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# 14 - CHAPTER 14 Multiple Integration Section 14.1 Iterated...

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Unformatted text preview: CHAPTER 14 Multiple Integration Section 14.1 Iterated Integrals and Area in the Plane . . . . . . . . . . . . . 275 Section 14.2 Double Integrals and Volume . . . . . . . . . . . . . . . . . . . 284 Section 14.3 Change of Variables: Polar Coordinates . . . . . . . . . . . . . 297 Section 14.4 Center of Mass and Moments of Inertia . . . . . . . . . . . . . 303 Section 14.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Section 14.6 Triple Integrals and Applications . . . . . . . . . . . . . . . . . 322 Section 14.7 Triple Integrals in Cylindrical and Spherical Coordinates . . . . 335 Section 14.8 Change of Variables: Jacobians Review Exercises . . . . . . . . . . . . . . . . . 341 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 CHAPTER 14 Multiple Integration Section 14.1 x Iterated Integrals and Area in the Plane 2xy 12 y 2 x 0 1. 0 2x y dy 32 x 2 x 2 2. x y dy x 1 y2 2x x2 x 1 x4 2x x2 x x2 x 2 1 2y 3. 1 y dx x 4 x2 2y cos y cos y y ln x 1 y ln 2y 0 y ln 2y, y > 0 4. 0 y dx yx 0 y cos y 5. 0 x 2y dy 122 xy 2 4 0 x2 4x 2 2 x x4 x 6. x 3 x2 y 3y2 dy x2y y ey y3 x3 x2 x x 3 x2x3 x3 3 x5 2 x3 2 x5 x9 7. e y y ln x dx x 1 y2 1 y ln2 x 2 1 y ln2 y 2 ln2 e y y ln y 2 1 1 3 2 y2 , (y > 0 8. 1 y2 x2 y 2 dx 13 x 3 y 2x 1 1 y2 y2 2 y2 32 y2 1 y2 12 21 3 y2 1 2y 2 x3 x3 x3 x3 9. 0 ye u yx dy dy, dv xye e yx 0 x 0 e xe yx dy x 4e x2 x 2e yx 0 x2 1 e x2 x 2e x2 y, du 2 yx dy, v yx 2 10. y sin3 x cos y dx y 1 cos2 x sin x cos y dx 1 cos3 x cos y 3 12 y 2 y3 3 2 1 2 cos x 1 2 1 cos y y 1 cos3 y cos y 3 1 11. 0 0 x y dy dx 0 xy dx 0 0 2x 2 dx x2 2x 0 3 1 2 1 2 1 12. 1 2 x2 y 2 dy dx 1 1 x2y 4x2 1 dx 2 1 2x2 4x3 3 sin x 8 3 1 1 2x2 4 3 8 dx 3 16 3 4 3 16 3 8 16 dx 3 16 x 3 sin x 13. 0 0 1 cos x dy dx 0 y y cos x 0 dx 0 sin x sin x cos x dx cos x 12 sin x 2 1 0 1 2 275 276 4 Chapter 14 x Multiple Integration 4 x 4 4 14. 1 1 2 ye x dy dx 1 y 2e x 1 dx 1 xe x e x dx xe x 1 4e 4 e 1 1 e 4 e4 1 x 1 x 1 15. 0 0 1 x2 dy dx 0 y1 x2 0 dx 0 x1 x2 dx 12 1 23 1 x2 32 0 1 3 4 x 2 4 x2 16. 40 64 x3 dy dx 4 4 y 64 64 4 x3 0 dx 2 64 9 4 4 x3 x2 dx x3 32 4 0 2 128 9 32 2048 9 2 2 4 2 17. 1 0 x2 2y 2 1 dx dy 1 2 1 13 x 3 64 3 2 2xy 2 8y 2 x 0 dy 4 3 2y 2 4 dy 19 1 6y 2 dy 4 19y 3 16 3 y 3 y4 2 2 2y 3 2 1 20 3 23 y 3 140 3 2 2y 18. 0 y 10 2x2 2y2 dx dy 0 2 10x 10y 0 2x3 3 14 3 y 3 2 2y2x y dy 0 20y 5y2 7y4 6 4y3 20 56 3 10y 8 2y3 dy 2y3 dy 0 1 1 0 y2 1 19. 0 x y dx dy 0 1 0 12 x 2 1 1 2 2y y2 xy 0 1 y2 dy 1 y 2 13 y 6 12 1 23 2 1 y2 y1 y 2 dy y2 32 0 2 3 2 2y 3y2 4 0 y2 2 2 20. 0 2 6y 3y dx dy 0 y2 3xy 3y2 2 6y dy 3 0 8y2 4y3 dy 2 3 83 y 3 2 y4 0 16 21. 0 2 4 y2 dx dy 0 2x 4 y2 4 0 y2 dy 0 2 dy 2y 0 4 2 2 cos 2 22. 0 0 r dr d 0 r2 2 2 cos 2 d 0 0 2 cos2 d 1 sin 2 2 2 0 2 2 sin 2 23. 0 0 r dr d 0 r2 2 2 sin 2 d 0 0 1 sin2 2 1 4 2 d 1 cos 2 4 2 2 1 4 4 cos cos 2 0 d 2 2 sin 2 0 32 1 8 4 cos 24. 0 0 3r 2 sin dr d 0 4 r 3 sin 0 d cos4 4 1 2x 4 0 cos3 sin d 0 1 4 1 2 1 2 1 2 4 1 3 16 1x 25. 1 0 y dy dx 1 y2 2 1x dx 0 1 2 1 1 dx x2 0 1 S ection 14.1 3 Iterated Integrals and Area in the Plane 3 0 277 26. 0 0 x2 1 y2 3 3 dy dx 0 x 2 arctan y 0 dx 0 x2 2 dx 2 x3 3 9 2 27. 1 1 1 dx dy xy 1 1 ln x y dy 1 1 1 y 1 0 y dy Diverges 1 ye 2 8 28. 0 0 xye x2 y2 dx dy 0 x2 y2 0 dy 0 1 ye 2 y2 dy 1 e 4 y2 0 1 4 y 8 3 8 3 8 29. A 0 3 0 8 dy dx 0 3 y 0 8 dx 0 3 3 dx 3x 0 3 24 8 6 A 0 0 dx dy 0 x 0 dy 0 8 dy 8y 0 24 4 2 x 2 4 6 8 3 3 3 3 3 3 y 30. A 1 3 1 3 dy dx 1 3 y 1 3 dx 1 3 2 dx 2x 1 3 4 3 A 1 1 dx dy 1 x 1 dy 1 2 dy 2y 1 4 2 1 x 1 2 3 2 4 0 x2 2 4 x2 y 31. A 0 2 dy dx 0 y 0 dx 4 3 y = 4 − x2 4 0 x 2 dx x3 3 4 y 2 0 2 1 x 1 2 3 4x 4 16 3 dx dy −1 A 0 4 0 4 0 x 0 y 4 4 dy 0 4 y dy 0 4 y 12 1 dy 2 4 3 5 y 32 4 0 2 8 3 16 3 5 1 0 5 x 1 5 1 x 1 5 32. A 2 12 dy dx 2 1 1 y 0 1 y2 dx 2 1 x 1 dx 2x y 1 2 2 A 0 12 2 dx dy 12 2 1 5 1 dx dy 1 y2 5 4 3 x 0 12 2 dy 12 1 x 2 dy y= 2 1 1 x−1 3 dy 0 12 12 1 y2 1 1 dy 2 12 x 1 2 3 4 5 3y 0 1 y y 278 Chapter 14 1 4 x2 Multiple Integration 2 4 0 2 x2 33. A 2x 1 2 4 dy dx x 2 2 y = 4 − x2 y 34. A 0 2 dy dx 4 0 2 y y= 4 − x2 y 2 1 x dx x2 x 12 x 2 x 2 dx −2 −1 3 2 (1, 3) x2 dx 1 y=x+2 1 x 1 2 4 2 1 4 0 2 cos 1 0 2 d x 1 2 2 2 x 2 dx 13 x 3 dx dy 1 2 4 2 9 2 4 0 4 4 y y cos 2 d 2 0 2x 3 y 2 dx dy x A dy 0 4 0 0 2 1 sin 2 2 2 A 0 3 4 y y 2 2 3 2 sin , dx 2 4 y2 2 cos d, 2 4 4 x2 y2 dy cos 2 2 cos dx dy 0 2 x 0 3 4 y dy 2 2y 4 2 4 3 2 3 x 2 3 3 4 4 y dy 4 0 cos2 2 d 2 0 2 0 1 d y 0 y dy y 32 0 12 y 2 4 4 3 y 32 3 9 2 1 sin 2 2 y 2 sin , dy 4 2x 2 cos d, 4 y2 2 cos 4 2 0 x 2 4 2 x 2 35. 0 dy dx 0 4 y 0 dx 4x 8 xx 3 x dx 36. A 0 4 x3 2 dy dx 2x 4 0 y 0 x3 2 dx x3 25 x 5 2 32 5 2 4x 4 0 0 2 y 2 x2 2 4 0 8 3 4 2x 0 dx 4 dx dy 8 3 x2 16 8 y 2 0 Integration steps are similar to those above. y 4 3 2 16 5 23 A 0 y2 dx dy 8 y = (2 − x )2 y2 0 x 1 2 3 4 3 y dy 2 y2 4 16 8 0 1 35 y 5 3 32 5 y 8 7 6 5 4 3 2 1 −1 3 16 5 (4, 8) y = 2x y = x 3/2 x 1234567 8 S ection 14.1 3 2x 3 5 5 0 5 x x Iterated Integrals and Area in the Plane 3 x 9 9x 279 37. A 0 3 0 dy dx 3 2x 3 5 dy dx y 3 5 0 38. A 0 0 3 dy dx 3 x 9 0 dy dx 9x 3 9 y 0 3 0 0 dx 5 3 dx y 0 0 3 dx 3 y 0 9 dx 0 x dx 3 9 dx x 2x dx 3 3 x dx 12 x 2 5 12 x 2 5 9 1 2 1 9 9 ln x 0 3 9 2 9 ln 9 ln 3 12 x 3 2 5 5x 0 y 3 ln 9 3 9y A 0 2 3y 2 5 dx dy y A 0 y 1 dx dy 1 9 3 y dx dy 9y x 0 2 3y 2 dy 3y dy 2 5y 52 y 4 2 x 0 1 y dy 1 x y 3 dy 9 y 5 0 2 y 9 0 y dy 1 y dy 12 y 2 3 1 5 0 y 4 3 2 5y dy 2 5 0 y 6 9y 12 y 2 1 9 ln y 0 9 1 2 ln 9 y=x y= 2x 3 4 y=5−x 2 (3, 3) y= 9 x (9, 1) x 1 x 1 2 3 4 5 2 4 6 8 −1 −2 39. A 4 a 0 0 a ba a2 x2 a ba a2 x2 2 y 4 2 dy dx 0 2 y 0 dx d 40. A 0 2 y2 dx dy 2 4 y2 dx dy 2 y dy 2 4 2 b a x a2 0 x 2 dx ab 0 cos2 0 y dy 2 2 a sin , dx ab 2 ab 4 2 a cos d cos 2 d ab 2 1 sin 2 2 2 0 y2 2 4 1 2 0 2y 3 dy dx y2 4 2 2 1 0 4 2x x A 0 2x 0 x dx x2 2 2 2 0 Therefore, A A 4 b 0 0 ab ab. b2 y2 y 4 3 2 dx dy ab 4 y = 2x Therefore, A above. y ab. Integration steps are similar to those 1 y=x x 1 2 3 4 b y= b a a2 − x2 a x 280 4 Chapter 14 y Multiple Integration 4 2 41. 0 0 f x, y dx dy, 0 ≤ x ≤ y, 0 ≤ y ≤ 4 4 0 y 42. 0 y f x, y dx dy, 2 x2 y ≤ x ≤ 2, 0 ≤ y ≤ 4 4 f x, y dy dx x y 4 3 2 f x, y dy dx 0 0 y = x2 3 2 1 x 1 2 3 4 1 x 1 2 3 4 2 4 x2 2 4 0 x 2 43. 20 2 0 y 3 f x, y dy dx, 0 ≤ y ≤ 4 4 y2 4 x, 2 2≤x≤2 44. 0 f x, y dy dx, 0 ≤ y ≤ 4 4 4 0 y x2, 0 ≤ x ≤ 2 dx dy y2 0 y 4 3 2 1 1 −2 −1 −1 x 1 2 −1 −1 1 2 f x, y dx dy x 3 4 10 ln y 2 e x 45. 1 0 f x, y dx dy, 0 ≤ x ≤ ln y, 1 ≤ y ≤ 10 ln 10 0 y 46. 10 f x, y dy dx, 0 ≤ y ≤ e x, e 2 1≤x≤2 ln y 10 2 e f x, y dy dx e x f x, y dx dy 0 y 1 e 2 f x, y dx dy 1 3 8 6 4 2 x 1 2 3 −1 x 1 2 2 1 1 2 cos x 47. 1 x2 f x, y dy dx, x 2 ≤ y ≤ 1, 1 0 y 4 3 2 1≤x≤1 48. 20 1 f x, y dy dx, 0 ≤ y ≤ cos x, arccos y 2 ≤x≤ 2 y f x, y dx dy y 0 y 2 3 2 f x, y dx dy arccos y 1 2 −2 −1 x 1 2 −π 4 π 4 x S ection 14.1 1 2 2 1 2 4 Iterated Integrals and Area in the Plane 4 2 281 49. 0 y dy dx 0 0 0 dx dy 2 50. 1 y 4 3 dx dy 2 2 1 dy dx 2 3 2 2 1 1 x 1 2 3 x 1 2 3 4 1 1 1 y2 1 1 0 x2 2 4 4 4 4 x2 2 51. 0 y2 y dx dy 1 dy dx 2 52. 2 2 2 x2 y2 dy dx 2 4 4 x2 4 x2 dx 4 dx dy y2 y 1 −1 x 1 1 x 1 −1 −1 2 x 4 4 0 x 2 4 y y y 53. 0 0 dy dx 2 dy dx 0 dx dy 4 3 2 1 x 1 2 3 4 −1 4 x2 6 6 0 6 2y x 4 0 54. 0 0 dy dx 4 2 0 y dy dx 2 x dx 2 6 6 y 6 4 x dx 6y 4 3y 2 2 2 2 6 0 6 6 5 4 3 2 1 −1 −1 x 1 2 3 4 5 6 dx dy 0 3y dy y= x 2 (4, 2) y = 6 − x 2 1 1 2y y 55. 0 x2 dy dx 0 0 dx dy 1 2 1 x 1 2 282 9 Chapter 14 3 9 Multiple Integration 9 2 0 y 56. 0 3 0 x y 0 2 dy dx 0 3 3 y2 dy 0 x dx y3 3 3 3x 9 23 x 3 27 18 9 dx dy 5 4 3 2 1 −1 −2 −3 −4 −5 y= x x 0 123456789 1 3 y 1 x 57. 0 y 2 dx dy 0 x 3 dy dx 5 12 2 4 y2 4 4 4 x 58. 20 y dx dy 0 x dy dx 32 3 x= 3 y y 2 2 x = y2 1 x 1 2 3 1 (1, 1) x −1 −2 x = 4 − y2 1 2 59. The first integral arises using vertical representative rectangles. The second two integrals arise using horizontal representative rectangles. 5 0 x 50 x2 5 y y= 50 − x 2 (0, 5 2 ) 5 x 2y 2 dy dx 0 12 x 50 3 x2 32 15 x dx 3 (5, 5) y=x x 5 15,625 24 5 0 y 5 2 0 50 y2 5 x 2y 2 dx dy 0 5 x 2y 2 dx dy 0 15 y dy 3 5 5 2 1 50 3 y2 3 2y2 dy 15,625 18 15,625 24 15,625 18 15,625 18 60 . The first integral arises using vertical representative rectangles. The second integral arises using horizontal representative rectangles. 2 0 2x 2 y x sin y dy dx x2 0 x cos 2x 1 cos 4 4 x cos x 2 dx 1 4 dy (2, 4) 4 3 2 1 x 1 2 1 sin 4 2 4 0 y 4 x sin y dx dy y2 0 1 y sin y 2 12 y sin y 8 1 sin 4 2 y3 dx dy 0 1 cos 4 4 2 2 2 y 1 4 2 61. 0 x x1 y3 dy dx 0 0 2 x1 1 y3 x2 2 y dy 0 2 1 2 1 0 y3 y 2 dy 1 2 1 3 2 1 3 y3 32 0 1 27 9 1 1 9 26 9 S ection 14.1 2 2 2 y Iterated Integrals and Area in the Plane 283 62. 0 x e y 2 dy dx 0 2 0 e y 2 dx dy 2 y xe 0 y2 0 dy 0 ye y2 dy 1 e 2 2 y2 0 1 e 2 4 10 e 2 1 1 2 1 e4 0.4908 1 1 1 x 1 x 63. 0 y sin x 2 dx dy 0 1 0 sin x 2 dy dx 0 y sin x 2 0 1 0 dx 1 cos 1 2 1 1 2 1 1 2 x sin x 2 dx 0 1 cos x 2 2 cos 1 0.2298 2 4 4 x 64. 0 y2 x sin x dx dy 0 4 0 x sin x dy dx x 0 4 4 y x sin x 0 dx 0 x sin x dx sin x x cos x 0 sin 4 4 cos 4 1.858 2 2x 65. 0 x 2 x3 3y 2 dy dx 1664 105 1 2y 15.848 66. 0 y sin x y dx dy sin 2 2 sin 3 3 0.408 4 y 0 67. 0 x 2 1y 1 dx dy ln 5 2 2.590 a a 0 x 68. 0 x2 y 2 dy dx a4 6 69. (a) x x 8 y3 ⇔ y x1 3 y 4 2y ⇔ x2 x1 3 32y ⇔ y x2 32 4 x = y3 2 (8, 2) (b) 0 x2 32 x 2y xy 2 dy dx −2 x 2 4 6 8 x = 4 2y (c) Both integrals equal 67,520 693 70. (a) y y 2 97.43 y 4 4 2 4 y 2 x2 ⇔ x x2 ⇔x 4 x2 xy y2 1 4 16 dx dy y2 4 4y 3 2 2 0 3 2 (b) 0 x2 xy y2 4 16 0 4y 1 dx dy 3 x2 xy y2 1 dx dy 1 x (c) Both orders of integration yield 1.11899. 2 4 0 x2 2 2 1 2 71. 0 e xy dy dx 20.5648 72. 0 x 16 x3 y3 dy dx 6.8520 2 1 0 cos 73. 0 6r 2 cos dr d 15 2 2 1 0 sin 74. 0 15 r dr d 45 2 32 135 8 30.7541 75. An iterated integral is a double integral of a function of two variables. First integrate with respect to one variable while holding the other variable constant. Then integrate with respect to the second variable. 284 Chapter 14 Multiple Integration 76. A region is vertically simple if it is bounded on the left and right by vertical lines, and bounded on the top and bottom by functions of x. A region is horizontally simple if it is bounded on the top and bottom by horizontal lines, and bounded on the left and right by functions of y. 77. The region is a rectangle. 79. True 78. The integrations might be easier. See Exercise 59-62. 80. False, let f x, y x. Section 14.2 For Exercise 1–4, 11 ,, 22 31 ,, 22 xi Double Integrals and Volume yi 1 and the midpoints of the squares are 71 ,, 22 13 ,, 22 33 ,, 22 53 ,, 22 73 ,. 22 4 3 2 y 51 ,, 22 1 x 1 2 3 4 1. f x, y 8 x y xi yi x 1 2 4 f xi, yi i 1 4 0 2 3 4 y2 2 2 2 3 4 4 5 24 4 y dy dx 0 xy dx 0 0 2x 2 dx x2 2x 0 24 0 2. f x, y f xi, yi i 1 4 0 2 0 12 xy 2 xi yi 12 x y dy dx 2 x2 y2 xi yi x2 2 4 10 4 4 1 16 4 0 9 16 x2y 2 4 25 16 2 49 16 4 3 16 x2 dx 0 27 16 x3 3 4 0 75 16 64 3 147 16 21.3 21 dx 0 3. f x, y 8 f xi, yi i 1 4 0 2 26 4 x2y 0 50 4 y3 3 10 4 2 18 4 4 34 4 2x 2 58 4 8 dx 3 52 2x3 3 8x 3 4 0 y 2 dy dx dx 0 0 0 160 3 4. f x, y 8 x 1 1y xi yi 1 1 4 9 4 15 dy dx 0 4 0 f xi, yi i 1 4 0 2 0 4 21 4 4 27 1 x 1 4 15 ln y 4 25 2 4 35 dx 0 4 45 7936 4725 1.680 x 1y 1 1 ln 3 dx x1 4 ln 3 ln x 1 0 ln 3 ln 5 1.768 S ection 14.2 4 4 Double Integrals and Volume 285 5. 0 0 f x, y dy dx 32 400 31 28 23 31 30 27 22 28 27 24 19 23 22 19 14 Using the corner of the ith square furthest from the origin, you obtain 272. 2 2 6. 0 0 f x, y dy dx 4 2 8 6 20 2 1 2 y 1 7. 0 0 1 2x 2y dy dx 0 2 y 2xy y 2 0 dx 3 2 2 0 2x dx 1 2 2x 8 2 x2 0 x 1 2 3 8. 0 0 sin2 x cos 2 y dy dx 0 12 sin x y 2 1 sin 2y 2 2 y dx 0 3 0 12 sin x dx 2 2 1 0 2 1 8 cos 2x dx 1 sin 2x 2 x 1 2 3 8 2 x 0 8 6 3 6 y 9. 0 y2 x y dx dy 0 6 0 12 x 2 9 2 3y 32 y 2 3 xy y2 dy 52 y dy 8 53 y 24 6 (3, 6) 6 4 2 9 y 2 36 4 y 4 x 0 2 4 6 10. 0 x y dx dy 1 2y 0 4 0 22 x3y 2 3 y 72 y y dy 1 2y 4 3 2 1 (2, 4) 3 y dy 24 y6 144 256 9 4 0 5 2y9 2 27 1024 27 x 1 2 3 4 256 27 286 a Chapter 14 a2 a2 x2 Multiple Integration a 11. a x2 x y dy dx a a xy 12 y 2 a2 a2 x2 x2 y dx a 2x a2 a x 2 dx a −a a x 22 a 3 1 0 1 1 0 y x2 32 a 0 1 −a 1 0 1 y 12. 0 y 1 ex y dx dy 0 ex y dx dy 0 1 ex e 0 y y 1 1 dy 0 ex y 0 dy 2 y e 2y 1 2y e 2 e 1 dy 1 y=x+1 y = −x + 1 ey 1 e 2 5 3 3 5 1 0 −1 1 x 13. 0 0 xy dx dy 0 3 0 0 xy dy dx 5 y 12 xy 2 3 5 dx 0 4 3 2 25 2 x dx 0 3 0 1 x 25 2 x 4 225 4 1 2 3 4 5 2 2 y 14. 0 sin x sin y dx dy 0 sin x sin y dy dx 2 5π 2 2π 3π 2 sin x cos y 0 dx π x sin x dx 0 2 y y2 4 2 y2 2 2x x 2 − 3π − π − π 2 2 π 2 π 3π 2 15. 0 y x2 y 2 dx dy y x2 y 2 2 dx dy y x2 ln x 2 y y 0 2 dy dx 4 3 y = 2x x=2 1 2 1 2 2x y2 x dx 2 0 2 1 y=x x 1 2 3 4 ln 5x 2 0 2 ln 2x 2 dx 15 ln 22 dx 0 2 1 5 ln x 2 2 ln 0 5 2 S ection 14.2 4 4 0 x 4 0 4 0 y y 4 3 2 Double Integrals and Volume 287 16. 0 xe y dy dx xe y dx dy For the first integral, we obtain: 4 4 x 4 xe y 0 0 dx 0 xe4 x x dx x 1 e4 5 4 4 4 y y 1 1 x e4 x2 2 e4 4 0 x 1 2 3 4 8 4 4 1 x2 13. y 17. 3 2y ln x dx dy 0 x 2y ln x dy dx 4 x2 4 3 (1, 3) ln x 0 1 y2 4 x 2 dx x2 4 x 2 2 ln x 4 0 dx 26 25 1 x 1 3 4 2 4 y2 18. 0 y 1 x 4 2 dx dy 0 0 4 0 4 0 x y 1 y2 x x2 x x2 y 2 dy dx 4 3 1 2 1 2 x 1 1 dx 0 y= 2 x dx 4 1 x 1 2 3 4 1 ln 1 4 4 3x 4 5 25 0 x2 x2 0 1 ln 17 4 3 25 4y 3 4 y2 y 19. 0 0 x dy dx 4 x dy dx 0 3 0 x dx dy 12 x 2 3 25 4y 3 y2 5 x= x= 4y 3 (4, 3) 25 − y 2 dy y 2 dy 3 2 1 25 18 9 0 x 1 2 3 4 5 25 9y 18 2 4 4 y2 13 y 3 3 25 0 20. 0 y2 2 x2 4 y 2 dx dy 4 y x2 x 20 2 2 y 2 dy dx x2 x=− 4 − y2 3 x= 4 − y2 x 2y 2 2 13 y 3 4 x2 x2 32 4 0 dx 1 4 3 1 x4 2 x2 3 2 x2 −2 −1 1 x 1 2 x2 2 dx 4 arcsin x 2 1 x4 12 x2 32 x 4 4 6x 4 x2 24 arctan x 2 2 4 2 288 4 Chapter 14 2 0 Multiple Integration 4 0 4 21. 0 y dy dx 2 y2 4 dx 2 4 2 4 2 dx 0 22. 0 0 6 2y dy dx 0 4 6y y2 0 dx 4 0 y 4 3 2 0 y 4 3 2 8 dx 32 1 x 1 2 3 4 1 x 1 2 3 4 2 y 2 23. 0 0 4 x y dx dy 0 2 4x 4y 0 x2 2 y2 2 y3 6 y 3 y y xy 0 dy 2 y 2 dy 1 32 0 y=x x 1 2 2y2 8 8 6 2 8 3 4 2 x 2 y 24. 0 0 4 dy dx 0 4x dx 2x2 0 8 2 y=x 1 x 1 2 6 2 3x 0 4 25. 0 12 2x 4 3y 6 dy dx 0 6 0 1 12y 4 12 x 6 2x x2 2xy 32 y 2 2 3x 0 4 y dx 5 4 y = − 2x + 4 3 6 dx 6 3 2 1 x 1 2 3 4 5 6 13 x 18 12 2 2 0 x 2 6x 0 −1 26. 0 2 x y dy dx 0 2y 21 0 xy x dx 2 y2 2 2 0 x y dx 2 y=2−x 2 2 1 1 x 6 2 2 3 0 4 3 x 1 2 S ection 14.2 Double Integrals and Volume 289 1 y 1 27. 0 0 1 xy dx dy 0 1 x x 2y 2 y 2 y 2 dy 0 28. 0 0 4 y 2 dx dy 0 4y y3 dy y4 4 2 0 y 0 y3 dy 2 y4 8 1 y 2y 2 4 y2 2 3 8 0 y 2 1 1 y=x y=x x 1 x 1 2 29. 0 0 1 x 1 2 y 1 2 dy dx 0 1 x 1 2 y 1 dx 0 0 1 x 1 2 dx 1 x 1 0 1 30. 0 0 e x y2 dy dx 0 2e x y2 0 dx 0 2e x2 dx 4e x2 0 4 2 4 0 x2 1 x 31. 4 0 4 x2 y 2 dy dx 8 32. 0 0 1 x2 dy dx 1 3 1 x 5 x 33. V 0 1 0 0 xy dy dx 12 xy 2 1 0 x 34. V 0 0 x dy dx 5 x 5 dx 0 1 2 1 x3 dx 0 xy 0 0 5 0 dx 0 x 2 dx 14 x 8 y 1 8 5 13 x 3 y 125 3 1 y=x 4 3 2 1 y=x x 1 x 1 2 3 4 5 2 4 y 35. V 0 2 0 x 2 dy dx 4 2 4 3 x 2y 0 0 2 0 dx 0 4x 2 dx 2 1 4x3 3 32 3 −1 x 1 2 3 290 Chapter 14 r r2 x2 Multiple Integration y 36. V 8 0 r 0 r2 y 0 r x2 y2 y2 dy dx r y= r2 − x2 4 r2 r2 x2 x 2 dx 13 x 3 r 0 r2 x2 arcsin y r2 x2 0 r2 x2 dx r x 4 2 2 0 r 2x 4 r3 3 37. Divide the solid into two equal parts. 1 x y y=x V 2 0 1 0 1 x 2 dy dx x 1 2 0 1 y1 x2 0 dx x 1 2 0 x1 2 1 3 x 2 dx 1 x2 32 0 2 3 y 2 4 0 x 2 2 4 0 x2 y 38. V 0 2 4 4 0 2 x2 dy dx x2 dx x4 dx 4 3 39. V 0 x 2 y dy dx 2 y= 4 − x2 x2 4 8x2 8 64 3 1 x3 3 32 5 y = 4 − x2 2 1 xy 0 2 x 12 y 2 x2 x2 4 0 x2 dx 12 x dx 2 1 16 0 x4 0 2 2x x 1 2 1 2 3 4 16x 32 x5 5 2 0 1 4 3 16 3 y 32 13 x 6 2 0 256 15 2 2 4 0 x2 40. V 0 2 0 1 y 2 dy dx 2 41. V 4 0 2 x2 x2 4 0 2 y 2 dy dx x2 1 4 3 32 cos4 3 x2 32 arctan y 0 2 0 0 dx 1 4 4 0 dx, x 2 sin 2 x 2 dx 1 2 x 16 cos2 32 3 3 16 d y 2 4 16 0 4 x 2 + y2 = 4 8 −1 1 x 1 −1 S ection 14.2 5 y Double Integrals and Volume 291 42. V 0 5 0 0 sin2 x dx dy dy 5 5 4 43. z V 9 3 x2 9 0 y 2, z x2 0 x2 y 2 dy dx 81 2 4 0 9 2 y 3 2 1 x 1 2 3 4 5 2 5 2 0 9 9 0 y 44. V 0 9 y dx dy 81 2 2 0.5x 0 1 45. V 0 1 2 x2 y2 dy dx 1.2315 16 4 0 y 46. V 0 ln 1 x y dx dy 38.25 47. f is a continuous function such that 0 ≤ f x, y ≤ 1 over a region R of area 1. Let f m, n and f M, N the maximum value of f over R. Then f m, n R the minimum value of f over R dA ≤ R f x, y dA ≤ f M, N R dA. f x, y dA ≤ f M, N 1 ≤ 1. R Since R dA 1 and 0 ≤ f m, n ≤ f M, N ≤ 1, we have 0 ≤ f m, n 1 ≤ f x, y dA ≤ 1. R Therefore, 0 ≤ 48. x a z V y b c1 z c x a z 1 a y b a b1 0 a xa R f x, y dA R 0 c1 c 0 a x a b1 0 y dy dx b xa a x a y y b1 0 xy a x a x a ab 3 y2 2b dx xb 1 a x a x3b 3a2 ab 6 b2 1 2b ab 1 6 abc 6 y c c c x a x a 2 dx 3a 0 ab 1 2 ab 2 2x 2 x 2b 2a ab 2 1 12 12 49. 0 y2 e x 2 dx dy 0 12 0 e x 2 dy dx 1 y = 2x 2xe 0 x 2 dx 12 1 2 e e 1 x2 0 14 1 2 1 x 1 0.221 e 14 292 Chapter 14 ln 10 10 e x Multiple Integration 10 1 10 1 10 ln y 0 50. 0 1 dy dx ln y 1 dx dy ln y ln y 1 arccos y 51. 0 0 sin x 1 2 cos x sin2 x dx dy x ln y dy dy 0 10 0 2 0 sin x 1 sin2 x dy dx y 1 9 0 1 1 2 y sin2 x 12 sin x cos x dx 2 1 y 10 8 6 4 2 x 1 2 3 4 5 2 1 3 sin2 x 32 0 1 22 3 1 y = ex 2 y = cos x 1 π 2 π x 2 2 2 2y 52. 0 1 2x 2 2 y cos y dy dx 0 0 y y cos y dx dy 53. Average 1 8 4 0 2 x dy dx 0 1 8 4 2x dx 0 x2 8 4 2 0 y cos y 2y dy 0 2 2 (2, 2) 2 0 y cos y dy 1 2 2 cos y 2 cos 2 4 0 y sin y 0 y = 1 x2 2 x 2 sin 2 2 1 4 1 2 54. Average 1 8 1 4 1 4 xy dy dx 0 1 8 2x dx 0 x2 8 4 2 0 2 0 2 0 2 55. Average x2 0 y 2 dx dy 2 56. Average 1 4 2 0 1 12 2 ex e2 e 1 0 1 1 1 ex x y dy dx 1 2 0 ex 12 e 2 1 e 2x dx e 1 2 x3 3 xy 2 0 dy 2 0 8 3 2y 2 dy 1 2x e 2 1 2 e2 0 18 y 43 23 y 3 8 3 2e 1 2 57. See the definition on page 992. 58. The second is integrable. The first contains sin y2 dy which does not have an elementary antiderivation. 59. The value of R f x, y dA would be kB. 60. (a) The total snowfall in the county R. (b) The average snowfall in R. 61. Average 1 1250 1 1250 325 300 325 250 100x 0.6y 0.4 dx dy 200 325 100y 0.4 300 x1.6 1.6 250 dy 200 128,844.1 1250 y 0.4 dy 300 103.0753 y1.4 1.4 325 25,645.24 300 S ection 14.2 1 150 60 45 50 Double Integrals and Volume 293 62. Average 192x 40 576y x2 5y 2 2xy 5000 dx dy 13,246.67 63. f x, y ≥ 0 for all x, y and 5 2 0 2 1 f x, y dA 0 2 1 dy dx 10 1 dy dx 10 5 0 2 0 1 dx 5 1 dx 10 1 1 . 5 P 0 ≤ x ≤ 2, 1 ≤ y ≤ 2 0 64. f x, y ≥ 0 for all x, y and 2 2 0 2 1 f x, y dA 0 1 1 xy dy dx 4 1 xy dy dx 4 2 0 1 0 x dx 2 3x dx 8 1 3 . 16 P 0 ≤ x ≤ 1, 1 ≤ y ≤ 2 0 65. f x, y ≥ 0 for all x, y and 3 6 3 f x, y dA 0 3 0 1 1 9 27 x y dy dx y2 2 6 3 1 9y 27 6 4 xy dx 3 0 1 1 2 2 4 27 1 x dx 9 x dx x 2 7 . 27 x2 18 3 1 0 P 0 ≤ x ≤ 1, 4 ≤ y ≤ 6 0 1 9 27 x y dy dx 0 66. f x, y ≥ 0 for all x, y and f x, y dA 0 0 b 0 1 b→ 1 b e x y dy dx lim e x y 0 dx 0 1 e x dx 1 b→ lim 1 e x 0 1 P 0 ≤ x ≤ 1, x ≤ y ≤ 1 0 x e 1 e 2 x y dy dx 0 1 e 1 0 x y x dx 0 1 e 1 2 1 e 2 2x e 1 2 x 1 dx 2x e x 1 e 2 2 e 1 0.1998. 67. Divide the base into six squares, and assume the height at the center of each square is the height of the entire square. Thus, V 4 3 6 7 3 2 100 2500 m3. (15, 5, 6) (25, 5, 4) 7 z (15, 15, 7) (5, 5, 3) (5, 15, 2) 20 y 30 x (25, 15, 3) 294 Chapter 14 Multiple Integration 1 2 68. Sample Program for TI-82: Program: DOUBLE : Input A : Input B : Input M : Input C : Input D : Input N :0 → V :B :D A M→G C N→H 69. 0 0 sin (a) 1.78435 (b) 1.7879 x y dy dx m 4, n 8 : For I, 1, M, 1 : For J, 1, N, 1 :A :C :V : End : End : Disp V 2 4 6 2 0.5G 2I 0.5H 2J sin x 1 →x 1 →y y G H→V 70. 0 0 20e x3 8 dy dx m 10, n 20 71. 4 0 y cos (a) 11.0571 (b) 11.0414 x dx dy m 4, n 8 (a) 129.2018 (b) 129.2756 4 2 72. 1 1 x3 (a) 13.956 (b) 13.9022 y3 dx dy m 6, n 4 73. V 125 (4, 0, 16) 16 z Matches d. (4, 4, 16) (4, 0, 0) 5 x (0, 4, 0) 5 y (4, 4, 0) 74. V 50 4 3 z 75. False 1 1 0 y2 Matches a. V 8 0 1 x2 y 2 dx dy x 3 3 y 76. True S ection 14.2 1 1 x 1 1 2 Double Integrals and Volume 1 e x 2 xy 1 295 77. Average 0 f x dx 0 1 0 t 1 e t dt dx 0 1 x 2 2 e t dt dx 2 2 78. 1 e Thus, xy dy e x e x 2x e t dx dt 0 1 0 0 te t dt 1 1 1 2 e e 0 x e x 2x 2 1 t2 e 2 t 1 e 2 dx 0 2 1 e xy dx dy e 1 2 0 xy dx dy 1 e xy 1 2 x 1 y 1 dy y dy 0 2 ln y 1 ln 2. 1 79. z 9 x 2 y 2 is a paraboloid opening downward with vertex 0, 0, 9 . The double integral is maximized if z ≥ 0. That is, R x, y : x 2 y2 ≤ 9 . 9 R 80. z x 2 y 2 4 is a paraboloid opening upward with vertex 0, 0, 4 . The double integral is minimized if z ≤ 0. That is, R x, y : x 2 y2 ≤ 4 . 8. The maximum value is x2 y 2 dA 81 . 2 The minimum value is 2 81. 0 tan 1 x 2 x x 2 y y 2 tan 1 1 1 x y2 y y2 1 1 1 x dx 7 6 y y = πx 0 y2 1 dy dx 2 2 y 5 4 3 0 y2 y y dx dy 2 2 1 1 y2 2 y dx dy −2 −1 2 1 1 2 y=x x 3 4 5 0 2 0 1 1 dy 2 x 1 2 y2 1 dy 2 y2 1 y 1 ln 1 ln 5 2 tan 1 y2 y2 dy 2 2 y 1 1 ln 1 2 4 2 y2 dy y2 2 2 1 1 1 2 1 1 2 1 ln 5 2 0.8274 3 9 0 y2 2 tan 0 1 y 2 tan 2 2 1 1 ln 1 2 2 2 tan 4 2 2 1 ln 5 2 2 tan 1 ln 1 2 82. 0 9 x2 y 2 dx dy 9 2 y2 z2 9 because this double integral represents the portion of the sphere x 2 in the first octant. V 14 83 3 3 9 2 296 Chapter 14 a b Multiple Integration 83. Let I 0 0 e max{b 2 x 2, a2 y2 dy dx. bx. y Divide the rectangle into two parts by the diagonal line ay b On lower triangle, b 2x 2 ≥ a 2y 2 because y ≤ x. a a bx a b ay b (a, b) b I 0 a 0 0 eb 2x2 dy dx 0 b 0 0 ea ay a2 y 2 e dy b b 0 2y2 dx dy ay = bx a x bx b2 x 2 e dx a a 0 1 22 eb x 2ab 1 b2a 2 e 2ab ea 2 b2 1 22 ea y 2ab 1 ea 2b2 1 1 ab 84. Assume such a function exists. 1 ux 1 x 1 u yu y 1 x dy; 1 1 > 1 ,0 ≤ x ≤ 1 2 x dy dx u x dx 0 0 dx 0 x u yu y Change the order of integration. 1 1 y u x dx 0 1 1 0 y 0 u yu y uy 0 x dx dy 1 0 uy y x dx dy dx. dy Hold y fixed and let z 1 x, dz uz dz 0 y y 1 0 uy 1 1 0 y uy 0 u z dz dy u y ,f 0 0, f 1 . Let f y 0 1 u z dz. Then f y f y f y dy 0 1 1 1 1 2 fy 2 21 0 2 1 f1 2 1 2 2 2 2 1 f0 2 2 0. For to exist, the discriminant of this quadratic must be nonnegative. b2 4ac > 4 8 ≥0⇒ ≤ 1 2 But, 1 , a contradiction. 2 Section 14.3 Change of Variables: Polar Coordinates 297 Section 14.3 Change of Variables: Polar Coordinates 2. Polar coordinates 4. Rectangular coordinates ≤ ≤2 1. Rectangular coordinates 3. Polar coordinates 5. R 7. R 2 6 r, r, : 0 ≤ r ≤ 8, 0 ≤ :0≤r≤3 6. R Cardioid 8. R π 2 r, r, : 0 ≤ r ≤ 4 sin , 0 ≤ : 0 ≤ r ≤ 4 cos 3 , 0 ≤ ≤ ≤ 3 sin , 0 ≤ 2 6 9. 0 0 3r 2 sin dr d 0 2 r3 sin 0 d 216 sin d 0 2 4 0 216 cos 0 0 4 4 4 10. 0 0 r 2 sin cos dr d 0 r3 sin cos 3 4 0 4 2 3 2 d 0 11. 0 2 9 r 2 r dr d 0 1 9 3 2 0 3 r2 32 2 d 64 sin2 3 2 16 3 π 2 55 3 55 6 π 2 0 0 1 2 3 4 1 2 3 2 3 2 12. 0 0 re r2 dr d 0 1 e 2 9 3 r2 0 2 d π 2 1 e 2 4 1 1 0 1 e9 0 1 2 3 2 1 0 sin 2 13. 0 r dr d 0 2 0 r2 2 1 1 2 1 0 sin d sin 2 d 1 2 1 cos 2 sin 1 2 12 sin 8 2 0 1 8 3 32 2 sin 9 8 cos 2 298 Chapter 14 2 1 0 cos Multiple Integration 2 14. 0 sin r dr d 0 2 0 sin sin 2 r2 2 1 1 0 cos π 2 d (x, y) = (0, 1) cos 2 2 d 0 1 1 6 a a2 0 y2 2 a cos 3 0 2 1 6 sin d a3 3 cos 0 1 15. 0 y dx dy 0 0 r 2 sin dr d a3 3 a3 3 2 0 a3 3 a3 3 a a2 0 x2 2 a 2 16. 0 x dy dx 0 0 r 2 cos dr d cos d 0 a3 sin 3 243 10 2 0 3 9 0 x2 2 3 17. 0 x2 y2 32 dy dx 0 0 r 4 dr d 243 5 2 d 0 2 8 y y2 4 2 0 2 18. 0 x2 y 2 dx dy 0 4 0 r 2 dr d 22 3 3 d 22 3 3 0 4 22 3 3 4 42 3 2 2x 0 x2 2 2 cos 2 19. 0 xy dy dx 0 0 r3 cos sin dr d 4 0 cos5 sin d 4 cos6 6 2 0 2 3 4 4y 0 y2 2 4 sin 2 20. 0 x 2 dx dy 0 0 2 r 3 cos2 dr d 0 64 sin4 cos2 d 2 64 0 2 x 2 sin4 2 0 8 x2 sin6 d 64 sin5 6 cos 4 2 0 4 2 sin3 cos 4 3 8 π 2 sin cos 0 2 21. 0 0 x2 y 2 dy dx 2 x2 y 2 dy dx 0 r 2 dr d 16 2 d 3 0 1 2 3 0 42 3 5 22 x 5 25 22 0 x2 4 5 22. 0 0 xy dy dx 5 xy dy dx 0 4 0 0 r 3 sin cos dr d 625 sin cos d 4 4 0 625 2 sin 8 625 16 S ection 14.3 2 4 0 x2 2 2 2 2 Change of Variables: Polar Coordinates 299 23. 0 x y dy dx 0 0 2 r cos 8 3 cos 0 r sin sin d r dr d 0 0 cos 2 sin 16 3 r 2 dr d 8 sin 3 cos 0 2 5 2 y 5 24. 2 0 e r 2 2r dr d 2 2 e 1 2 r2 2 0 d d 2 e 25 2 25 2 5 4 3 2 1 −5 − 4 −3 −2 −1 −2 −3 −4 −5 x 1234 1 e 1 2 e 25 2 1 2 4 1 y2 y2 25. 0 y arctan dx dy x 2 1 2 4 0 4 0 y 2 4 y2 y arctan dx dy x π 2 ( 1 , 2 1 2 ( ( 2, 2) r dr d 1 3 d 2 2 3 32 4 4 0 32 64 0 1 2 3 9 0 x2 26. 0 9 x2 y 2 dy dx 0 2 0 0 3 9 r 2 r dr d 2 9r 0 r 3 dr d 0 92 r 2 14 r 4 3 d 0 81 4 2 d 0 81 8 2 1 27. V 0 0 2 0 r cos 1 2 1 r sin r dr d 1 8 2 r 3 sin 2 dr d 0 sin 2 d 0 1 cos 2 16 1 2 0 1 8 2 5 2 2 1 2 28. V 4 0 0 r2 3 r dr d 4 4 0 2 0 r4 4 3r 2 2 7 2 d 0 29. V 0 0 r 2 dr d 0 125 d 3 250 3 7 d 4 2 2 30. R ln x2 y 2 dA 0 2 1 2 ln r 2 r dr d 2 0 2 1 r ln r dr d r2 4 ln 4 0 2 2 0 2 1 2 ln r 1 d 2 4 3 d 4 3 4 ln 4 300 Chapter 14 2 4 cos Multiple Integration 2 31. V 2 0 0 2 16 r 2r dr d 2 0 1 3 16 r2 3 4 cos d 0 2 0 2 3 64 3 9 2 64 sin3 0 64 d 128 3 2 1 0 sin 1 cos2 d 128 3 cos cos3 3 2 4 4 2 32. V 0 1 16 r 2 r dr d 0 1 3 1 3 16 r2 3 4 d 1 0 5 15 d 10 15 2 4 2 33. V 0 a 16 r 2 r dr d 0 16 3. r2 3 4 d a 1 3 16 a2 3 2 One-half the volume of the hemisphere is 64 2 16 3 16 16 a2 a2 32 64 3 32 322 16 3 32 a2 a2 a 322 44 3 16 2 x2 83 2 4 y2 23 2 a2 2.4332 r2 23 2 a2 34. x 2 V y2 8 z2 2 0 2 a2 ⇒ z a a2 0 r 2 r dr d 22 a 3 8a3 3 2 4 a (8 times the volume in the first octant) 8 0 2 1 2 a3 d 3 r2 32 0 2 d 4 a3 3 8 0 0 35. Total volume V 0 2 0 25e r2 4r dr d 4 50e 0 2 r2 4 0 d 50 e 0 4 1d 308.40524 1 e 4 100 Let c be the radius of the hole that is removed. 1 V 10 2 0 2 c 2 c 25e 0 r2 4 r dr d 0 50e r2 4 0 d 50 e 0 c2 4 1 d ⇒ 30.84052 ⇒e c2 4 100 1 0.90183 0.10333 0.41331 0.6429 2c e c2 4 c2 4 c2 c ⇒ diameter 1.2858 S ection 14.3 9 y2 9 4r 2 36 ≤z≤ Change of Variables: Polar Coordinates 301 36. 4 x2 (a) 9 4 x2 9 9 y2 36 9 ; 1 1 ≤r≤ 1 4 2 cos2 −1 0.7 ≤z≤ 1 4r 2 r2 (b) Perimeter r dr d 1 1 2 cos cos2 sin 2 0 2 1 21 14 cos 2 dr d 1 2 2 d. − 0.7 1 cos2 2 1 z Perimeter (c) V 2 0 1 1 4 9 4r 2 1 cos2 2 cos2 sin2 d 5.21 x 1 y 36 r dr d 0.8000 6 cos 37. A 0 0 r dr d 0 18 cos 2 d 9 0 1 cos 2 d 9 1 sin 2 2 9 0 2 4 2 38. A 0 2 r dr d 0 6d 12 2 1 0 cos 39. 0 r dr d 1 2 1 2 2 1 0 2 2 cos cos2 1 d cos 2 2 1 2 1 2 1 2 4 2 1 0 2 cos d 2 sin 1 sin 2 2 2 0 3 2 cos 2 2 2 2 0 sin 40. 0 r dr d 1 2 2 2 0 sin 4 cos 2 d 1 2 1 2 2 4 0 4 sin 2 0 sin2 1 8 2 4 d 4 0 4 sin 1 d 1 4 2 3 2 sin 3 1 sin 2 4 3 9 2 3 0 41. 3 0 0 r dr d 3 2 3 4 sin2 3 d 0 3 0 1 cos 6 d 3 1 sin 6 6 4 3 cos 2 4 4 42. 8 0 0 r dr d 4 0 9 cos2 2 d 18 0 1 cos 4 d 18 1 sin 4 4 4 0 9 2 43. Let R be a region bounded by the graphs of r r g2 , and the lines a and b. g1 and 44. See Theorem 14.3. When using polar coordinates to evaluate a double integral over R, R can be partitioned into small polar sectors. 45. r-simple regions have fixed bounds for . -simple regions have fixed bounds for r. 46. (a) Horizontal or polar representative elements (b) Polar representative element (c) Vertical or polar 302 Chapter 14 Multiple Integration 47. You would need to insert a factor of r because of the r dr d nature of polar coordinate integrals. The plane regions would be sectors of circles. 48. (a) The volume of the subregion determined by the point 5, 8 12 , you obtain volumes, ending with 45 10 V 10 8 57 9 9 5 15 8 35 12 5 150 4 (b) 57 24013.5 (c) 7.48 24103.5 2 5 16, 7 is base 8 25 10 45 9 14 height 15 14 5 11 12 10 8 7 . Adding up the 20 10 15 11 555 1250 2135 2025 18 16 5 6115 4 10 ft3 24013.5 1,368,769.5 pounds 179,621 gallons 49. 4 0 r1 r 3 sin dr d 2 56.051 5 Note: This integral equals 4 sin d 0 r1 r3 dr . 4 4 50. 0 0 5e r r dr d 87.130 51. Volume base 8 height 12 300 16 z Answer (c) 6 x 4 46 y 52. Volume base 9 4 height 3 21 6 4 2 z 53. False Let f r, r 1 where R is the circular sector 0 ≤ r ≤ 6 and 0 ≤ ≤ . Then, y 4 2 4 x Answer (a) r R 1 dA > 0 but r 1 0 for all r. 54. True 2 2 2 55. (a) I 2 (b) Therefore, I e x2 y2 2 dA 4 0 0 e r2 2 r dr d 4 0 e r2 2 0 d 4 0 d 2 2. 56. (a) Let u 2x, then e x2 dx e u2 2 1 du 2 1 2 1 2 2 . (b) Let u 2x, then e 4x 2 dx e u2 1 du 2 . S ection 14.4 7 49 49 x2 2 7 2 Center of Mass and Moments of Inertia 7 303 57. 7 4000e x2 0.01 x 2 y 2 dy dx 0 0 4000e 2 0.01r 2 r dr d 0 200,000e 400,000 1 2 r2 0 0.01r 2 d 0 200,000 e 2 0.49 1 k e 2 e k d 2 0.49 486,788 2 58. 0 0 ke x2 y2 dy dx 0 0 ke r2 r dr d 0 d 0 k 4 For f x, y to be a probability density function, k 4 k 1 4 . 4 y y 59. (a) 2 2 y 3 f dx dy 5 y= 3x y=x 3x 4 2 3 x 3x 4 4 4 (b) 2 32 3 4 csc f dy dx f r dr d 4 2 csc f dy dx 3x f dy dx 3 ( ( 1 2 (4, 4) 4 ,4 3 ( x (c) 1 (2, 2) 2 ,2 3 3 ( 4 5 y 2 2 2 4 y2 (x − 2) 2 + y 2 = 4 60. (a) 4 0 2 4 0 2 4 cos x 2 f dx dy 2 2 1 (b) 4 0 f dy dx −1 −2 x 1 3 (c) 2 0 0 f r dr d 61. A r22 2 r12 2 r1 2 r2 r2 r1 r r Section 14.4 4 3 Center of Mass and Moments of Inertia 4 1. m 0 4 0 0 xy dy dx 0 xy 2 2 36 3 3 9 0 x 2 3 0 3 0 dx 0 2. m 0 xy dy dx xy 2 2 x9 9 0 x2 dx x2 2 2 9 x dx 2 9x2 4 4 0 dx 33 0 1 9 x2 4 3 1 0 12 2 2 2 2 93 243 4 3. m 0 0 r cos r sin r dr d 0 2 0 cos sin 4 cos 0 r 3 dr d sin d 2 4 sin2 2 2 0 304 Chapter 14 3 3 3 9 x2 Multiple Integration 3 4. m 0 xy dy dx 0 3 0 x y2 2 3 3 9 x2 dx 9 x2 x2 81 4 32 x 3 2 3 x2 9x 9x2 2 9 dx x3 dx x4 4 3 0 1 2 1 2 6x 9 0 29 1 81 22 a b 54 297 8 a b 5. (a) m 0 a 0 b k dy dx kab kab2 2 ka2b 2 a 2 b 2 (b) m 0 a 0 b ky dy dx kab2 2 kab3 3 ka2b2 4 a 2 2 b 3 Mx 0 a 0 b ky dy dx Mx 0 a 0 b ky 2 dy dx My 0 0 kx dy dx My m Mx m ab , 22 a b My 0 0 kxy dy dx My m Mx m a2 ,b 23 ka2b2 4 kab2 2 kab3 3 kab2 2 x y x, y ka2b 2 kab kab2 2 kab x y x, y a (c) m 0 0 a kx dy dx b k 0 xb dx ka2b2 4 ka3b 3 12 ka b 2 Mx 0 a 0 b kxy dy dx My 0 0 kx 2 dy dx My m Mx m ka3b 3 ka2b 2 ka2b2 4 ka2b 2 2b a, 32 a b x y x, y 2 a 3 b 2 6. (a) m 0 a 0 b kxy dy dx ka2b2 4 ka2b3 6 ka3b2 6 2 a 3 2 b 3 a b (b) m 0 a 0 b k x2 y 2 dy dx kab 2 a 3 kab2 2 2a 12 ka2b 2 3a 12 b2 Mx 0 a 0 b kxy2 dy dx Mx 0 a 0 b k x 2y y3 dy dx 3b2 My 0 0 kx 2y dy dx My m Mx m ka3b2 6 ka2b2 4 ka2b3 6 ka2b2 4 My 0 0 k x3 My m Mx m xy 2 dy dx 2b2 2b2 b2 3b2 b2 x y x y ka2b 12 3a2 kab 3 a2 kab2 12 2a2 kab 3 a2 2b2 b2 3b2 b2 a 3a2 4 a2 b 2a2 4 a2 S ection 14.4 k bh 2 b by symmetry 2 b2 2hx b b 2h x bb h Center of Mass and Moments of Inertia 305 7. (a) m x Mx y y= 2hx b y=− 2 h (x − b ) b ky dy dx 0 0 b2 0 ky dy dx x b kbh2 12 y x, y (b) m 0 b2 0 kbh2 12 kbh2 6 kbh 2 kbh2 6 h 3 Mx m bh , 23 b2 2hx b b 2h x bb ky dy dx b2 0 b 2h x 2hx b ky dy dx bb kbh2 6 kbh3 12 kb2h2 12 Mx 0 b2 0 2hx b ky 2 dy dx b2 0 b 2h x bb ky 2 dy dx My 0 0 kxy dy dx b2 0 kxy dy dx x y x, y My m Mx m kb2h2 12 kbh2 6 b 2 h 2 kbh3 12 kbh2 6 bh , 22 b2 2hx b b 2h x bb (c) m 0 0 kx dy dx b2 0 kx dy dx 12 kb h 4 b 2h x bb 12 kb h 12 b2 2hx b 12 kb h 6 kxy dy dx Mx 0 0 kxy dy dx b2 0 1 22 kh b 32 b2 2hx b 5 22 kh b 96 kx dy dx 2 1 22 kh b 12 b b2 0 2h x bb My 0 0 kx2 dy dx 73 kb h 48 13 kb h 32 x y x, y My m Mx m 11 3 kb h 96 7kb3h 48 kb2h 4 kh2b2 12 kb2h 4 7h b, 12 3 7 b 12 h 3 306 Chapter 14 a2k 2 a a 0 x Multiple Integration 8. (a) m Mx y ky dy dx 0 ka3 6 a y=a−x My x (b) m Mx by symmetry y a 0 a Mx m a 0 x ka3 6 ka2 2 x2 a 3 a x y 2 dy dx a 0 x a x 2y 0 a a 0 x y3 3 x3 dx 0 ax 2 x3 1 a 3 x 3 dx a4 6 My 0 a xy 2 dy dx 13 ax 3 2a 5 a2x2 ax3 14 x dx 3 1 3 a ax3 0 x4 a3x 0 3a2x2 6ax3 4x4 dx a5 15 x y My m a5 15 a4 6 2a by symmetry 5 a 5 b 9. (a) The x-coordinate changes by 5: x, y (b) The x-coordinate changes by 5: x, y a 2 a 2 5, 5, b 2 2b 3 (c) m 5 a 5 0 b kx dy dx kxy dy dx 5 a 5 0 b 1 ka 2 1 ka 4 1 ka 3 5b 5b 3 22 2 25 kb 2 25 2 kb 4 125 kb 3 Mx My 5 0 kx 2 dy dx My m Mx m 5b x y 2 a2 15a 75 3 a 10 b 2 10. The x-coordinate changes by h units horizontally and k units vertically. This is not true for variable densities. a a2 x2 11. (a) x m Mx 0 by symmetry a2k 2 a a0 a2 x2 (b) m a0 a a2 x2 ka y y dy dx a4k 16 24 a5k 15 120 0 3 Mx yk dy dx 2a3 3 4a 3 k My a0 a0 a a2 x2 ka y y 2 dy dx 32 kx a My m Mx m 0 a 15 5 16 32 3 y y dy dx y Mx m 2a3k 3 2 a2k x y S ection 14.4 a a2 0 a2 0 a x2 x2 Center of Mass and Moments of Inertia a a2 0 2 a x2 307 12. (a) m 0 a k dy dx k a2 4 (b) m 0 k x2 y 2 dy dx ka4 8 y 2 y dy dx ka5 5 My 0 kx dy dx 0 a 0 a2 0 2 0 a kr3 dr d x2 k 0 x a2 k2 a 3 x 2 dx a Mx 0 k x2 x2 32 0 ka3 3 My x kr 4 sin dr d 0 x y My m 3 k a2 4 ka3 4a 3 Mx by symmetry y My m ka5 5 8 ka4 8a 5 4a by symmetry 3 4 x 13. m 0 4 0 kxy dy dx x 32k 3 256k 21 32k 3 8 7 2 x 0 x 0 x 0 3 2 14. m 0 2 3 kx dy dx 0 kx 4 dx 16k 32k 3 5 3 5 2 32k 5 Mx 0 4 0 x kxy dy dx 2 Mx 0 2 3 kxy dy dx kx2 dy dx 0 My 0 0 kx 2y dy dx My m Mx m 32k 1 256k 21 3 32k 3 32k My x y My m Mx m x y y 3 32k 3 16k 5 32k 5 32k y= 2 x 1 x 1 2 3 4 −1 4 4x 15. x m 0 by symmetry 1 11 x 2 16. m 1 0 4x kx 2 dy dx 4 30k k dy dx 10 1 11 x 2 k 2 k 2 8 2 4 Mx 1 0 4x 4 kx 2 y dy dx 24k Mx 10 ky dy dx Mx m y My 1 0 kx3 dy dx My m My m 84k 30k 24k 30k 14 5 84k y 4 y k 2 8 2 k x y 2 4 5 3 2 y= 4 x y= 1 1 + x2 1 x 1 2 3 4 −1 x 1 308 17. y m Chapter 14 Multiple Integration 18. 8192k 15 524,288k 105 64 7 x m 30 3 9 x2 0 by symmetry 4 16 y2 0 by symmetry 3 9 x2 kx dx dy 40 4 16 y2 ky 2 dy dx 23,328k 35 139,968k 35 6 My 40 kx 2 dx dy My m 524,288k 105 15 8192k Mx 30 ky 3 dy dx Mx m y x y 8 y 139,968k 35 35 23,328k x = 16 − y 2 12 4 x 4 −4 −8 8 y = 9 − x2 6 3 x 3 6 −6 −3 19. x m L by symmetry 2 L 0 L sin x L L2 cos x L 20. m 0 0 cos x L k dy dx L2 kL kL 8 L2 2 L kL 2 2 ky dy dx 0 sin x L kL 4 4kL 9 Mx 0 L2 0 cos x L ky dy dx kx dy dx 0 0 Mx 0 0 ky 2 dy dx Mx m 4kL 9 4 kL 16 9 My x y My m Mx m t 2k 2 y y L2 2 kL 8 2 2k 2 y = sin π x L kL 8 1 1 x L 2 y = cos π x L L x L 2 21. m Mx a2k 8 4 a π 2 ky dA R 0 4 0 a kr 2 sin dr d ka3 2 6 ka3 2 6 2 y=x r=a My R kx dA 0 0 kr 2 cos dr d 8 a2k 2 6 4a 2 3 8 a2k 4a 2 3 a 0 x y My m Mx m ka3 2 6 ka3 2 2 S ection 14.4 4 a Center of Mass and Moments of Inertia 309 22. m R k x2 y 2 dA 0 0 4 kr 2 dr d a ka3 12 ka4 2 8 ka4 2 8 2 π 2 Mx R k x2 y 2 y dA 0 4 0 a kr3 sin d y=x r=a My R k x2 My m Mx m 2 y 2 dA 0 0 kr 3 cos d 3 2a 2 12 ka3 32 2 2a a 0 x ka4 2 8 ka4 2 8 e 0 e 0 e 0 x x x 12 ka3 2 y 23. m 0 2 ky dy dx ky 2 dy dx 0 2 k 1 4 k 1 9 k1 4e4 k e4 e ln x 0 ln x 0 ln x 0 e e 4 24. m 1 e 6 k dy dx x k y dy dx x k x dy dx x 2 k 2 k 2 1 3 k 2 k 6 k Mx My 0 Mx 1 e kxy dy dx My m Mx m y 5e 8 1 4 My 1 x y k e4 5 8e4 k e6 1 9e6 e4 2 e4 4 e6 9 e6 5 1 1 e2 0.46 0.45 x y y My m Mx m k 1 k 6 4e4 k e4 1 2 3 1 y = e −x 2 y = ln x 1 x 1 2 1 2 e3 x 25. y m 0 by symmetry 6 2 cos 3 k dA R 6 0 kr dr d k 3 My R kx dA 6 6 2 cos 3 kr 2 cos dr d 0 27 3 k 40 1.17k x π 2 My m 81 3 40 1.12 π θ= 6 r = 2 cos 3θ 0 1 π θ =−6 310 26. y m Chapter 14 Multiple Integration π 2 0 by symmetry 2 1 0 2 1 0 cos cos r = 1 + cos θ k dA R 0 kr dr d 3k 2 0 1 My R kx dA 0 2 kr 2 cos dr d k 3 k 3 cos 0 2 1 3 cos 3 1 2 3 cos2 cos3 d 1 1 4 cos 0 cos2 3 cos 1 sin2 cos 2 2 d 5k 4 x My m 5k 4 2 3k 5 6 b h 0 h 0 h 0 hx b hx b hx b 27. m Ix bh b 0 b h 28. m 0 dy dx b bh 2 bh3 12 b3h 12 b 6 h 6 6 b 6 6 h 6 y2 0 h dy dx bh3 3 b3h 3 Ix 0 b y2 dy dx Iy 0 0 x 2dy Iy m Ix m dx Iy b2 3 h2 3 b 3 h 3 3 b 3 3 h 3 0 x2 dy dx Iy m Ix m a2 2 a x y b3h 3 bh3 3 1 bh 1 bh x y b3h 12 bh 2 bh3 12 bh 2 29. m Ix a2 2 a 30. m y 2 dA r3 sin2 0 2 0 a dr d R a4 4 a4 4 Ix R y 2 dA 0 0 a r3 sin2 dr d a4 8 a4 8 Iy R x 2 dA 0 0 r3 cos2 a4 4 a4 4 a4 4 a4 2 1 a2 dr d Iy R x 2 dA 0 0 r3 cos2 a4 8 a4 8 a4 8 a4 4 2 a2 dr d I0 x Ix y Iy I0 a 2 x Ix y Ix Ix m Ix m a 2 31. m Ix a2 4 2 a y 2 dA R 0 2 0 a r3 sin2 dr d a4 16 a4 16 Iy R x 2 dA 0 0 r3 cos2 a 16 4 dr d I0 x Ix y Iy Ix m a4 16 a4 16 a4 8 4 a2 a 2 S ection 14.4 32. m Ix 4 0 a 0 Center of Mass and Moments of Inertia 311 ab a ba a2 x2 y 2 dy dx b3 2 a 3a3 4b3 3a3 a 4 0 x2 32 dx a2 a2 0 x2 x 2 a2 a2 a2 x 2 dx x2 a4 arcsin x a a 0 4b3 a2 x a2 3a3 2 b ab 0 b2 y2 x2 a2 arcsin a 3b x a 1 x 2x 2 8 a b3 4 Iy I0 x y 4 0 x 2 dx dy Ix Iy m Ix m a3b 4 a3b 4 ab3 4 ab3 4 1 ab 1 ab 4 ab 4 a 2 b 2 34. ky a a2 0 x2 Iy a2 b2 33. m ky a b k 0 a 0 b y dy dx kab2 2 kab4 4 ka3b2 6 2kb2a3 12 ka3b2 6 kab2 2 kab4 4 kab2 2 a2 3 b2 2 a 3 b 2 3 a 3 2 b 2 m 2k 0 a y dy dx 2ka3 3 4ka5 15 2ka5 15 Ix Iy I0 x y k 0 a 0 b y3 dy dx k 0 a a2 a2 x 2 dx x2 k 0 0 x 2yy dy dx Iy Iy m Ix m 3kab4 Ix Iy I0 x y k a0 a a2 x2 y3 dy dx Ix k a0 x 2y dy dx Iy Iy m Ix m 2ka5 5 2ka5 15 2ka3 3 4ka5 15 2ka3 3 Ix a2 5 2a2 5 a 5 2a 10 35. m kx 2 4 0 4 0 4 0 x2 x2 x 2 36. x dy dx 0 2 kxy 1 x k 4k 32k 3 16k 3 m k 0 1 x 2 xy dy dx x k 2 k 4 k 2 3k 80 1 x3 0 1 x5 dx k 24 k 60 k 48 Ix k 0 2 xy dy dx 2 Ix Iy I0 k 0 1 x 2 xy3 dy dx x x5 0 1 x9 dx Iy I0 x k 0 x3 dy dx Iy Iy m Ix m 16k 16k 3 4k 32k 3 4k k 0 x 2 x3y dy dx Iy Iy m Ix m 9k 240 x5 0 x7 dx Ix Ix 4 3 8 3 2 3 4 6 23 3 26 3 x y k 48 k 24 k 60 k 24 1 2 2 5 y 312 37. m Chapter 14 kxy 4 0 4 0 x x Multiple Integration 38. x2 1 y2 x kxy dy dx 32k 3 16k 512k 5 m 0 1 x 2 x2 x y 2 dy dx y 2 y 2 dy dx y 2 x 2 dy dx 6 35 158 2079 158 2079 Ix 0 4 0 x kxy3 dy dx Ix 0 1 x 2 x2 x Iy 0 x2 x 2 Iy 0 0 kx3y dy dx 592k 5 512k 5 16k 1 I0 x Ix Iy Iy m Ix m I0 x Ix Iy Iy m Ix m 316 2079 158 2079 x 35 6 395 891 3 32k 3 32k 3 2 48 5 4 15 5 6 2 40. y 395 891 y 39. m kx 1 0 1 x 2 ky 2 4x x kx dy dx x 3k 20 3k 56 k 18 m 2 0 2 x 3 ky dy dx 4x 512k 21 32,768k 65 2048k 45 Ix 0 1 kxy 2 x 2 dy dx Ix Iy I0 x y 2 0 2 x 3 ky3 dy dx 4x x Iy 0 x2 kx3 dy dx 2 0 x 3 kx 2 y dy dx Iy Iy m Ix m 321,536k 585 2048k 45 I0 x Ix Iy Iy m Ix m b 55k 504 k 18 3k 56 b2 0 b x2 Ix 20 3k 20 3k 30 9 70 14 b 21 512k 21 512k 28 15 8 1365 65 2 105 15 y 32,768k 65 41. I 2k b b x a 2 dy dx 2k b x a 2 b2 x2 dx b 2k b x 2 b2 b4 8 0 x 2 dx a2b2 2 2a b x b2 4a2 x2 dx a2 b b2 x 2 dx 2k k b2 2 b 4 4 2 4 42. I 0 0 kx 6 dy dx 0 2 2k x 6 dx 2 2k x 3 6 3 4 0 416k 3 4 2 0 4 x 4 43. I 0 0 kx x 6 2 dy dx 0 kx x x 2 12x 36 dx k 29 x 9 2 24 7 x 7 2 72 5 x 5 42,752k 315 S ection 14.4 a a2 x2 Center of Mass and Moments of Inertia 313 44. I a0 a ky y y4 4 14 a 4 2ay3 3 2a2x 2 2a2x3 3 a2 25 a 3 x2 a 2 dy dx a2y 2 2 x4 x5 5 a2 15 a 5 x2 k a a a2 0 x2 dx a2 2 a 2 k a 2a 2 2 aa 3 2a a2 x a2 32 a4 arcsin 4 x2 x2 x 2 a2 a2 arcsin a2 2 ax 2 a3 a 2 2 x2 x a a a x2 dx k 14 ax 4 1 x 2x 2 8 x a 4 x3 3 a 3 3 2k 15 a 4 a2 0 a 2a a 34 a 16 a a2 0 a2 0 2k 7a5 15 a a5 8 k a 4 ka5 56 15 60 x2 a x2 45. I 0 ka k 4 k 4 k 4 a4 0 a yy a 2 dy dx 0 ka y4 x2 y dy dx 0 3 y 4 0 a2 dx 4a3y 6a2 y2 4ay3 dx a4 0 a 4a3 a2 x2 6a2 a2 x2 4a a2 x2 a2 x2 a4 2a2x2 x4 a4 dx 7a4 0 8a 2x 2 8a2 3 x 3 85 a 3 x4 x5 5 8a3 a2 x2 4ax 2 a 2 x a x 2 dx a x 2x 2 2 x a a 0 k 7a4x 4 k 7a5 4 4a3 x a2 15 a 4 x2 a2 arcsin 7 16 a2 a2 x2 a4 arcsin 15 a 5 2a5 a5k 17 15 2 4 x2 2 46. I 20 ky k 3 2 2 2 dy dx 2 k y 3 1 3 4 0 x2 2 dx 2 k 2 3 17 x 7 x2 2 2 8 dx 16 2 12x 2 192 5 6x 4 128 7 x6 dx 1408k 105 k 16x 3 4x3 65 x 5 2k 32 3 47. x, y ky 32 48. x, y k2 x y will increase. 49. x, y kxy 50. x, y will be the same. x, y k4 x4 y Both x and y will increase. Both x and y will decrease. 314 51. Let Chapter 14 Multiple Integration x, y be a continuous density function on the planar lamina R. The movements of mass with respect to the x- and y-axes are Mx R y x, y dA and My R x x, y dA. If m is the mass of the lamina, then the center of mass is x, y My Mx , . mm 52. Ix R y2 x, y dA, Moment of inertia about x-axis x2 x, y dA, Moment of inertia about y-axis R 53. See the definition on page 1014. Iy 54. Orient the xy-coordinate system so that L is along the y-axis and R is in the first quadrant. Then the volume of the solid is V R 2 x dA y 2 R x dA L x dA 2 R R ( x, y ) dA dA R x R 2 x A. By our positioning, x L ,A 2 b L r. Therefore, V L 2 2 rA. a ,A 2 b a 55. y bL, h L 2 2 56. y ab, h a 2 2 L a 2 a3b 12 a 3L 3 2L 2a a Iy 0 b 0 0 y y Iy hA dy dx 3L Iy 0 0 y a 2 dy dx L2 3 L 2 bL ,h 2 L dx 0 L3b 12 L 3 ya a3b 12 L a 2 ab ya y L3b 12 L 2 bL L 3 2L 3 3L 2 57. y 2L ,A 3 b2 58. y Iy 0, A a a2 a2, h x2 L y 2 dy dx a 2 a a2 x2 Iy 2 0 2Lx b b2 y 2L 3 dy dx 2 3 2 3 r3 sin2 dx 0 2 0 dr d y 0 b2 0 2L x b L 27 2Lx b 2L 3 2L 3 3 dx L3b 36 ya 0 a4 2 sin 4 d 2 L3x 3 27 ya 2L 3 3 b 2Lx 8L b L b 36 L2b 6 L 2 4 b2 0 a4 4 a4 4 L a2 a2 4L Section 14.5 Surface Area 315 Section 14.5 1. f x, y R fx 1 2 Surface Area 2y 2. f x, y fx 1 fy 2 2x 15 2x 3 3y triangle with vertices 0, 0 , 2, 0 , 0, 2 2, fy fx 2 0 2 x 2, fy fx 3 3 2 2 3 2 fy 2 14 3 S 0 0 14 dy dx 0 y 3 14 dx 9 14 S 0 3 dy dx 3 2x y 3 0 2 x dx 3 x2 2 2 6 0 R 2 1 2 y = −x + 2 R x 1 2 3 1 x 1 2 3. f x, y R fx 1 2 8 2x 2y y2 ≤ 4 4. f x, y R fx 10 x, y : x 2 2x 3y x, y : x 2 2, fy fx 2 4 4 y y2 ≤ 9 3 2 fy x 2 2, fy 1 fx 3 2 9 9 3 2 3 2 2 fy x2 2 14 S 2 x2 3 dy dx 0 0 3r dr d 12 S 3 2 x2 14 dy dx 14 r dr d 0 0 y y = 4 − x2 9 14 R −1 1 y = 9 − x2 x 1 −1 2 R −2 −1 1 x y = − 4 − x2 −1 −2 1 y = − 9 − x2 5. f x, y R fx 1 3 9 x2 3 y square with vertices, 0, 0 , 3, 0 , 0, 3 , 3, 3 2x, fy fx 3 2 R 0 fy 1 2 2 1 4x 2 3 1 S 0 0 4x 2 dy dx 0 31 1 4x 2 dx 3 x 1 2 3 3 2x 1 4 4x 2 ln 2x 4x 2 0 3 6 37 4 ln 6 37 316 Chapter 14 y2 Multiple Integration y 6. f x, y R fx 1 3 square with vertices 0, 0 , 3, 0 , 0, 3 , 3, 3 0, fy fx 3 2 3 R 2y fy 1 2 2 1 4y 2 3 1 S 0 0 4y 2 dx dy 0 31 1 4y 2 dy 3 x 1 2 3 3 2y 1 4 x3 4y 2 ln 2y 4y 2 0 3 6 37 4 ln 6 37 7. f x, y R fx 1 3 2 2 4 y rectangle with vertices 0, 0 , 0, 4 , 3, 4 , 3, 0 3 12 x , fy 2 fx 4 0 2 R 3 0 fy 4 2 9x 32 0 2 2 1 3 9 x 4 4 0 4 2 9x 2 8 dx 9x 1 x 1 2 3 4 S 0 9x dy dx 3 4 4 4 27 4 31 31 27 8. f x, y fx 1 S 0 0 2 0, fy fx2 2 2 y 23 y 3 y1 fy2 1 2 2 2 y y=2−x R 1 y 2 1 y dx dy 0 1 2 y2 y dy 1 2 x 21 2 12 5 9. f x, y R fx 1 S 0 0 y 2 32 2 1 5 35 2 y 2 52 0 33 3 2 5 8 5 2 5 ln sec x x, y : 0 ≤ x ≤ 4 , 0 ≤ y ≤ tan x 2 y y = tan x 1 tan x, fy fx 4 2 tan x 0 fy 2 1 tan2 x 4 sec x 4 R π 4 π 2 x sec x dy dx 0 sec x tan x dx sec x 0 2 1 S ection 14.5 10. f x, y fx 1 2 Surface Area 317 9 x2 2y 2 y2 y 2x, fy fx 2 fy 1 2 1 dr d 2 32 0 4x 2 4y 2 R −1 1 x 1 −1 S 0 2 0 2 0 0 4r 2 r 4r 2 2 1 1 12 1 173 12 x2 d 17 17 1 1d 6 11. f x, y R 0≤ fx 1 1 y2 1 y x, y : 0 ≤ f x, y ≤ 1 x2 x x2 fx 2 x 2 + y2 = 1 y 2 ≤ 1, x 2 y , fy 2 fy x2 2 y2 ≤ 1 x y x2 1 2 1 y2 x2 x2 1 y2 y2 x2 y2 2 2 1 1 S 1 x2 2 dy dx 0 0 2r dr d 12. f x, y R fx 1 4 xy x, y : x 2 y 2 ≤ 16 2 y x 2 + y 2 = 16 y, fy fx 2 x fy x2 2 1 y2 y2 x2 −2 −2 x 2 16 16 4 S 4 2 0 x2 1 x 2 dy dx 2 17 17 3 1 0 r 2 r dr d 1 13. f x, y R fx 1 b a2 x, y : x 2 x x2 2 x2 y2 a b y y 2 ≤ b2, 0 < b < a y2 fy 2 x 2 + y 2 ≤ b2 a2 fx , fy 1 a x2 a2 a2 y2 y x2 x2 x2 y2 y2 2 b 0 a2 y2 x2 y2 a2 a x2 −b b −b a x y2 a2 b2 b 2 x 2 2 S b b x 2 a2 dy dx 0 a r dr d a2 r 2 2 aa 14. See Exercise 13. a a2 a2 x2 x2 S a a2 a x2 2 a 0 y2 dy dx 0 a r dr d a2 r 2 2 a2 318 15. z 1 Chapter 14 24 fx 8 Multiple Integration 2y fy 12 2 y 3x 2 3 2x 0 14 14 dy dx 48 14 16 12 8 S 0 4 x 4 8 12 16 16. z 1 16 fy 2 4 x2 y2 fy2 1 1 4x 2 4 x2 4y 2 y 2 dy dx 6 y 16 0 x 2 y= 4 16 − x 2 S 0 2 0 4 2 1 0 4r 2r dr d 24 65 65 1 x 2 4 6 17. z 1 25 fx 3 2 x2 y2 fy 2 y x 2 + y2 = 9 1 5 x2 r dr d x2 25 x2 dy dx y2 25 y2 x2 y2 25 5 x2 2 y2 −2 −1 1 9 9 3 0 x 2 x S 2 3 2 x2 −1 −2 1 2 25 y2 20 2 0 5 25 y2 fy2 r2 18. z 1 S 2 x2 fx2 2 0 2 y x2 + y2 = 4 1 4 4x2 x2 5 y2 x2 4y2 y2 5 1 5r dr d 0 −1 −1 x 1 19. f x, y R 1 1 2y x2 1 y triangle with vertices 0, 0 , 1, 0 , 1, 1 y=x fx x 2 fy 5 2 5 4x 2 1 27 12 55 R 1 S 0 0 4x 2 dy dx x 20. f x, y R 1 2 2x y2 3 y triangle with vertices 0, 0 , 2, 0 , 2, 2 fx x 2 fy 5 2 5 4y 2 5 8 21 ln 4 5 37 21 4 55 12 y=x 2 1 R S 0 0 4y 2 dy dx x 1 2 3 S ection 14.5 21. f x, y R 0≤4 fx 1 2 Surface Area 319 4 x2 y2 22. f x, y R y ≤4 2 x2 y2 x, y : 0 ≤ f x, y x 2 x, y : 0 ≤ f x, y ≤ 16 x2 2x, fy 1 4 y,x 2 2 0≤ fx y 2 ≤ 16 2y 2 16 16 4 x 2 2x, fy fx 2 4 4 2 x 2 2y fy 2 1 4x 2 4x 2 4y 2 S fx fy 2 1 4x 2 4x 2 4y 2 S 2 2 0 x2 1 4y 2 dy dx 17 17 6 1 1 4 2 0 x2 4y 2 dy dx 65 65 6 1 1 0 y 4r 2 r dr d 1 0 y 4r 2 dr d x 2 + y2 = 4 x 2 + y 2 = 16 1 2 −1 −1 x 1 −2 −2 x 2 23. f x, y R fx 1 1 4 x2 y2 24. f x, y R fx 1 4y 2 4x 2 dy dx 4y 2 1.8616 S 0 0 2 32 3x cos x x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 2x, fy fx 1 2 x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 x1 1 1 2 2y fy 1 2 sin x, fy 2 0 1 x 2 fx 1 fy 1 2 x sin x 2 S 0 0 4x2 sin x dy dx 1.02185 25. Surface area > 4 Matches (e) z 10 6 24 26. Surface area Matches (c) z 3 2 9 x 5 x 5 y 3 3 y 27. f x, y R fx 1 1 ex x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 28. f x, y R fx fy 1 2 2 52 5y x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0, fy 1 1 e x, fy fx 1 2 0 1 e 2x y3 2 2 fx 1 fy 1 2 1 y3 S 0 1 0 e 2x dy dx S 0 1 0 y 3 dx dy y 3 dy 1.1114 1 0 e 2x 2.0035 1 0 320 Chapter 14 x3 3xy Multiple Integration y3 1, 1 , 3x 9 y2 1, 3y 2 1 , 1, 3 y2 1 x 30. f x, y R fx 1 x2 3xy y2 29. f x, y R fx S 1 square with vertices 1, 1 , 3x 2 1 1 x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ x 2x fx 4 x 3y 1 1 3 x2 9 x2 y , fy y 2 3y, fy fy 2 3x 1 1 2y 2x 13 x 2 y 2 dy dx 3x 3y 2 2y 3x 2y 2 x 2 dy dx y2 S 0 0 1 13 x 2 31. f x, y fx 1 S 2 2 e e fx2 x x sin y e 1 1 e x sin y, fy fy2 4 4 x2 cos y e 2x 2x sin2 y e 2x cos2 y 1 e 2x dy dx x2 32. f x, y R fx 1 S cos x 2 x, y : x 2 2x sin x 2 fx 2 2 2 ( ( y2 y2 ≤ 2 2y sin x 2 1 x2 y 2 , fy fy 2 2) 2) y2 y2 4y 2sin2 x 2 y 2 dy dx y2 1 4 sin2 x 2 y 2 x2 y2 4x 2 sin2 x 2 4 x2 1 x2 y 2 sin2 x 2 33. f x, y R fx 1 4 exy x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ 10 ye xy, fy fx 10 2 xe xy fy 1 2 1 y 2e2xy y 2 dy dx x 2e2xy 1 e2xy x 2 y2 S 0 0 e2xy x 2 34. f x, y R fx 1 4 e x sin y 35. See the definition on page 1018. x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ x e x sin y, fy 2 e 2 x cos y 1 1 e e 2x 2x fx x fy sin2 y e 2x cos2 y S 0 0 1 e 2x dy dx 36. (a) Yes. For example, let R be the square given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and S the square parallel to R given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, z 1. 37. f x, y S R 1 1 1 x 0 x 2; fx fx2 1 1 x x2 x 2 x 12 fy2 dA dy dx x2 , fy 0 (b) Yes. Let R be the region in part (a) and S the surface xy. given by f x, y (c) No 16 0 1 1 16 0 1 dx 16 1 x2 12 0 16 S ection 14.5 38. f x, y 1 S R Surface Area 321 k x2 fx 2 y2 fy fx 2 1 2 k2x 2 x y2 2 k2y 2 x y2 2 k2 k2 1 1 R 1 fy 2 dA R k2 1 dA dA A k2 1 r2 k2 1 50 502 0 x2 39. (a) V 0 20 xy 100 x 5 y dy dx 50 20 502 0 x2 x2 x 502 200 502 arcsin x 50 x2 x 5 25 2 x 4 502 x4 800 x2 502 x 2 dy 10 x2 32 10 x 50 30,415.74 (b) z 1 S 1 100 1 100 13 y 75 20 fx xy 100 2 1 502 15 250x x3 30 50 0 ft3 fy 502 0 2 50 2 1 x2 y2 1002 x2 x2 1002 y 2 dy dx 1002 x2 100 y2 50 0 1002 1002 0 0 r 2r dr d 2081.53 ft2 40. (a) z 42 y 25 13 y 75 12 y 25 15 16 y 15 42 y 25 8 y 25 fy2 15 25 (b) V 2 50 0 13 y 75 42 y 25 16 y 15 25 dy 100 266.25 (c) f x, y fx S 0, fy 50 26,625 cubic feet 16 y 15 16 15 25 (d) Arc length 30.8758 2 50 30.8758 3087.58 sq ft Surface area of roof 2 0 0 1 fx2 dy dx 3087.58 sq ft y 41. (a) V R f x, y 24 20 8 R 2 625 25 x2 y 2 dA where R is the region in the first quadrant 16 12 R 8 8 0 4 2 625 2 625 3 609 609 609 cm3 r 2r dr d 25 4 x 4 8 12 16 20 24 4 0 r2 32 4 d 8 0 3 812 2 —CONTINUED— 322 Chapter 14 Multiple Integration 41. —CONTINUED— (b) A R 1 fx 2 fy 2 dA 8 R 2 25 4 1 625 25 625 x2 x2 r dr d y2 625 y2 x2 y2 dA 8 R 25 625 x 2 200 625 y dA 2 b 8 0 r2 b → 25 lim r2 4 2 100 609 cm2 Section 14.6 3 2 0 1 Triple Integrals and Applications 3 2 0 2 0 1. 0 0 x y z dx dy dx 0 3 0 12 x 2 1 2 y 1 xy xz 0 dy dx 3 z dy dz 0 1 y 2 12 y 2 2 3 yz 0 dz 3z z2 0 18 1 1 1 1 2. 1 1 x 2y 2z2 dx dy dz 1 3 2 3 1 1 1 1 1 1 x3y 2z2 1 1 1 dy dz 2 9 1 1 y 2z2 dy dz 1 y3z2 1 1 dz 4 9 1 z2 dz 1 43 z 27 1 1 8 27 1 x 0 xy 1 x xy 3. 0 0 x dz dy dx 0 1 0 9 y3 0 0 y 2 9x 2 0 x xz 0 dy dx 1 x 2y dy dx 0 9 0 9 y3 0 x 2y 2 2 x 1 dx 0 0 x4 dx 2 x5 10 1 0 1 10 4. 0 z dz dx dy 1 2 1 2 y2 0 9x 2 dx dy y3 xy 2 0 3x3 0 dy 2 18 4 1 9 y 3 dy 0 14 y 36 9 0 729 4 4 1 0 x 4 1 5. 1 0 2ze x2 x dy dx dz 1 4 0 2ze x2 y 0 dx dz 1 4 0 2zxe x2 dx dz z2 2 4 1 1 ze 1 x2 0 dz 1 z1 e 1 dz 1 e 1 15 1 2 1 e 4 e2 1 1 xz 4 e2 1 xz 4 e2 1 6. 1 0 ln z dy dz dx 1 4 1 1 ln z y 0 2 e2 dz dx 1 4 ln z dz dx xz 4 1 ln z x2 4 0 4 0 2 dx 1 1 2 dx x x 2 ln x 1 2 ln 4 4 2 0 1 0 x 1 4 2 7. 0 x cos y dz dy dx x1 0 x cos y z 0 2 dy dx 0 4 0 x1 x1 0 x cos y dy dx x2 2 x3 3 4 x sin y 0 dx x dx 8 0 64 3 40 3 S ection 14.6 2 y2 0 1y 2 y2 0 Triple Integrals and Applications 2 0 323 8. 0 0 sin y dz dx dy 0 sin y dx dy y 1 2 128 15 2 sin y dy 0 1 cos y 2 1 2 2 4 4 2 x2 x2 2 x2 x2 2 4 4 2 x2 9. 0 0 x dz dy dx 0 y2 x2 x3 dy dx x2 4 2 0 10. 0 0 2x 2 y 2 y dz dy dx 0 4y 2x 2y 2y3 dy dx 16 2 15 2 4 0 x2 4 1 11. 0 x 2 sin y dz dy dx z 2 0 2 0 4 x2 4 x 2 sin y ln z 1 4 x2 dy dx 2 x 2 ln 4 0 3 2 0 (2y 3) 6 0 2y 3z 6 (6 0 3 0 cos y 0 x) 2 (6 0 (x 2) x dx 0 2y) 3 x 2 ln 4 1 cos 4 x 2 dx 2.44167 12. 0 ze x2 y 2 dx dz dy 0 6 0 ze 16 2 2 x 2y 2 dz dy dx x 2y 2 x 3 2y 2 e dy dx 2.118 4 4 0 x 4 0 x y 3 2x 0 9 0 x 3 9 9 x2 x2 9 0 x 2 y 2 13. 0 dz dy dx 14. 0 dz dy dx 15. 3 dz dy dx 16. z x2 4 4 12 x 2 y2 16 16 y2 ⇒ 2z z2 x 2 x2 y2 2z 80 0⇒ z 8z 10 0⇒z 8 ⇒ x2 y2 2z 16 2z 80 z2 x2 y 2 80 ⇒ z2 dz dy dx y2 x 2 1 2x 2 2 4 y2 x 2 4 y2 17. 20 0 dz dx dy 20 x dx dy 1 2 2 2 4 2 y 2 2 dy 0 16 8y 2 y 4 dy 16y 83 y 3 15 y 5 2 0 256 15 1 1 0 xy 1 1 1 18. 0 0 dz dy dx 0 0 xy dy dx 0 x dx 2 a2 x2 x2 4 1 0 1 4 a a2 0 x2 0 a2 x2 y2 a 19. 8 0 dz dy dx 8 0 a 0 a2 x2 y 2 dy dx y a2 a 0 a2 x2 4 0 y a2 a x2 y2 a2 x 2 arcsin 13 x 3 x2 dx 0 4 2 a2 0 x 2 dx 2 a2x 43 a 3 324 Chapter 14 6 36 0 x2 36 0 x 2 Multiple Integration y 2 6 36 0 x2 6 20. 4 0 dz dy dx 4 0 6 36 36 36 0 x2 x2 y2 dy dx x2 36 x2 x 6 4 0 36y 1 36 3 x2 x2 x2y 32 y3 3 dx 6 36 0 x2 dx 4 4 9x 36 x2 324 arcsin 1 x 36 6 32 0 4 162 648 2 4 0 x2 4 0 x2 2 2 21. 0 dz dy dx 0 4 x 2 2 dx 0 16 8x 2 x 4 dx 16x 83 x 3 15 x 5 2 0 256 15 2 2 0 x 9 0 x2 2 2 0 x 2 22. 0 dz dy dx 0 2 9 x 2 dy dx 0 9 x2 2 x dx 92 x 2 23 x 3 14 x 4 2 0 18 0 9x 2x 2 x3 dx 18x 50 3 z y2 6 9 6 z 23. Plane: 3x 3 0 (12 0 4z) 3 6y (12 0 4z 4z 12 3x) 6 3 z 24. Top plane: x y dy dx dz Side cylinder: x2 3 0 2 3 y 9 0 y2 6 0 x y dz dx dy 4 x 6 x 3 3 y 6 25. Top cylinder: y 2 Side plane: x 1 0 x 0 0 1 y 2 z2 y 1 z 26. Elliptic cone: 4x 2 4 1 z2 y2 5 4 3 2 1 z 4 z 0 y2 z2 2 dx dy dz 0 dz dy dx 1 x 1 3 2 1 y x 5 y 27. Q x, y, z : 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤ 3 3 1 0 1 3 1 0 1 0 1 0 1 0 1 0 3 0 3 0 1 y x 0 x y xyz dV 0 Q y xyz dx dy dz 0 0 1 xyz dy dx dz 1 y=x xyz dx dz dy y x R x 1 xyz dy dz dx 0 3 xyz dz dx dy 0 3 xyz dz dy dx 0 9 16 S ection 14.6 28. Q x, y, z : 0 ≤ x ≤ 2, x 2 ≤ y ≤ 4, 0 ≤ z ≤ 2 2 4 2 x Triple Integrals and Applications 325 x 4 z xyz dV 0 Q 4 0 2 0 2 0 2 0 4 0 0 2 0 2 0 (2 0 2 0 z x y 2 0 x2 0 xyz dz dy dx 2 x xyz dz dx dy 4 x 2 4 y (2, 4) 4 xyz dy dz dx x2 4 xyz dy dx dz x2 z)2 0 y 0 y y 2 4 (2 2 2 y 2 z)2 0 2 0 z z xyz dx dy dz 0 4 xyz dx dy dz xyz dx dz dy 0 dx dz dy 104 21 z 5 29. Q x, y, z : x2 4 y2 ≤ 9, 0 ≤ z ≤ 4 3 3 3 3 4 9 9 9 9 9 9 9 9 4 y2 x2 xyz dV Q 0 4 0 3 x2 y2 xyz dy dx dz xyz dx dy dz y2 4 3 3 4 y y2 x xyz dx dz dy 30 3 3 3 y2 4 xyz dz dx dy y2 0 9 9 9 9 x2 x2 xyz dy dz dx 30 3 3 x2 4 xyz dz dy dx x2 0 0 30. Q x, y, z : 0 ≤ x ≤ 1, y ≤ 1 1 1 0 1 0 6 0 1 0 6 0 1 0 0 1 0 1 0 x2 x2 0 1 y 1 y x2 6 x2, 0 ≤ z ≤ 6 6 z xyz dV 0 Q 1 0 1 0 6 0 1 0 6 0 0 xyz dz dy dx 6 xyz dz dx dy 0 y xyz dx dz dy xyz dx dy dz 2 x 1 1 2 y xyz dy dz dx xyz dy dx dz 3 2 326 31. Q 1 0 Chapter 14 Multiple Integration y 2, 0 ≤ z ≤ 1 y x, y, z : 0 ≤ y ≤ 1, 0 ≤ x ≤ 1 1 0 y 2 y z=1− 1−x 1 z 1 0 y 1 0 1 0 1 0 1 0 1 0 0 2z 0 1 1 1 0 1 0 1 x 1 0 dz dx dy z 2 dz dy dx 1 0 1 1 y 1 0 z 1 0 y 2 y+z=1 z 1 0 z 1 2z 1 0 z 2 1 0 1 x dy dx dz 1 0 x dy dx dz 1 0 x 1 x 1 dy dz dx x0 y 2 dy dz dx y x = 1 − y2 dx dz dy dx dy dz 5 12 32. Q 3 0 x 0 x, y, z : 0 ≤ x ≤ 3, 0 ≤ y ≤ x, 0 ≤ z ≤ 9 9 0 x 2 x2 z = 9 − x2 9 z 3 3 y 9 0 9 0 x 2 x 2 dz dy dx 0 3 0 9 0 9 0 3 0 0 0 9 y 2 dz dx dy x z = 9 − y2 dy dz dx 0 9 0 9 z y 9 y z z x dy dx dz 0 9 z x 3 y=x 3 y dx dy dz dx dz dy 81 4 6 4 0 (2x 3) 2 0 ( y 2) (x 3) 5 5 0 5 0 x 1 5 15 0 3x 3y 33. m k 0 dz dy dx 34. m Mxz k 0 5 x 1 5 15 0 3x 3y y dz dy dz y 2 dz dy dx 0 125 k 8 125 k 4 8k 6 4 0 (2x 3) 2 0 ( y 2) (x 3) k Mxz m 2 Myz k 0 x dz dy dx y 12k x Myz m 4 4 0 4 12k 8k 4 0 x 3 2 4 4 b a1 0 a[1 0 2 ( y b)] ( y b) c1 0 c[1 0 ( y b) (x a)] ( y b) (x a) 35. m k 0 x dz dy dx 4x 0 4 4 0 4 4 0 x k 0 0 x4 x dy dx 36. m Mxz k 0 b dz dx dy y dz dx dy 0 kabc 6 kab2c 24 4k x dx 2 128k 3 4 4 k Mxz m kab c 24 kabc 6 b 4 Mxy k 0 xz dz dy dx k 0 0 x 128k 3 4 2 x 2 dy dx y 2k 0 16x 1 8x 2 x3 dx z Mxy m S ection 14.6 b b 0 b 0 b 0 b 0 b Triple Integrals and Applications a b 0 b 0 b 0 b 0 c 327 37. m k 0 b 0 b xy dz dy dx kb5 4 kb6 6 kb6 6 kb6 8 38. m Mxy Myz Mxz x y z k 0 a 0 c z dz dy dx z2 dz dy dx 0 a 0 c kabc2 2 kabc3 3 ka2bc2 4 kab2c2 4 Myz Mxz Mxy x y z k 0 b 0 b x y dz dy dx 2 k k 0 a 0 c k 0 b 0 b xy dz dy dx 2 xz dz dy dx yz dz dy dx 0 0 k Myz m Mxz m Mxy m k 0 0 xyz dz dy dx kb6 6 kb5 4 kb6 6 kb5 4 kb6 8 kb5 4 2b 3 2b 3 b 2 Myz m Mxz m Mxy m ka2bc2 4 kabc2 2 kab2c2 4 kabc2 2 kabc3 3 kabc2 2 a 2 b 2 2c 3 39. x will be greater than 2, whereas y and z will be unchanged. 40. z will be greater than 8 5, whereas x and y will be unchanged. 41. y will be greater than 0, whereas x and z will be unchanged. 42. x, y and z will all be greater than their original values. 2 4 0 x2 y 43. m x Mxy 1 k r 2h 3 y 4k 0 0 r 0 r 0 h r2 x2 x2 y2 r 44. m 2k 0 2 0 dz dy dx 16k 3 x dz dy dx 2 0 4 0 4 0 x2 x2 0 y 2 0 by symmetry r r2 x2 h k 0 4 2 4 x 2 dx x2 y z dz dy dx Myz x2 y 2 dy dx Mxz x2 32 k 0 2kh2 r2 4kh2 3r 2 r2 2k 0 0 y 2 y dz dy dx 2k r2 0 dx Mxy x 2k z dz dy dx 0 0 k k r 2h2 4 z Mxy m k r 2h2 4 k r 2h 3 3h 4 Myz m Mxz m Mxy m 0 16k 3 2k 16k 3 k 16k 3 0 3 8 3 16 y z 328 Chapter 14 128k 3 y 42 4 Multiple Integration 45. m x z Mxy 0 by symmetry x2 4 0 42 0 2 x2 2 y2 x 2 42 0 x2 y2 4k 0 4 z dz dy dx 4 2k 0 42 x2 y 2 dy dx 2k 0 16y x 2y 13 y 3 42 0 x2 dx 4k 3 4 4 0 2 x2 32 dx 1024k 3 64 k z Mxy m 0 2 cos4 0 d let x 4 sin by Wallis’s Formula 64k 1 3 128k 3 2 46. x m 1 0 1 0 1 ( y 2 1) 2 1 0 2 2k 0 2 0 1 ( y 2 1) dz dy dx 2k 0 1 y2 1 0 2 1 y dy dx 2k 4 2 dx 0 k Mxz Mxy 2k 0 2 0 1 0 1 0 y dz dy dx 1 ( y 2 1) 2k 0 y2 1 dy dx k 0 ln 2 dx k ln 4 2k 0 2 0 z dz dy dx 1 y2 k ln 4 k k 1 2 4 1 2 2 dy dx k 0 k 0 y 2 y2 1 1 arctan y 2 1 dx 0 k 1 4 2 8 dx 0 k 1 2 4 y z Mxz m Mxy m 5 y 12 20 ln 4 2 4 k 47. f x, y m k y 20 (3 5)x 0 (3 5)x 0 (3 5)x 0 (3 5)x 0 12 (5 12)y dz dy dx 0 20 0 12 (5 12)y 200k 16 12 8 y = − 3 x + 12 5 Myz Mxz Mxy x y z k 0 20 0 12 (5 12)y x dz dy dx 1000k 4 x 4 8 12 16 20 k 0 20 0 12 (5 12)y y dz dy dx 1200k k 0 0 z dz dy dx 1000k 200k 1200k 200k 250k 200k 5 6 5 4 250k Myz m Mxz m Mxy m S ection 14.6 1 60 15 5 (3 5)x 0 (3 5)x 0 (3 5)x 0 (3 5)x 0 3 3 0 (1 15)(60 0 12x 20y) 3 Triple Integrals and Applications 329 48. f x, y m k 12x 3 20y (1 15)(60 12x 20y) 5 y dz dy dx 0 5 0 (1 15)(60 0 (1 15)(60 12x 20y) 12x 20y) 10k 25k 2 15k 2 10k 4 3 2 1 x 1 2 3 4 5 y = 3 (5 - x) 5 Myz Mxz Mxy x y z k 0 5 x dz dy dx k 0 5 y dz dy dx k 0 z dz dy dx 25k 2 10k 15k 2 10k 10k 10k a a 0 a a Myz m Mxz m Mxy m 5 4 3 4 1 a a 49. (a) Ix k 0 0 y2 13 y 3 Iz a a 0 a 0 a z2 dx dy dz a a ka 0 0 y2 az 2 z2 dy dz dz ka 13 az 3 13 az 3 a 0 ka 0 z2y 0 dz ka 0 13 a 3 2ka 3 5 Ix (b) Ix Iy k 0 2ka5 3 y2 0 by symmetry z2 xyz dx dy dz y 2z3 2 a ka2 2 a a 0 a y3z 0 yz3 dy dz ka4 a2z2 8 2 2z4 4 a 0 ka2 2 Ix Iy y 4z 4 ka8 8 a2 a2 5 dz 0 ka4 8 a2z 0 2z3 dz ka8 8 Iz a2 by symmetry a2 50. (a) Ixy Ixz Ix (b) Ixy k a2 a2 z2 dz dy dx ka by symmetry 12 Iz a2 ka5 12 Iyz Iy k ka5 12 a2 a2 a2 a2 a2 ka5 12 z2 x 2 ka5 6 y 2 dz dy dx a3k 12 ka a2 a2 a2 a2 a2 a2 x2 a2 a2 y 2 dy dx a2 a2 a2 a2 a7k 72 7ka7 360 Ixz Iyz Ix Iy Iz k a2 a2 y 2 x2 y 2 dz dy dx x 2y 2 y 4 dy dx Ixz by symmetry Ixy Ixy Iyz Ixz Iyz Ixz a7k 30 a7k 30 7ka7 180 330 Chapter 14 4 4 0 4 0 Multiple Integration x 4 4 51. (a) Ix k 0 4 y2 y3 4 3 32 4 3 4 4 0 4 4 0 x z2 dz dy dx y 4 3 4 k 0 0 4 y2 4 k 0 x x 1 4 3 4 4 3 x 3 dy dx 3 k 0 x x 2 x x 3 0 4 4 0 dx 64 4 3 x dx k Iy k 0 1 4 3 256k 4 4 x2 4x2 0 4 4 0 4 0 x z2 dz dy dx 1 4 3 x 3 k 0 0 x2 4 4k 4 x 14 x 4 y2 4 1 4 3 1 4 12 x 3 dy dx 4 4 0 4k Iz k 0 4 x3 x2 dx k 43 x 3 4 x 512k 3 y 2 dz dy dx 4 x2 0 0 4 x dy dx x dx 256k x x 3 k 0 4 4 0 x 2y 4 0 x y3 4 3 y y2 x 0 dx k 0 4x 2 4 4 64 4 3 y3 4 x x (b) Ix k 0 4 z2 dz dy dx y2 4 6 2 4 3 x 4 k 0 0 4 1 y4 3 8 4 3 dy dx dx k 0 y4 4 4 32 4 x x x 2 x 4 3 0 4 0 dx k 0 64 4 3 k 4 2048k 3 4 4 4 0 4 4 0 Iy k 0 y x2 4x 2 0 4 4 0 4 0 x z2 dz dy dx 1 4 3 x 3 k 0 0 x 2y 4 8k 4 x 1 y4 3 1 4 12 x dx x dx x x 4 3 dy dx 1024k 3 8k Iz k 0 4 x3 y x2 dx k 43 x 3 4 14 x 4 y3 4 64 4 4x 2 4 0 y 2 dz dy dx 4 x 2y 0 0 4 k 0 4 x 2y 2 2 32 8x 4 0 2 y2 y4 4 4 4x 2 x 0 dx k 0 8x 2 8k 0 4 x3 dx 4 2 0 8k 32x 43 x 3 14 x 4 4 0 2048k 3 2 0 52. (a) Ixy k 0 z3 dz dy dx 4 0 4 k 0 1 4 4 y 2 4 dy dx y8 dy dx y9 9 2 4 k 4 k 4 Ixz k 256 0 256y 2 256y3 3 96y4 96y5 5 k 0 16y6 16y7 7 4 2 0 256y 0 4 0 4 2 0 2 0 2 0 2 0 4 0 y2 dx 0 k 0 16,384 dx 945 65,536k 315 y 2z dz dy dx 1 16y 2 2 4 0 y2 12 y4 2 k 2 4 0 y 2 2 dy dx 16y3 3 8y5 5 y7 7 2 k 0 4 8y4 y6 dy dx 4 2 0 dx 0 k 2 4 0 1024 dx 105 2048k 105 Iyz k 0 4 x 2z dz dy dx 12 x 16 2 8y 2 k 0 12 x4 2 k 2 4 y 2 2 dy dx x 2 16y 8y3 3 Ixz y5 5 2 k 0 y4 dy dx Iyz Ixy dx 0 0 k 2 4 0 256 2 x dx 15 8192k 45 Ix Ixz Ixy 2048k , Iy 9 8192k , Iz 21 Iyz 63,488k 315 —CONTINUED— S ection 14.6 52. —CONTINUED— 4 2 0 4 2 0 2 0 4 2 0 2 0 2 0 4 0 4 0 4 0 4 0 4 0 4 0 y2 4 2 0 4 0 y2 y2 y2 4 2 0 4 0 y2 y2 y2 4 2 0 4 0 y2 y2 Triple Integrals and Applications 331 (b) Ixy 0 z2 4 z dz dy dx 32,768k 105 65,536k 315 32,768k 315 k 0 4 4z2 dz dy dx k 0 z3 dz dy dx Ixz 0 y2 4 z dz dy dx 1024k 15 2048k 105 1024k 21 k 0 4 4y 2 dz dy dx k 0 y 2z dz dy dx Iyz k 0 4 x2 4 z dz dy dx 4096k 9 Ixz Iyz 8192k 45 11,264k 35 4096k 15 k 0 4x 2 dz dy dx Ixy 48,128k , Iy 315 a2 a 2 k 0 x 2z dz dy dx Ixy 118,784k , Iz 315 a a Ix Ixz Iyz L2 a a x2 L2 53. Ixy k L2 x 2 z2 dz dx dy L2 k L2 22 a 3 1 x 2x 2 8 x2 a2 a2 x2 x 2 dx dy a2 a4 arcsin x a a 2 3 2k 3 Since m k L2 L2 a2 x a2 2 a4 4 a4 16 x2 dy a2 arcsin a4 Lk 4 x a dy a 2 L2 a2Lk, Ixy L2 a a a 2 ma2 4. x 2 L2 a Ixz k L2 L2 a2 x2 y 2 dz dx dy y2 x a2 2 a a a2 a2 x2 2k L2 a y 2 a2 x a L2 a x 2 dx dy L2 2k L2 L2 x2 a2 arcsin 2k dy a a k a2 L2 y 2 dy x 2 dx dy ka4 4 L2 2k a2 L3 3 8 1 mL2 12 Iyz k L2 L2 x2 x 2 dz dx dy 1 x 2x 2 28 ma2 4 ma2 4 mL2 12 a2 mL2 12 ma2 4 ma2 4 a2 x2 x 2 a2 L2 a 2k L a4 arcsin L2 x a a dy a dy L2 ka4 L 4 ma2 4 Ix Iy Iz Ixy Ixy Ixz Ixz Iyz Iyz m 3a2 12 ma2 2 m 3a2 12 L2 332 Chapter 14 c2 a2 a2 a2 a2 a2 a2 Multiple Integration b2 54. Ixy c2 c2 b2 b2 z2 dz dy dx b3 12 b c2 c2 c2 a2 dy dx a2 a2 12 b abc 12 ba3 12 c2 1 mb2 12 dx ba3c 12 1 mc2 12 12 a abc 12 1 ma2 12 Ixz c2 c2 b2 b2 y 2 dz dy dx y 2 dy dx c2 c2 a2 c2 Iyz c2 b2 x 2 dz dy dx Ixy Ixy Ixz Ixz Iyz Iyz 1 m a2 12 1 m b2 12 1 m a2 12 b2 c2 c2 ab c2 x 2 dx abc3 12 12 c abc 12 Ix Iy Iz 1 1 1 x 1 1 1 x2 4 x2 y2 55. 1 10 x2 y2 x2 y2 z2 dz dy dx 56. 1 x2 0 kx 2 x 2 y 2 dz dy dx 57. kz 2 4 4 x2 4 x 2 z y 2 (a) m 2 x2 0 kz dz dy dx y Mxy m 2 32k 3 4 z = 4 − x2 − y2 (b) x z Iz 0 by symmetry 1 m 4 4 2 2 x2 4 4 4 x 2 x2 4 x 2 y 2 kz 2 dz dy dx x2 0 y 2 2 2 y x2 2 x2 0 y 2 kz dz dy dx 32k 3 2 x x2 + y2 = 4 58. kxy 5 25 0 x 2 z 25 0 x 2 y 2 5 x 2 + y 2 + z 2 = 25 (a) m 0 kxy dz dy dx Myz m 1 m 5 0 0 25 x2 0 25 x2 y2 625 k 3 25 32 5 y 5 x (b) x y z (c) Iz x kxy dz dy dx x by symmetry Mxy m 5 0 0 1 m 25 5 0 x2 0 0 25 x2 0 25 x2 y2 z kxy dz dy dx 25 x2 y2 25 16 62500 k 21 60. 6 x 2 + y 2 = 25 x2 y 2 kxy dz dy dx 59. See the definition, page 1024. See Theorem 14.4, page 1025. 61. (a) The annular solid on the right has the greater density. (b) The annular solid on the right has the greater movement of inertia. (c) The solid on the left will reach the bottom first. The solid on the right has a greater resistance to rotational motion. 62. Because the density increases as you move away from the axis of symmetry, the moment of inertia will increase. S ection 14.6 63. V 1 (unit cube) 1 V Q 1 0 1 0 1 1 0 1 1 Triple Integrals and Applications 333 64. V f x, y, z dV 27 (cube with sides of length 3) 1 V Q Average value Average value f x, y, z dV 3 0 3 0 3 0 3 0 3 0 3 z2 0 4 dx dy dz 4 dy dz 1 27 1 27 1 27 xyz dx dy dz 0 z2 0 9 yz dy dz 2 z2 0 4 dz 1 81 z dz 4 27 8 z3 3 1 3 1 base 3 11 22 32 f x, y, z Plane: x x y 4 4z 0 1 729 27 8 13 3 65. V height 2 z (0, 0, 2) 2 y z 1 V z 2 4 3 x + y+ z = 2 (2, 0, 0) x 2 (0, 2, 0) 2 y Average value f x, y, z dV Q 2 0 2 0 2 0 0 0 2 x 2 x 0 2 x y 3 4 3 4 3 4 x 1 2 2 4x x y yx z dz dy dx y 2 dy dx 1 x 6 3 2 2 2 dx 3 2 4 4 3 8 3 y 1 V Q 66. V 2 x 3 2 2 z x2 + y2 + z2 = 2 f x, y, z Average value f x, y, z dV x 2 2 y 3 8 2 2 2 2 2 x2 x2 2 2 x2 x2 y2 −2 x y2 y dz dy dx 0 (by symmetry) 334 67. 1 Chapter 14 2x 2 2x 2 Q 1 1 2 2 Multiple Integration 3z 2 ≥ 0 3z 2 ≤ 1 y2 1 2x 2 y2 y2 z 1 2x 2 + y 2 + 3z 2 = 1 x, y, z : 2x 2 1 1 2x 2 3z 2 ≤ 1 ellipsoid y 2 2 3 1 2x 2 2x 2 y2 3z 2 dz dy dx 0.684 1 x −1 1 y 1 2x y 2 3 46 Exact value: 45 68. 1 x2 Q 1 1 x2 y2 y2 z2 ≥ 0 z2 ≤ 1 y2 x2 x2 y2 x, y, z : x 2 1 1 x2 x2 1 1 z 2 ≤ 1 sphere 1 x2 y2 z 2 dz dy dx 1.6755 y2 Exact value: 8 15 a y2 4 a a y2 x y2 69. 14 15 1 0 1 0 1 3 0 3 0 dz dx dy 4 4 0 1 70. x 2 y2 b2 z2 9 1 x ax a3 12 a 2 32 16 a y2 x2 2 a 3 0 a dx dy a y2 By symmetry, the volume in the first octant is 1 16 8 1 b 0 1 2. x2 0 31 x2 y 2 b2 y2 y2 dy y2 3 a 2 y2 2 dy 1 dz dy dz 0 4 0 By trial and error, b 4. x2 a2 y2 b2 z2 c2 1 is 4 abc. 3 94 15 Hence, 3a 2 a 11a 3 22a 2 3a Note: Volume at ellipsoid 0 0 2, 16 . 3 71. Let yk 2n x1 1 ... xk. xn 2n 2 n y1 y1 y2 ... ... yn yn 2n Hence, 1 1 1 I1 0 0 1 1 0 0 ... 0 0 cos2 0 2n 2n 2n 1 . 2 x1 y1 x1 ... ... ... xn yn xn dx1 dx2 . . . dxn dy1 dy2 . . . I2 dyn ... 1 1 1 sin2 1 ... 0 0 sin2 dx1 dx2 . . . dxn I1 I2 1 ⇒ I1 n→ Finally, lim I1 1 . 2 Section 14.7 Triple Integrals in Cylindrical and Spherical Coordinates 335 Section 14.7 4 2 0 2 Triple Integrals in Cylindrical and Spherical Coordinates 4 0 4 0 0 0 2 2 1. 0 0 r cos dr d dz r2 cos 2 2 d dz 0 4 2 4 2 cos d dz 0 2 sin 0 dz 0 2 dz 8 4 2 0 2 0 r 4 2 0 4 2. 0 rz dz dr d 0 rz2 2 2 2 0 r dr d 4r 2 r 3 dr d 1 2 4 1 2 2 2 cos2 0 4 0 r2 4r 0 0 2r 2 0 4r 3 3 2 r4 4 2 d 0 2 3 4 d 0 6 2 0 2 2 cos2 3. 0 r sin dz dr d 0 r4 8 cos4 0 r 2 sin dr d 0 2r 2 8 cos5 5 r4 sin 4 4 cos9 9 2 2 cos2 d 0 2 0 4 cos8 sin d 52 45 2 2 2 3 4. 0 0 0 e 2 ddd 0 0 1 e 3 2 0 0 4 2 3 2 dd 0 0 0 1 1 3 1 12 e 8 dd 6 4 1 e 8 2 4 0 cos 2 5. 0 0 sin ddd 1 3 2 cos3 sin dd cos4 0 0 d 8 4 4 0 cos 2 6. 0 0 sin cos ddd 1 3 1 3 1 3 4 0 4 0 4 0 0 4 cos3 4 sin cos sin cos d cos cos dd 1 sin3 3 sin2 4 sin sin 0 4 dd cos sin d 0 4 0 52 36 4 z 0 0 2 0 5 2 sin2 36 2 2 sin 52 144 7. 0 rer d dr dz 2 er e4 3 8. 0 0 0 2 cos 2 ddd 8 9 2 3 0 2 3 9. 0 0 r dz dr d 0 2 0 2 0 0 re 1 e 2 1 1 2 e 9 r2 z dr d 3 r2 3 d 0 9 1 2 3 x 2 1 3 y e d 4 1 336 2 Chapter 14 3 0 3 0 r2 Multiple Integration 2 3 z 10. 0 r dz dr d 0 2 0 2 0 0 r3 3r 2 2 9 d 4 r4 4 9 2 r 2 dr d 3 4 d 0 2 3 x 2 3 y 2 2 6 4 2 11. 0 0 sin ddd 64 3 64 3 2 0 2 2 z sin 6 dd 2 4 cos 0 2 6 d x 4 4 y 32 3 3 64 3 3 2 5 2 0 0 2 d 0 12. sin ddd 117 3 117 3 468 3 2 z sin 0 2 0 dd d 0 7 x 7 r=5 r=2 cos 0 156 7 y 2 2 0 4 13. 0 2 0 r2 r 2 cos dz dr d arctan 1 2 0 4 sec 3 0 0 2 2 arctan 1 2 cot 0 csc 3 0 sin2 cos d d d sin2 cos d d d 0 2 2 0 0 6 0 16 r2 14. 0 2 0 4 3 0 r 2 dz dr d sin2 ddd 8 3 2 2 2 0 3 2 6 2 csc 3 4 sin2 ddd 8 3 2 2 2 3 2 a 0 4 2 0 a a a2 r2 1 0 0 2 0 1 r2 15. 0 2a cos r 2 cos dz dr d 3 0 a sec 0 16. 0 2 1 3 0 0 r r2 sin z2 dz dr d ddd 8 8 sin2 cos d d d 0 2 a cos 0 2 0 a2 r2 2 a cos 17. V 4 0 r dz dr d 43 a 3 4 0 0 r a2 1 cos 3 r 2 dr d 2 43 a 3 1 0 sin3 d sin2 2 0 43 a 3 2 2 3 2a 3 3 9 4 S ection 14.7 2 2 0 2 r 2 4 2 Triple Integrals in Cylindrical and Spherical Coordinates 16 20 r2 337 18. V 2 3 4 3 4 0 0 r dz dr d 0 r dz dr d 7 z (Volume of lower hemisphere) V 128 3 128 3 128 3 128 3 2 2 0 2 0 2 4(Volume in the first octant) 2 4 4 0 r 2 dr d 0 2 r 16 2 4 r 2 dr d x 7 7 y 4 4 82 3 82 3 1 16 3 r2 32 2 2 d 82 3 64 3 r2 64 2 3 a cos a2 0 2 2 2 2 0 2 r 4 r2 19. V 2 0 0 a cos r dz dr d 20. V 0 2 r dz dr d 2 0 0 r a2 12 a 3 1 0 r 2 dr d 0 a cos 0 32 0 2 r4 d 0 r2 r 2 dr d r3 3 2 2 0 r2 1 4 3 2 r2 32 d 0 2a3 3 2a3 3 sin3 cos 4 d cos3 3 8 2 3 0 2a3 3 9 2 2 0 2 9 0 r cos 2r sin 2 2 0 2 12e r 2 2 21. m 0 2 0 2 kr r dz dr d kr 2 9 0 22. 0 0 k r dz dr d 0 2 0 12ke r2 r dr d 2 0 r cos r4 cos 4 4 cos 2r sin r4 sin 2 8 sin 2 2 dr d 0 6ke 2 r2 k 3r 3 0 2 d 0 0 6ke 3k 1 e 4 6k d k 24 0 d 4 k 24 k 48 4 sin 8 8 8 cos 0 48k 338 Chapter 14 h r0 2 Multiple Integration h r r0 0 r dz dr d 23. z V h 4 0 x2 r0 0 2 h r0 0 r0 y2 r r0 r 24. x m Mxy y 0 by symmetry 1 r 2hk from Exercise 23 30 2 r0 0 2 0 h r0 0 r0 r r0 4h r0 4h r0 r0r 0 2 0 0 r2 dr d 4k 0 z r dz dr d r02r 0 r03 d 6 2 1 r 2h 30 z 2kh2 r02 2r0r 2 k r02 h2 12 3 r02hk h 4 r3 dr d 4h r03 r0 6 2kh2 r04 r02 12 Mxy m kz x y 4k 0 0 2 k r02h2 12 25. x m k x2 y 4k 0 0 y2 kr 26. 0 by symmetry 2 r0 h(r0 0 r) r0 0 by symmetry 2 r0 h(r0 0 r) r0 r 2 dz dr d m zr dz dr d 1 k r03h 6 2 r0 0 h(r0 0 r) r0 1 k r02h2 12 2 r0 0 h(r0 0 r) r0 Mxy 4k 0 r 2 z dz dr d Mxy 4k 0 z2r dz dr d 1 k r03 h2 30 z Mxy m 2 1 k r02h3 30 h 5 z Mxy m k r02h3 30 k r02h2 12 2h 5 k r03h2 30 k r03h 6 r0 0 2 h(r0 0 r0 r) r0 27. Iz 4k 0 r3 dz dr d 28. Iz Q x2 2 r0 0 2 y2 h r0 0 r0 0 x, y, z dV r r0 4kh r0 r0r3 0 0 r 4 dr d 4k 0 r 4 dz dr d r0 r0 r r 4 dr d 4kh r05 r0 20 1 k r04h 10 2 4kh 0 2 4kh 0 r5 5 r05 5 15 rd 30 0 r 6 r0 d 6r0 0 r05 d 6 r02h from Since the mass of the core is m kV k Exercise 23, we have k 3m r02h. Thus, Iz 1 k r04h 10 1 10 3m r02h r04h 1 3 2 4kh 0 2 4kh 0 4kh 15 r 30 0 2 3 mr 2. 10 0 15 r kh 15 0 S ection 14.7 29. m Iz k b2h 2 b a 2 Triple Integrals in Cylindrical and Spherical Coordinates 30. m Iz k a2h 2 2a sin 0 h 339 a2h h k h b2 a2 4k 0 0 b r 3 dz dr d 2k 0 0 r 3 dz dr d 4kh 0 2 a r3 dr d 3 k a4h 2 32 ma 2 kh 0 b4 a4 h 2 a4 d k b4 k b2 a2 b2 2 b2 a2 h 1 m a2 2 2 4 sin 2 31. V 0 0 0 sin d d d 16 2 4 2 0 b 2 32. V 8 0 a sin 2 ddd (includes upper and lower cones) z 83 b 3 4 b3 3 4 b3 3 1 4 a3 0 0 4 sin dd b a3 0 sin cos d x b a a b y 4 a3 0 24 b3 23 2 2 0 2 a 3 0 0 2 a3 2 2 3 2 b3 a3 2 2 0 2 a 3 33. m 8k sin ddd 34. m 8k 0 0 2 sin2 ddd 2ka4 0 2 0 sin dd 2ka4 0 2 0 sin2 dd k a4 0 sin cos d 2 k a4 0 sin2 1 2 1 k 4 d 1 sin 2 4 2a4 2 0 k a4 k a4 k a4 0 k a4 4 340 Chapter 14 2 k r3 3 y 4k 0 0 2 0 2 0 Multiple Integration 35. m x Mxy 36. x m y k 4k 0 0 by symmetry 23 R 3 2 0 by symmetry 2 2 r 3 0 2 23 r 3 2 0 r 2 R 3 2 k R3 3 cos 2 r3 ddd cos sin ddd Mxy sin 14 kr 2 kr 4 4 sin 2 d d sin 2 d 0 1 k R4 2 1 k R4 4 r4 0 0 2 sin 2 d d sin 2 d 0 2 r4 1 k r 4 cos 2 8 z Mxy m k r4 4 2k r3 3 2 0 1 k r4 4 z 1 k R4 8 Mxy m 2 r 4 cos 2 0 1 k R4 4 r4 r3 r4 3r 8 k R4 2k R3 2 0 r 2 R 4 r4 4 r3 3 3 R4 8 R3 2 2 0 cos 4 37. Iz 4k 4 0 2 sin3 ddd dd sin 2 4 38. Iz 4k 0 sin3 2 ddd sin3 dd cos2 cos3 3 d 2 0 4 k 5 2 k 5 2 k 5 k 192 39. x y z r cos r sin z 2 4 0 2 cos5 cos5 4 sin3 cos2 4k 5 R 5 d 2k 5 2k 5 R5 R5 r5 0 0 2 1 r5 0 sin cos 1 1 cos6 6 1 cos8 8 r5 r5 4k R5 15 x2 y2 tan z r2 y x z 40. x y z sin sin cos cos sin 2 x2 y x x2 y2 z2 tan cos z y2 z2 2 g2 g1 h2 r cos , r sin 41. 1 f r cos , r sin , z r dz dr d h1 r cos , r sin 2 2 2 42. 1 1 1 f sin cos , sin sin , cos 2 sin ddd 43. (a) r z r0: right circular cylinder about z-axis 0: plane parallel to z-axis (b) 0: sphere of radius cone 0 0: plane parallel to z-axis 0: z0: plane parallel to xy-plane 44. (a) You are integrating over a cylindrical wedge. (b) You are integrating over a spherical block. S ection 14.8 a a2 0 a x2 0 a2 0 2 a 0 2 a 0 2 a 0 2 0 x2 0 a2 r2 a2 x2 y2 0 a2 x2 y2 a2 x2 y 2 z2 Change of Variables: Jacobians 341 45. 16 0 dw dz dy dx 16 0 a2 a2 0 x2 y2 z2 dz dy dx 16 16 0 r2 z2 z2 dz r dr d a2 r 2 arcsin z a2 r2 0 a2 r2 1 z 2 a2 a2 r2 r dr d 8 0 2 r 2 r dr d r4 4 a4 2 2 a 4 0 2 a2r 2 2 d d 0 a4 0 46. 2 x2 e 0 0 2 k→ 0 y2 22 z2e x2 y2 z2 dx dy dz sin d d d 2 k 3e lim 0 0 0 sin d d d 2 k lim k 0 sin d 0 2 d 0 3e 2k 2 d 22 4 1 2 k→ lim 2 1e 2 0 Section 14.8 1. x y 1 u 2 1 u 2 xy uv v yx uv v Change of Variables: Jacobians 2. x y au cu xy uv bv dv yx uv ad cb 1 2 1 2 1 2 1 2 1 2 3. x y u u xy uv v2 v yx uv 11 1 2v 1 2v 4. x y uv uv xy uv 2u yx uv v 2u vu 2u 342 5. x y Chapter 14 u cos u sin xy uv v sin v cos yx uv Multiple Integration 6. x y sin2 1 u v xy uv a a yx uv 11 00 1 cos2 7. x y eu sin v eu cos v xy uv u v u xy uv v yx uv 2v 1 1 v 1 u v2 1 v u v2 u v2 v yx uv eu sin v eu sin v eu cos v eu cos v e2u 8. x y v 9. x y v u 3u 3v y 3 x 3 x 3 x, y 0, 0 3, 0 2, 3 u, v 0, 0 1, 0 0, 1 1 (0, 1) 2v 2y 9 x 2y3 3 (1, 0) u 1 10. x y u v 1 4u 3 1 u 3 x x y 4y v v x, y 0, 0 4, 1 2, 2 6, 3 u, v 1 v 0, 0 3, 0 0, 3, 6 6 (0, 0) 1 2 (3, 0) 4 5 6 −1 u −1 −2 −3 −4 −5 −6 (0, − 6) (3, − 6) 11. x y 1 u 2 1 u 2 xy uv 4 x2 v v v yx uv y 2 dA 1 1 1 (− 1, 1) (1, 1) u 1 2 1 1 1 2 1 4 u 4 1 u2 1 1 2 v 1 2 2 1 2 1 u 4 1 (− 1, − 1) (1, − 1) v 2 R 1 dv du 2 1 du 3 2 u3 3 u 3 1 1 1 v 2 dv du 1 2 u2 8 3 S ection 14.8 1 u 2 1 u 2 xy uv Change of Variables: Jacobians 343 12. x y v, v, 11 22 u v x x 1 2 y y 1 2 1 2 x, y 0, 1 2, 1 1, 2 1, 0 u, v 1, 1 1, 3 1, 3 1, 1 yx uv 60xy dA R 1 3 60 11 1 3 1 u 2 15 2 v 2 v 1 u 2 v 1 dv du 2 (−1, 3) v (1, 3) 2 u2 dv du 3 11 1 1 1 1 15 23 15 2u2 2 v3 u2v 1 du (−1, 1) −2 −1 −1 1 (1, 1) u 2 26 du 3 26 u 3 1 1 15 2 3 u 23 2 15 3 13. x y u u xy uv yx R 26 3 120 v 4 v yx uv y dA 3 10 3 0 4 11 uv 1 dv du 0 1 2 3 1 8u du 0 36 u 1 2 3 4 14. x y 1 u 2 1 u 2 xy uv 4x R v v u 1 2 v −1 v=u−2 yx uv y ex y 1 2 2 0 −2 dA 0 2 u 2 4uev 2u 1 0 1 dv du 2 2 eu du 2 u2 2 2 ue u 2 eu 2 0 21 e 2 344 Chapter 14 Multiple Integration 15. R e R: y xy 2 dA 2x, y 1 ,y x 4 x y ,v x v1 2 2 2 2 4 y 1 y= x y = 2x x ,y 4 v u, y x u y u 3 y= 4 x 2 1 y=4x x uv ⇒ u x v y v 1 2 u3 1 v1 2 u1 xy R 1 x 1 2 3 4 x, y u, v 1 1 2 u1 2v1 1 u1 2 2 v1 2 2 11 4u 1 u 1 2u v Transformed Region: y y y y 1 ⇒ yx x 4 ⇒ ux x 2x ⇒ y x 1⇒v 4⇒v 2⇒u 1 ⇒u 4 2 4 1 4 3 S 2 u 2 1 4 v2 1 3 4 x y ⇒ 4 x e R xy 2 dA 14 1 e e 2 1 dv du 2u 2 12 2 14 e v2 4 2 u 2 du 1 14 e ln 2 ln 2 e 12 1 du u e 2 e ln u 14 e e 12 1 4 e 12 ln 8 0.9798 16. x y u v v yx uv 1 v 4 4 xy uv y sin xy dA R 1 1 v sin u 1 dv du v 0 4 v 4 4 3 sin u du 1 3 cos u 1 3 cos 1 cos 4 3.5818 17. u u x x x 1 u 2 x, y u, v x y y v 1 2 4, 8, v v y x x 1 u 2 y y y 6 x−y=0 4 x+y=8 2 x+y=4 2 4 x−y=4 x 6 8 4 y ex y dA 4 0 8 uev 1 2 u e4 4 R 1 dv du 2 1 du 124 ue 4 8 1 4 12 e4 1 S ection 14.8 18. u u x x x 1 u 2 x, y u, v x R Change of Variables: Jacobians y 345 y y , 2, v, 1 2 v v y x x 1 u 2 y y v 0 3π 2 π π 2 x−y=0 x + y = 2π x+y=π π 2 π x− y=π 3π 2 x 2 y 2 sin2 x y dA 0 u2 sin2 v 1 u3 1 23 0 5 v 1 du dv 2 cos 2v 2 2 dv 0 73 v 12 1 sin 2v 2 0 74 12 19. u u x x x 1 u 5 4y 4y 0, 5, 4v , v v y 1 5 1 5 x x 1 u 5 y y y x−y=0 2 x + 4y = 5 1 −1 x 3 −1 4 xy uv x R yx uv yx 1 5 5 5 4 5 uv 1 5 1 du dv 5 5 2 x + 4y = 0 −2 x−y=5 4y dA 0 5 0 0 12 3 u 53 x x 3v 1 8 8 16 v 0 dv 2 52 3 v 33 5 2 0 100 9 20. u u x 3x 3x 1 u 4 2y 2y v, yx uv 0, 16, v v y 2y 2y 1 u 8 1 8 1 4 0 8 (−2, 3) 5 y 2y − x = 8 (2, 5) 3x + 2y = 16 3 2 (4, 2) 2y − x = 0 xy uv 3x R 13 48 x 32 3x + 2 y = 0 −2 −1 −1 1 2 3 x 4 (0, 0) 2y 2y dA 0 8 0 u v3 2 1 du dv 8 2 16v 5 5 8 2 0 16v 3 2 dv 0 4096 5 2 21. u x y, v yx uv x y dA x y, x 1 2 a u 1 u 2 v ,y 1 u 2 v xy uv u 0 u R y 1 dv du 2 v a u u du 0 25 u 5 a 2 0 25 a 5 2 v=u a a x+y=a u 2a x −a a v = −u 346 22. u u x Chapter 14 x x u, xy uv yx uv 1, 4, v v y 1 u Multiple Integration xy xy u v 1 4 4 3 y x=1 xy = 4 2 x=4 1 x 1 2 3 4 R xy dA 1 x 2y 2 4 1 4 1 4 1 v 1 1 dv du v2 u 4 xy = 1 1 ln 1 2 v2 1 du 1u 1 ln 17 2 4 ln 2 ln u 1 1 17 ln ln 4 2 2 23. au 2 a2 x2 a2 y2 b2 bv 2 b2 1, x 1 1 1 y au, y bv u2 (a) x2 a2 v2 y2 b2 u2 v2 v 1 (b) x, y u, v x u ab y v yx uv 00 ab 1 b R x (c) A S u ab dS S 1 a ab 1 2 ab 24. (a) f x, y R: x2 16 16 x2 y2 y2 ≤1 9 f x, y dA R V Let x 16 R 4u and y x2 3v. 1 1 1 1 u2 y 2 dA 1 2 0 u2 16 16u2 9v2 12 dv du Let u r cos , v r sin . 16 0 2 16r 2 cos2 9r 2 sin2 94 2 r sin 4 91 4 12 39 4 12r dr d 1 2 12 0 2 8r2 4r 4 cos2 1 cos 2 2 2 0 d 0 12 0 8 4 cos2 2 92 sin 4 7 cos 2 8 d 12 0 8 39 8 4 cos 2 2 117 d 12 0 39 8 d 12 —CONTINUED— 7 sin 2 16 S ection 14.8 24. —CONTINUED— x2 a2 y2 b2 Change of Variables: Jacobians 347 (b) f x, y x2 a2 A cos y2 ≤1 b2 au and y 2 R: Let x bv. 1 1 1 u2 f x, y dA R 1 u2 A cos r sin . r r dr d Aab 2 u2 v 2 ab dv du Let u 2 r cos , v 1 Aab 0 0 cos 2r 2 sin 2 r 2 0 4 2 cos r 2 4 2 1 2 0 2 Aab 0 4 2 Aab 25. Jacobian x, y u, v v, y 1 v1 vw xy uv uv 1 v w yx uv uvw 0 uv uv 1 1 u2v v u2v 1 v u2v 26. See Theorem 14.5. 27. x u1 x, y, z u, v, w w, z u u1 w uw w u uv 2 u2vw u uv2 1 w uv2w 28. x 4u x, y, z u, v, w v, y 4 0 1 4v 1 4 0 w, z 0 1 1 sin u 17 w 29. x sin x, y, z ,, cos , y sin cos sin sin cos cos cos 2 2 2 2 sin , z sin sin sin cos 0 cos cos cos 2 sin cos sin sin cos cos2 sin sin2 sin cos2 sin2 sin2 cos2 sin2 sin2 sin sin cos2 cos2 cos cos 2 sin2 sin2 sin3 cos2 sin sin sin2 2 sin 30. x r cos , y x, y, z r, , z cos sin 0 r sin , z r sin r cos 0 z 0 0 1 1 r cos2 r sin2 r 348 Chapter 14 x ,v 3 y 3 2 Multiple Integration x, y u, v 30 01 x ⇒v 2 v 31. Let u y⇒ 3, y 3u . 2 y= x 2 1 v = 3u 2 A′ x 3 −1 u 1 A −3 −1 −2 −3 1 x2 + y2 = 1 9 −1 u2 + v2 = 1 y 3 2 v 1 v = 3mu B′ y = mx B x 1 3 −3 −1 −2 −3 −1 u 1 x2 + y2 = 1 9 −1 u2 + v2 = 1 Region A is transformed to region A , and region B is transformed to region B . A B⇒ 2 3 3m ⇒ m 2 9 Note: You could also calculate the integrals directly. Review Exercises for Chapter 14 x2 x2 1. 1 x ln y dy xy 1 ln y 1 x3 1 ln x 2 x x x3 x3 ln x 2 2y 2. y x2 y 2 dx x3 3 2y xy 2 y 10 y 3 3 1 1 0 x 1 1 x 1 3. 0 3x 2y dy dx 0 3xy y2 0 dx 0 4x 2 5x 1 dx 43 x 3 43 x 3 52 x 2 14 x 2 1 x 0 29 6 2 0 2 2x 2 2x 2 4. 0 x2 x2 2y dy dx 0 x 2y y2 x2 dx 0 4x 2 2x3 2 x 4 dx 25 x 5 88 15 3 9 0 x2 3 5. 0 4x dy dx 0 4x 9 x 2 dx 4 9 3 3 x2 32 0 36 3 2 2 4 4 y2 3 6. 0 y2 dx dy 2 0 4 y 2 dy y4 y2 4 arcsin y 2 3 3 0 4 3 3 3 0 x3 1 3 0 3y 7. 0 dy dx 0 1 3 0 3y dx dy 1 A 0 dx dy 0 3 3y dy 3y 32 y 2 1 0 3 2 R eview Exercises for Chapter 14 2 x 3 6 0 2x 2 6 y y2 349 8. 0 0 dy dx 2 2 6 y y2 dy dx 0 dx dy 2 A 0 dx dy 1 2 4 6 0 3y dy 1 6y 2 4 3 32 y 2 2 3 0 3 25 25 x2 25 25 y2 5 25 25 y2 9. 5 x2 3 dy dx 5 25 x2 y2 3 dx dy 4 25 y2 dx dy 4 y2 dx dy x 5 3 5 A 2 50 dy dx 2 5 25 x2 dx x 25 x2 25 arcsin 25 2 12 25 arcsin 3 5 67.36 4 6x x2 x2 0 1 1 1 y 8 1 3 1 9 y 9 3 3 9 9 y 10. 0 2x dy dx 11 4 6x x 2 dy dx y 4 0 y x2 dx dy 8 y dx dy 4 0 A 0 2x dy dx 0 8x 2x2 dx 4x2 23 x 3 64 3 1 1 x 0 1 x2 1 11. A 4 0 12 1 1 dy dx 1 1 4y 2 4y 2 4 0 2 x1 x2 dx 4 1 3 x2 32 0 4 3 A 4 0 2 dx dy 2 y2 1 1 2 5 2 12. A 0 0 dx dy 0 0 dy dx 1 x 1 dy dx 14 3 9 2 9 2 y 5 x x 3 1 2 x 0 1 2 y 3 13. A 2 dy dx 2 1 dy dx 1 y2 1 dx dy 3 2y y y2 0 1 x 1 x 1 1 1 1 1 x 14. A 0 dx dy 3 dy dx 0 x dy dx 15. Both integrations are over the common region R shown in the figure. Analytically, 1 0 2 0 2 2y x2 2 2 y2 x y dx dy 4 3 2 0 4 3 8 2 y= 1x 2 2 1 (2, 1) R y=1 2 8 − x2 x x2 2 x 0 y dy dx 2 x y dy dx 5 3 4 3 2 1 3 4 3 4 3 2 −1 1 2 3 16. Both integrations are over the common region R shown in the figure. Analytically, 2 0 3 0 5 y y ex 3y 2 2x 3 y dx dy 2 5 5 85 e 5 5 0 x 5 4 3 (3, 2) e 0 x y dy dx 3 e x y dy dx 35 e 5 e 3 2 5 e 5 e 3 85 e 5 2 5 2 1 x 1 2 3 4 5 350 Chapter 14 4 x2 4 Multiple Integration 3 x 17. V 0 0 x2 y 4 dy dx 18. V 0 3 0 x y dy dx 12 y 2 x 4 x2y 0 4 0 12 y 2 4x2 43 x 3 x2 4 xy dx 0 dx 0 4y 0 14 x 2 8 dx 4 3 2 3 x2 dx 0 15 x 10 19. Volume 8x 0 3296 15 z 6 13 x 2 3 0 27 2 base height 9 3 2 27 2 4 2 20. Matches (c) 3 2 1 z Matches (c) y y 4 2 2 2 x x (3, 3) 4 (3, 0) 21. 0 0 k xye x y dy dx 0 k xe k xe 0 x x y y 1 0 dx 22. 1 0 x 1 k xy dy dx 0 0 1 k xy 2 2 kx 3 dx 2 1 0 x dx 0 dx 1e x 0 kx Therefore, k 1 1 k 0 1. x ye x y kx 4 8 Since k 8 k 8 8. P 0 0 dy dx 0.070 1, we have k 1 1 2 2 23. True 24. False, 0 0 x dy dx 1 1 x dy dx 1 1 0 25. True 26. True, 0 1 1 x2 1 1 0 y 2 dx dy < 0 1 1 x2 dx dy 4 h x 4 h sec 27. 0 0 x2 y 2 dy dx 0 0 4 r2 dr d h3 3 h3 sec tan 6 4 sec3 d 0 ln sec tan 0 h3 6 1 1 z2 2 ln 2 1 4 16 0 y2 2 4 h 2 0 2 28. 0 x2 y2 dx dy 0 2 0 2 0 r3 r4 4 64 d 0 dr d 4 29. V 4 0 h r dr d dz d 0 2 0 0 h 1 z2 1 d dz 32 z 2 dz 0 13 z 3 h 0 h3 3 R eview Exercises for Chapter 14 2 R 351 30. V 8 0 b 2 R2 r 2r dr d R 31. (a) x2 r2 2 y2 2 9 x2 y2 r 2 sin2 sin2 9 cos 2 9 r 2 cos2 9 cos2 3 cos 2 4 8 3 82 R 3 4 3 R2 0 r2 2 32 b d r2 r b2 32 0 d −6 R2 b2 32 6 −4 4 3 0 cos 2 (b) A 4 0 4 3 0 cos 2 r dr d 9 r 2 r dr d (c) V 4 0 9 20.392 32. tan 12 13 8 13 arctan 3 2 0 4 3 ⇒ 2 0.9828 ≤ 0.9828. Hence, The polar region is given by 0 ≤ r ≤ 4 and 0 ≤ r cos 0 y r sin r dr d 288 . 13 (8/ 13, 12/ 13) 4 3 2 1 x 2 + y 2 =16 y=2 x 3 θ x 1 8/ 13 4 1 2x 1 2x 33. (a) m Mx My x y y k 0 1 2x 3 xy dy dx 2x k 4 16k 55 8k 45 (b) m Mx My x y k 0 1 2x 3 x2 2x y2 dy dx y2 dy dx y2 dy dx 17k 30 392k 585 156k 385 k 0 1 2x 3 xy2 2x dy dx dy dx k 0 1 2x 3 y x2 2x k 0 2x 3 x2y 32 45 64 55 k 0 2x 3 x x2 936 1309 784 663 My m Mx m y = 2x My m Mx m 2 1 y = 2x 3 x 1 2 352 Chapter 14 L h2 2 0 h2 2 0 L Multiple Integration xL x2 L2 34. m Mx k 0 L xL x2 L2 dy dx y dy dx 0 2 kh 2 L 2 0 x L x2 dx L2 7khL 12 h y 2 x y = h 2 − L − x2 2 L k kh 8 2 0 L ( ( x x L 4x L 2x 2 L xL x2 L2 2 dx L kh2 8 4 0 3x2 L2 x3 L2 x2 L2 2x3 L3 x4 2L3 x dy dx x4 dx L4 x5 5L4 L 0 kh2 4x 8 L kh2 8 17L 10 17kh2L 80 h2 2 0 L My k 0 kh 2 x y My m Mx m 2x 0 x2 L x3 dx L2 5L 14 51h 140 b kh 2 x 2 x3 3L x4 4L2 L 0 kh 2 5L2 12 5khL2 24 5khL2 24 17kh2L 80 12 7khL 12 7khL a 35. Ix R y2 x, y dA 0 a 0 b kxy2 dy dx 1 32 kb a 6 1 kba4 4 3a2 2 4 0 4 0 x 2 36. Ix R y2 x2 R x, y dA 0 2 x 2 ky3 dy dx kx2y dy dx 0 16,384 k 315 512 k 105 512 k 9 Iy R x2 x, y dA 0 0 kx3 dy dx 1 kba 4 4 a b Iy I0 m R x, y dA 16,384k 315 2 I0 m Ix Iy 1 32 kb a 6 ka2b 2 2b 12 1 kba2 2 a2 2 b2 3 Ix Iy 512k 105 4 0 x 2 17,920 k 315 x, y dA 0 ky dy dx 512k 105 128k 15 16,384k 315 128k 15 4 7 x, y dA R 0 0 kx dy dx 1 4 kba 4 1 2 kba2 1 6 kb3a2 1 2 kba2 2 128 k 15 x Iy m Ix m x a2 2 y b3 3 Iy m Ix m 128 21 y 37. S R 4 1 16 0 2 4 x2 fx fy 2 dA 4 0 1 4x2 4y2 dy dx 4 0 0 1 13 65 3 4r 2 r dr d 2 2 1 0 6 65 65 1 R eview Exercises for Chapter 14 38. f x, y R fx 1 S 0 y 353 16 x y2 x, y : 0 ≤ x ≤ 2, 0 ≤ y ≤ x 1, fy fx 2 2 2 2y fy 2 2 2 4y2 dx dy 4y 2 2 22 0 4y2 y2 1 2 12 4y2 dy 2 1 2y 2 2 1 4 18 2 62 ln 4 4y2 2 ln 4 2 ln 2y 18 92 2 2 4y2 4y2 ln 2 32 0 1 18 18 12 ln 2 2 6 22 12 ln 2 2 3 32 52 3 39. f x, y fx S R 3 0 3 9 y2 2y 1 fx2 1 fy2 dA 0, fy y 4y2 dx dy y y 1 0 3 4y2 x y dy 12 1 43 3 2y 1 0 4y2 dy 4y2 32 0 1 37 6 32 1 40. (a) Graph of f x, y z 25 1 over region R z 50 (b) Surface area R 1 fx x, y 2 fy x, y 2 dA e x2 y 2 1000 cos2 x2 y 2 1000 Using a symbolic computer program, you obtain surface area 4540 sq. ft. R 50 x 50 y 3 9 9 x2 9 2 3 0 3 9 41. 3 x2 x2 y2 x2 y 2 dz dy dx 0 2 0 r2 r 2 dz dr d 2 2 9r 2 0 r 4 dr d 0 3r 3 r5 5 3 d 0 162 5 d 0 324 5 354 2 Chapter 14 4 4 x2 x2 (x2 0 Multiple Integration y2) 2 2 2 0 r2 2 42. 2 x2 y 2 dz dy dx 0 0 r3 dz dr d 1 2 2 0 2 r 5 dr d 0 16 3 2 d 0 32 3 a b 0 c a b 0 43. 0 0 x2 y2 z2 dx dy dz 0 a 0 13 c 3 cy 2 13 bc 3 cz2 dy dz 1 abc 3 3 5 0 2 13 bc 3 bcz 2 dz 13 ab c 3 13 a bc 3 1 abc a 2 3 b2 c2 5 25 0 x2 0 25 x2 y2 44. 0 1 1 x2 y2 z2 2 2 0 dz dy dx 0 1 2 sin ddd 2 0 2 0 2 5 arctan 0 sin dd 2 5 0 1 1 1 x2 x 2 arctan 5 r2 r2 cos 0 d 8 15 2 5 arctan 5 1 1 x2 y2 2 1 0 1 1 45. 1 x x2 2 y2 dz dy dx 0 r 3 dz dr d y 2 2 4 0 x2 0 4 x2 y2 46. 0 xyz dz dy dx 4 3 2 2 cos 2 2 cos 0 2 0 4 r2 47. V 4 0 r dz dr d 4 4 3 cos 2 cos 4 0 0 2 r4 1 0 r 2 dr d d r2 32 0 d 2 0 0 32 3 sin3 32 3 2 1 cos3 3 16 0 r2 32 32 2 2 3 2 sin 2 sin 0 48. V 2 0 2 r dz dr d 2 0 2 0 r 16 r 2 dr d 2 0 32 sin2 2 sin 2 4 sin4 13 sin 4 d cos 8 0 8 sin2 31 42 sin4 d 2 0 84 1 sin 2 4 29 2 2 2 0 cos 2 49. m 4k 4 0 2 sin ddd dd sin 2 k 3 ddd 1 k 2 2 2 4 k 3 Mxy 4k 2 4 2 4 2 0 2 0 cos3 2 cos sin 3 cos3 4 sin d 2 k 3 1 cos4 4 2 4 k 24 cos 0 k 4 0 cos5 k k 96 24 sin 1 4 dd cos5 4 sin d 1 k cos6 12 2 4 k 96 z x Mxy m y 0 by symmetry R eview Exercises for Chapter 14 2 a 0 2 a 0 2 a 0 cr sin 2 a 2 355 50. m Mxz Mxy x y z 2k 0 0 cr sin r dz dr d 2kc 0 0 r 2 sin dr d 2 a 2 kca3 3 dr d sin d 0 2 kca3 3 sin2 d 1 kca4 8 1 k c2a4 16 2k 0 0 cr sin r 2 sin dz dr d rz dz dr d 0 0 2kc 0 2 a 0 r3 sin2 r 3 sin2 dr d 1 kca4 2 2 2 0 2k 0 Mxz m Mxy m 2 kc2 0 0 1 24 kc a 4 sin2 0 d kca4 8 2kca3 3 kc2a4 16 2kca3 3 2 0 2 2 0 a 2 3a 16 3 ca 32 k a3 6 ddd 3a 8 500 3 3 3 0 2 51. m Mxy x k 0 0 a sin cos ddd 2 k 0 0 sin k a4 16 y z Mxy m 3 0 2 0 k a4 6 16 k a3 25 r2 52. m 500 3 500 3 2 0 2 3 0 3 0 r dz d dr 1 25 3 r 25 0 r2 4r d dr 125 3 500 3 14 3 r2 32 2r 2 0 500 3 2 64 3 18 162 x Mxy y by symmetry 4 2 5 3 25 25 r2 2 3 zr dz dr d 0 2 0 25 r2 0 r2 zr dz dr d 0 3 0 2 0 2 0 8 81 8 1 25 2 81 4 r2 r dr d 0 13 r 2 9 r dr d 2 1 8 14 r 8 92 r 4 d 0 z Mxy m 2 4 3 2 4 81 1 4 162 16 0 r2 2 a 2 53. Iz 4k 0 r3 dz dr d 16r 3 0 3 54. Iz 833 k 3 z k 0 0 0 sin2 2 sin ddd 4k r 5 dr d 4k a6 9 55. z f x, y a2 a2 x2 r2 h2 y2 a−h a 0≤r≤ h 2ah x a a y —CONTINUED— 356 Chapter 14 Multiple Integration 55. —CONTINUED— (a) Disc Method a V a h a2 a2y a3 y3 3 a3 3 y 2 dy a y a3 a h a3 3 a3 3 a2h a2 a ah2 h h3 3 a 3 h 3 2a y= a a2 − x2 a3 a2h ah2 h3 3 12 h 3a 3 h a−h −a x a Equivalently, use spherical coordinates. 2 cos 0 cos 0 1 1 a ha a 2 V 0 2 a a ha a h sec sin ddd (b) Mxy 0 a h sec cos 12 h 4 Mxy V 2a h 2 2 sin ddd z 12 h 2a 4 12 h 3a 3 0, 0, 3 2a 4 3a h h h2 h 3 a. 8 2 3 2a 4 3a h2 h Centroid: (c) If h a, z 3a2 4 2a Centroid of hemisphere: (d) lim z h →0 3 0, 0, a 8 3 4a2 12a a 3 2a h → 0 4 3a lim 2 cos 0 h2 h (e) x2 Iz y2 2 0 sin2 1 a ha a 2 a h sec sin2 3a2 2 sin ddd h3 20a2 30 15ah 3h2 (f) If h a, Iz a3 20a2 30 15a2 45 a. 15 2 6 sin 2 0 0 0 56. x2 Iz y2 z2 a2 x2 Q 1 y2 dV z2 z2 a2 a2 1 1 y 2 z2 y 2 z2 a2 57. sin ddd a a 1 1 x2 a2 y 2 dx dy dz 6 sin represents (in the yz-plane) a circle of Since radius 3 centered at 0, 3, 0 , the integral represents the volume of the torus formed by revolving 0 < < 2 this circle about the z-axis. 8 a 15 2 1 0 r2 58. 0 0 r dz dr d Since z 1 r 2 represents a paraboloid with vertex 0, 0, 1 , this integral represents the volume of the solid below the 4 x2 in the xy-plane. paraboloid and above the semi-circle y P roblem Solving for Chapter 14 x, y u, v 1 x, y u, v x 1 u 2 xy uv 3 xy uv v ,y yx uv 23 xy vu 1 u 2 9 1 2 1 2 11 22 x y, v 1 2 x y x, y u, v xy uv 2u 2v y 357 59. 60. yx uv 2u 2v 8uv 61. 3 y=x+1 y = −x + 5 v ⇒u 2 Boundaries in xy-plane x x x x y y y y ln x R Boundaries in uv-plane u u 3 5 1 1 v 3 ln u 1 u 1 u 2 3 v 1 dv du 2 5 ln 5 3 ln 3 5 3 1 1 y = −x + 3 y=x−1 x 3 5 1 1 5 1 1 2 3 v v y dA 3 1 ln 5 ln 5 1 u 2 5 1 ln u dv du 12 2.751 y 5 5 ln u du 3 u ln u u 3 2 62. x x, y u, v u, y xy uv v ⇒u u xy vu x, v 1 1 u xy 0 5 4 x=1 3 2 y= 1 x x=5 Boundary in xy-plane x x xy xy 1 1 5 1 5 x dA x2y2 5 1 5 1 Boundary in uv-plane u u v v 1 1 5 1 5 2 1 x 1 y= 1 x 4 5 R u u2 v u 5 1 1 du dv u 5 1 5 1 1 1 v2 5 du dv 1 4 1 v2 dv 4 arctan v 4 arctan 5 Problem Solving for Chapter 14 z 1. (a) V 16 R 4 1 1 x2 dA 1 16 0 0 4 0 1 1 cos2 r 2 cos2 r dr d 1 cos cos2 tan 0 32 16 3 1d 1 1 y 16 sec 3 82 2 4 x R y=x 4.6863 (b) Programs will vary. 358 Chapter 14 1 d c Multiple Integration 2. z fx 1 S ax by b c Plane a ,f cy fx2 fy2 1 R 1 a2 c2 a2 c2 b2 c2 a2 b2 c c2 R b2 dA c2 dA a2 b2 c c2 AR 3. (a) du a2 Then u2 2 22 1 u arctan a a 1 u2 2 2 u2 v2 dv c. Let a2 1 2 v 2 u 2 u 2 u2 u2 u2 u2 2 u2, u v 2 v. C. arctan u u2 (b) I1 0 22 0 22 0 arctan du u 2 2 4 2 u2 u2 arctan arctan u 2 u2 du arctan du u2 2 2 sin2 2 cos2 . Let u I1 4 0 2 sin , du 6 2 cos d , 2 2 sin 2 cos 42 2 v 2 u 2 2 2 u2 u 6 1 2 cos arctan tan arctan 2 cos d 2 2 6 4 0 2 d 2 0 u 6 2 18 du (c) I2 22 2 22 2 22 2 2 2 2 4 2 arctan u2 arctan 2 2 u2 u du u2 u2 arctan u 2 2 u2 du u2 arctan Let u I2 4 6 2 sin . 2 1 2 cos arctan 1 arctan sin cos 1 1 2 2 sin 2 cos 2 cos d 2 4 6 d cos cos 2 2 2 (d) tan 1 22 1 1 1 sin sin sin cos2 2 1 1 sin sin 1 1 sin sin cos —CONTINUED— P roblem Solving for Chapter 14 3. —CONTINUED— 2 359 (e) I2 4 6 arctan 2 1 sin cos d 2 d 2 4 6 arctan tan 1 22 d 4 6 1 22 2 2 2 6 2 2 2 d 2 2 2 2 18 2 96 72 1 xy 2 6 4 2 8 4 36 ... 2 12 2 72 2 1 1 1 0 1 9 xy < 1 . . . dx dy 1 (f ) xy 1 0 xy 1 2 1 1 xy 1 dx dy 0 1 0 0 1 1 xy xy 2 xy K dx dy 0K 1 K 0 0 0 K 0 0 xK K 1 2 0 1y K 1 1 dy 0 yK K 1 1 dy K 0 yK 1 K1 K 0 K 1 2 n 1 n2 1 y (g) u u u x y ,v 2 y x 2 u 2 u 2 1 1 2 2 1 v 2 1 v v x, y u, v 2x ⇒x 2 2y ⇒y 2 1 1 S 0, 0 1 , 2 2 2 v R v x R 0, 0 ↔ 1, 0 ↔ 0, 1 ↔ 1, 1 ↔ 1 0 1 0 ( 1 , 2 1 2 ) ( 2, 0) u 3 4 S 1 2 1 −1 −2 2 1 1 , 2 2 2, 0 22 u u ( 1 −1 , 2 2 ) 1 1 xy dx dy 0 1 2 1 u2 2 2 2 u 2 2 v 2 9 2 dv du 22 u 1 1 u2 2 v2 2 dv du 2 I1 I2 18 6 360 Chapter 14 2 5 4 2 Multiple Integration r2 r dr d 160 r2 r dr d 160 1333 960 523 960 4. A: 0 r 16 10 4.36 ft3 1.71 ft3 5. Boundary in xy-plane y y y y x 2x 12 x 3 12 x 4 x, y u, v 1 3 2 3 v u v u 23 Boundary in uv-plane u u v v 2 3 1 3 u v u v 13 B 0 9 r 16 1 2 3 4 The distribution is not uniform. Less water in region of greater area. In one hour, the entire lawn receives 2 0 10 0 r 16 r2 160 r dr d 125 12 32.72 ft3. 13 23 1 3 1 3 A R 2 2 0 2 4 0 2 2 2 0 2 sec 8 r2 1 dA S 1 x, y dA u, v 6. (a) V 0 2 r dz dr d sin 8 42 3 ddd 5 5 (b) V 8 42 3 z 3 2x 0 6 x x 7. V 0 dy dz dx 18 (3, 3, 6) 6 5 4 (0, 0, 0) 3 x 2 (3, 3, 0) (0, 6, 0) 6 y 1 1 1 8. 0 0 x n y n dx dy 0 1 0 xn 1 n y n1 1 1 1 dy 0 9. From Exercise 55, Section 14.3, e Thus, 0 x2 x2 2 n y n dy 1 2 0 dx dx 2 2 and 2 1 xe 2 x2 0 yn 1 n1 1 n 1 n→ 1 e x2 2 e 0 x2 dx x2 2 x2e 1 2 0 dx 1 2 e 0 dx lim 0 0 xn yn dx dy n→ lim 1 n 1 2 0 1 22 4 10. Let v ev 1 ln 1 ,x x 1 , dv x e v, dx 0 dx . x e v v, 2u du ue 0 v dv v ln 1 x dx 0 e dv 0 ve v dv Let u 1 v, u2 dv. u2 ln 1 x dx 0 2u du 2 0 u2e u2 du 2 4 2 (See Problem Solving #9.) P roblem Solving for Chapter 14 ke 0 x ya 361 11. f x, y x ≥ 0, y ≥ 0 elsewhere ke 0 0 xa x ya 12. Essay f x, y dA dx dy k 0 e dx 0 e ya dy These two integrals are equal to b e 0 xa dx b→ lim ae xa 0 a. Hence, assuming a, k > 0, you obtain 1 ka2 or a 1 k . 13. A l w ∆x cos θ x cos P y sec xy 14. The greater the angle between the given plane and the xyplane, the greater the surface area. Hence: z2 < z1 < z4 < z3 ∆y θ ∆x ∆y Area in xy-plane: xy 1 1 0 1 0 15. Converting to polar coordinates, 1 0 0 2 16. 0 x x x x y dx dy y3 y dy dx y3 1 2 1 2 1 x2 y2 2 dx dy 0 0 1 1 r2 2 2 r d dr 1 0 r 0 1 t r2 4 1 2 dr 2 t→ lim 0 r2 1 r2 t 0 2r dr The results are not the same. Fubini’s Theorem is not valid because f is not continuous on the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. t→ lim 4 1 4 17. The volume of this spherical block can be determined as follows. One side is length .Finally, the third side is given by the length of an arc of angle Another side is circle of radius sin . Thus: V 2 . in a z ρi sin φi ∆ θi ∆ ρi ρi ∆φi sin sin x y ...
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