Maxima and minima of functions of two independent variables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    243
  169. CHAPTER XIX
    ASYMPTOTES. SINGULAR POINTS

  170. 150.
    Rectilinear asymptotes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    249
  171. 151.
    Asymptotes found by method of limiting intercepts
    ................................................................................................................................................................................................................................................................................................................................................................................................
    249
  172. 152.
    Method of determining asymptotes to algebraic curves
    ................................................................................................................................................................................................................................................................................................................................................................................................
    250
  173. 153.
    Asymptotes in polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    254
  174. 154.
    Singular points
    ................................................................................................................................................................................................................................................................................................................................................................................................
    255
  175. 155.
    Determination of the tangent to an algebraic curve at a given point by inspection
    ................................................................................................................................................................................................................................................................................................................................................................................................
    255
  176. 156.
    Nodes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    258
  177. 157.
    Cusps
    ................................................................................................................................................................................................................................................................................................................................................................................................
    259
  178. 158.
    Conjugate or isolated points
    ................................................................................................................................................................................................................................................................................................................................................................................................
    260
  179. 159.
    Transcendental singularities
    ................................................................................................................................................................................................................................................................................................................................................................................................
    260
  180. CHAPTER XX
    APPLICATIONS TO GEOMETRY OF SPACE

  181. 160.
    Tangent line and normal plane to a skew curve whose equations are given in parametric form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    262
  182. 161.
    Tangent plane to a surface
    ................................................................................................................................................................................................................................................................................................................................................................................................
    264
  183. 162.
    Normal line to a surface
    ................................................................................................................................................................................................................................................................................................................................................................................................
    266
  184. 163.
    Another form of the equations of the tangent line to a skew curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    268
  185. 164.
    Another form of the equation of the normal plane to a skew curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    269
  186. CHAPTER XXI
    CURVES FOR REFERENCE


    INTEGRAL CALCULUS

    CHAPTER XXII
    INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

  187. 165.
    Integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    279
  188. 166.
    Constant of integration. Indefinite integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    281
  189. 167.
    Rules for integrating standard elementary forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    282
  190. 168.
    Trigonometric differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    298
  191. 169.
    Integration of expressions containing or by a trigonometric substitution
    ................................................................................................................................................................................................................................................................................................................................................................................................
    304
  192. CHAPTER XXIII
    CONSTANT OF INTEGRATION

  193. 170.
    Determination of the constant of integration by means of initial conditions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    307
  194. 171.
    Geometrical signification of the constant of integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    307
  195. 172.
    Physical signification of the constant of integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    309
  196. CHAPTER XXIV
    THE DEFINITE INTEGRAL

  197. 173.
    Differential of an area
    ................................................................................................................................................................................................................................................................................................................................................................................................
    314
  198. 174.
    The definite integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    314
  199. 175.
    Calculation of a definite integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    316
  200. 176.
    Calculation of areas
    ................................................................................................................................................................................................................................................................................................................................................................................................
    318
  201. 177.
    Geometrical representation of an integral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    319
  202. 178.
    Mean value of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    320
  203. 179.
    Interchange of limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    320
  204. 180.
    Decomposition of the interval
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  205. 181.
    The definite integral a function of its limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  206. 182.
    Infinite limits
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  207. 183.
    When is discontinuous
    ................................................................................................................................................................................................................................................................................................................................................................................................
    322
  208. CHAPTER XXV
    INTEGRATION OF RATIONAL FRACTIONS

  209. 184.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    325
  210. 185.
    Case I
    ................................................................................................................................................................................................................................................................................................................................................................................................
    325
  211. 186.
    Case II
    ................................................................................................................................................................................................................................................................................................................................................................................................
    327
  212. 187.
    Case III
    ................................................................................................................................................................................................................................................................................................................................................................................................
    329
  213. 188.
    Case IV
    ................................................................................................................................................................................................................................................................................................................................................................................................
    331
  214. CHAPTER XXVI
    INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

  215. 189.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    335
  216. 190.
    Differentials containing fractional powers of only
    ................................................................................................................................................................................................................................................................................................................................................................................................
    335
  217. 191.
    Differentials containing fractional powers of only
    ................................................................................................................................................................................................................................................................................................................................................................................................
    336
  218. 192.
    Change in limits corresponding to change in variable
    ................................................................................................................................................................................................................................................................................................................................................................................................
    336
  219. 193.
    Differentials containing no radical except
    ................................................................................................................................................................................................................................................................................................................................................................................................
    338
  220. 194.
    Differentials containing no radical except
    ................................................................................................................................................................................................................................................................................................................................................................................................
    338
  221. 195.
    Binomial differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    340
  222. 196.
    Conditions of integrability of binomial differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    341
  223. 197.
    Transformation of trigonometric differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    343
  224. 198.
    Miscellaneous substitutions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    345
  225. CHAPTER XXVII
    INTEGRATION BY PARTS. REDUCTION FORMULAS

  226. 199.
    Formula for integration by parts
    ................................................................................................................................................................................................................................................................................................................................................................................................
    347
  227. 200.
    Reduction formulas for binomial differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    350
  228. 201.
    Reduction formulas for trigonometric differentials
    ................................................................................................................................................................................................................................................................................................................................................................................................
    356
  229. 202.
    To find and
    ................................................................................................................................................................................................................................................................................................................................................................................................
    359
  230. CHAPTER XXVIII
    INTEGRATION A PROCESS OF SUMMATION

  231. 203.
    Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    361
  232. 204.
    The fundamental theorem of Integral Calculus
    ................................................................................................................................................................................................................................................................................................................................................................................................
    361
  233. 205.
    Analytical proof of the Fundamental Theorem
    ................................................................................................................................................................................................................................................................................................................................................................................................
    364
  234. 206.
    Areas of plane curves. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    365
  235. 207.
    Area when curve is given in parametric form
    ................................................................................................................................................................................................................................................................................................................................................................................................
    368
  236. 208.
    Areas of plane curves. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    370
  237. 209.
    Length of a curve
    ................................................................................................................................................................................................................................................................................................................................................................................................
    372
  238. 210.
    Lengths of plane curves. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    373
  239. 211.
    Lengths of plane curves. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    375
  240. 212.
    Volumes of solids of revolution
    ................................................................................................................................................................................................................................................................................................................................................................................................
    377
  241. 213.
    Areas of surfaces of revolution
    ................................................................................................................................................................................................................................................................................................................................................................................................
    381
  242. 214.
    Miscellaneous applications
    ................................................................................................................................................................................................................................................................................................................................................................................................
    385
  243. CHAPTER XXIX
    SUCCESSIVE AND PARTIAL INTEGRATION

  244. 215.
    Successive integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    393
  245. 216.
    Partial integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    395
  246. 217.
    Definite double integral. Geometric interpretation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    396
  247. 218.
    Value of a definite double integral over a region
    ................................................................................................................................................................................................................................................................................................................................................................................................
    400
  248. 219.
    Plane area as a definite double integral. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    402
  249. 220.
    Plane area as a definite double integral. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    406
  250. 221.
    Moment of area
    ................................................................................................................................................................................................................................................................................................................................................................................................
    408
  251. 222.
    Center of area
    ................................................................................................................................................................................................................................................................................................................................................................................................
    408
  252. 223.
    Moment of inertia. Plane areas
    ................................................................................................................................................................................................................................................................................................................................................................................................
    410
  253. 224.
    Polar moment of inertia. Rectangular coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    410
  254. 225.
    Polar moment of inertia. Polar coördinates
    ................................................................................................................................................................................................................................................................................................................................................................................................
    411
  255. 226.
    General method for finding the areas of surfaces
    ................................................................................................................................................................................................................................................................................................................................................................................................
    413
  256. 227.
    Volumes found by triple integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    417
  257. CHAPTER XXX
    ORDINARY DIFFERENTIAL EQUATIONS

  258. 228.
    Differential equations. Order and degree
    ................................................................................................................................................................................................................................................................................................................................................................................................
    421
  259. 229.
    Solutions of differential equations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    422
  260. 230.
    Verifications of solutions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    423
  261. 231.
    Differential equations of the first order and of the first degree
    ................................................................................................................................................................................................................................................................................................................................................................................................
    424
  262. 232.
    Differential equations of the order and of the first degree
    ................................................................................................................................................................................................................................................................................................................................................................................................
    432
  263. CHAPTER XXXI
    INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

  264. 233.
    Mechanical integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    443
  265. 234.
    Integral curves
    ................................................................................................................................................................................................................................................................................................................................................................................................
    443
  266. 235.
    The integraph
    ................................................................................................................................................................................................................................................................................................................................................................................................
    445
  267. 236.
    Polar planimeter
    ................................................................................................................................................................................................................................................................................................................................................................................................
    446
  268. 237.
    Area swept over by a line
    ................................................................................................................................................................................................................................................................................................................................................................................................
    446
  269. 238.
    Approximate integration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    448
  270. 239.
    Trapezoidal rule
    ................................................................................................................................................................................................................................................................................................................................................................................................
    448
  271. 240.
    Simpson's rule (parabolic rule)
    ................................................................................................................................................................................................................................................................................................................................................................................................
    449
  272. 241.
    Integrals for reference
    ................................................................................................................................................................................................................................................................................................................................................................................................
    451
  273. ................................................................................................................................................................................................................................................................................................................................................................................................
    461