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# ODD13 - C H A P T E R 13 Multiple Integration Section 13.1...

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Unformatted text preview: C H A P T E R 13 Multiple Integration Section 13.1 Iterated Integrals and Area in the Plane . . . . . . . . . . . . . 133 Section 13.2 Double Integrals and Volume . . . . . . . . . . . . . . . . . . . 137 Section 13.3 Change of Variables: Polar Coordinates . . . . . . . . . . . . . 143 Section 13.4 Center of Mass and Moments of Inertia . . . . . . . . . . . . . 146 Section 13.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Section 13.6 Triple Integrals and Applications . . . . . . . . . . . . . . . . . 157 Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates . . . . 162 Section 13.8 Change of Variables: Jacobians . . . . . . . . . . . . . . . . . . 166 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 C H A P T E R 13 Multiple Integration Section 13.1 Iterated Integrals and Area in the Plane Solutions to Odd-Numbered Exercises x 1. 0 2x y dy 2xy 12 y 2 x 0 32 x 2 2y 3. 1 y dx x 2y y ln x 1 y ln 2y 0 y ln 2y 4 x2 5. 0 x 2y dy 122 xy 2 4 0 x2 4x 2 2 x4 y 7. e y y ln x dx x 1 y ln2 x 2 y ey 1 y ln2y 2 x3 x3 ln2ey y ln y 2 2 y2 x3 x3 9. 0 ye u 1 yx dy dy, dv xye e yx 0 x 0 e xe yx dy x4 e x2 x 2e yx 0 x2 1 e x2 x 2e x2 y, du 2 yx dy, v 12 y 2 yx 1 2 1 1 11. 0 0 x y dy dx 0 xy dx 0 0 2x 2 dx x2 2x 0 3 1 x 1 x 1 13. 0 0 1 x2 dy dx 0 y1 x2 0 dx 0 x1 x2 dx 12 1 23 1 x2 32 0 1 3 2 4 2 15. 1 0 x2 2y 2 1 dx dy 1 2 1 13 x 3 64 3 4 2xy 2 8y 2 x 0 dy 4 3 2 4 dy 19 1 6y 2 dy 4 19y 3 2y 3 2 1 20 3 1 1 0 y2 1 17. 0 x y dx dy 0 1 0 12 x 2 1 1 2 xy 0 1 y2 dy 1 y 2 13 y 6 12 1 23 1 y2 y1 y 2 dy y2 32 0 2 3 2 4 0 y2 19. 0 2 4 y 2 dx dy 2 0 2x 4 y2 sin 4 0 y2 2 2 dy 0 2 dy 2y 0 4 2 sin 2 21. 0 0 r dr d 0 r2 2 2 2 d 0 0 1 sin2 2 1 4 2 d 1 cos 2 4 2 2 1 4 cos 2 0 d 2 2 sin 2 0 32 1 8 133 134 Chapter 13 1x Multiple Integration 1x 23. 1 0 y dy dx 1 y2 2 dx 0 1 2 1 1 dx x2 1 2x 0 1 1 2 1 2 25. 1 1 1 dx dy xy 1 1 ln x y dy 1 1 1 y 1 0 y dy Diverges y 8 3 8 3 8 8 27. 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 2 4 0 x2 2 4 x2 y 29. A 0 2 dy dx 0 y 0 dx 4 3 y = 4 − x2 4 0 x2 x3 3 4 y 2 0 dx 16 3 dx dy 2 1 x 1 2 3 4x 4 −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 4 2 0 x y 32 4 0 2 8 3 4 16 3 2 x 2 2 1 4 x2 31. A 2x 1 2 4 dy dx x2 y = 4 − x2 y 33. 0 3 dy dx 0 4 y 0 dx (1, 3) y 2 1 x 2 dx 4 2 4x 8 xx 3 x dx x2 2 4 0 y=x+2 1 0 4 2 1 x 2 x 2 dx 4x −2 −1 x 1 2 8 3 4 2 0 y 2 2 2 x 12 x 2 x 2 dx 13 x 3 dx dy 1 2 4 dx dy 0 8 3 2x 3 y 9 2 4 0 4 y Integration steps are similar to those above. y 4 2 A 0 3 4 y y 2 2 3 4 dx dy y 3 2 y = (2 − x )2 x 0 3 4 y dy 2 3 x 0 4 dy 1 x 1 2 3 4 y 0 2 2y 4 2 4 3 y dy 3 2 3 4 4 4 3 y dy 4 12 y 2 y 32 0 y 32 3 9 2 Section 13.1 3 2x 3 5 5 0 5 x x Iterated Integrals and Area in the Plane a 0 0 a ba a2 x2 a ba a2 x2 135 35. A 0 3 0 dy dx 3 2x 3 5 dy dx y 3 5 0 A 37. 4 dy dx 0 2 y 0 dx d y 0 3 0 0 dx 5 3 dx x b a a2 0 x 2 dx ab 0 cos2 2x dx 3 3 a sin , dx ab 2 2 a cos d cos 2 d ab 2 1 sin 2 2 2 0 x dx 12 x 2 5 1 0 12 x 3 2 5 5x 0 y 5 3 ab 4 Therefore, A A 4 b 0 0 ab A 0 2 3y 2 5 dx dy y ab. b2 y2 x 0 2 3y 2 dy 3y dy 2 5y 52 y 4 2 dx dy ab 4 5 0 2 y Therefore, A above. y ab. Integration steps are similar to those 5 2 y 5y dy 2 5 0 b y= b a a2 − x2 a x 4 3 2 y= 2x 3 y=5−x 1 x 1 2 3 4 5 −1 4 y 2 4 x2 39. 0 0 f x, y dx dy, 0 ≤ x ≤ y, 0 ≤ y ≤ 4 4 0 y 41. 20 2 f x, y dy dx, 0 ≤ y ≤ 4 4 y 3 4 x2, 2≤x≤2 4 y2 f x, y dy dx x 0 y2 dx dy 3 2 1 x 1 2 3 4 −2 −1 −1 1 x 1 2 10 ln y 1 1 43. 1 0 f x, y dx dy, 0 ≤ x ≤ ln y, 1 ≤ y ≤ 10 ln 10 0 y 45. 1 x2 f x, y dy dx, x 2 ≤ y ≤ 1, 1 ≤ x ≤ 1 1 y 10 f x, y dy dx e x f x, y dx dy 0 y 4 y 8 3 6 4 2 x 1 2 3 −2 −1 x 1 2 2 136 1 Chapter 13 2 2 1 Multiple Integration 1 1 1 y2 1 1 0 x2 47. 0 y dy dx 0 0 0 dx dy 2 49. 0 y2 y dx dy 1 dy dx 2 3 1 2 −1 x 1 x 1 1 2 3 2 x 4 4 0 x 2 4 y y 2 1 1 2y 51. 0 y dy dx 0 2 dy dx 0 dx dy 4 53. 0 y dy dx x2 0 0 dx dy 1 3 2 2 1 x 1 2 3 4 1 −1 x 1 2 1 3 y 1 x 55. 0 y2 dx dy 0 x3 dy dx 5 12 2 x= 3 y y x = y2 1 (1, 1) x 1 2 57. The first integral arises using vertical representative rectangles. The second two integrals arise using horizontal representative rectangles. 5 0 x 50 x2 5 x 2y 2 dy dx 0 12 x 50 3 x2 32 15 x dx 3 15625 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 32 y2 dy 15625 18 15625 18 15625 18 15625 24 y y= 50 − x 2 (0, 5 2 ) 5 (5, 5) y=x x 5 Section 13.2 2 2 2 y 2 Double Integrals and Volume 137 59. 0 x x1 y3 dy dx 0 0 2 x1 1 2 y3 dx dy 0 1 1 2 1 3 2 1 3 x y3 x2 2 32 y dy 0 2 1 0 y3 y 2 dy y3 0 1 27 9 1 1 9 26 9 1 1 1 x 1 61. 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 4 x sin x 2 dx 0 1 cos x 2 2 1 1 2 y 0 1 1 2 cos 1 0.2298 2 2x 63. 0 x 2 x3 3y 2 dy dx 1664 105 15.848 65. 0 x 2 1y 1 dx dy ln 5 2 2.590 67. (a) x x 8 y3 ⇔ y x1 3 y 4 2y ⇔ x2 x 13 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 67520 693 97.43 2 4 0 x2 2 1 0 cos 69. 0 exy dy dx 20.5648 71. 0 6r 2 cos dr d 15 2 73. 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. 75. The region is a rectangle. 77. True Section 13.2 For Exercise 1–3, 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 0 138 Chapter 13 x2 y2 xi yi x2 0 0 Multiple Integration 3. f x, y 8 f xi, yi i 1 4 2 2 4 10 4 4 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 0 160 3 4 4 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 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 6 3 6 x2 0 x 1 2 3 9. 0 y2 x y dx dy 0 6 0 12 x 2 9 2 3y 32 y 2 3 xy y2 dy 6 y (3, 6) 52 y dy 8 53 y 24 6 0 4 9 y 2 36 a a2 a2 x2 2 x 2 4 6 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 5 3 3 5 x2 32 a 0 −a 13. 0 0 xy dx dy 0 3 0 0 x y dy dx 5 y 12 xy 2 3 5 dx 0 4 3 2 25 2 x dx 0 3 0 1 x 1 2 3 4 5 25 2 x 4 225 4 Section 13.2 2 y y2 Double Integrals and Volume y 139 15. 0 y x2 y 4 2 dx dy 2 2 y2 y x2 y 2 2 dx dy 0 2x x 2 y x2 ln x 2 y 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 0 5x 2 2 ln 2x 2 dx 15 ln 22 dx 0 2 5 1 ln x 2 2 ln 0 5 2 y 4 4 4 y y 1 4 4 1 x2 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 1 ln x 4 0 dx 1 3 4 x 26 25 4 3x 4 5 25 0 x2 3 25 4y 3 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 4 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 4 2 0 13 y 3 3 25 0 21. 0 y dy dx 2 4 0 4 y2 4 2 y dx 0 4 3 dx 0 4 2 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 xy 0 dy 2 y y2 dy y3 3 4 2 0 1 2 1 2y2 8 8 6 y=x x 8 3 140 6 Chapter 13 2 3x 0 4 Multiple Integration 6 2 3x 0 4 y 25. 0 12 2x 4 3y dy dx 0 6 0 1 12y 4 12 x 6 2x x2 2xy 32 y 2 dx 5 4 y = − 2x + 4 3 6 dx 6 3 2 1 x 1 2 3 4 5 6 13 x 18 12 1 y 1 6x 0 −1 y 27. 0 0 1 xy dx dy 0 1 x x 2y 2 y dy 0 1 y 0 y3 dy 2 y4 8 1 0 y=x x 1 y2 2 3 8 1 0 0 29. x 4 0 x2 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 2 31. 4 0 4 x2 y 2 dy dx 8 1 x 2 4 33. V 0 1 0 0 xy dy dx 12 xy 2 1 0 x 35. V 0 0 x 2 dy dx 2 4 2 dx 0 1 2 1 x3 dx 0 0 x 2y 0 2 0 dx 0 4x 2 dx 14 x 8 y 1 8 4x3 3 y 32 3 4 1 y=x 3 2 1 x 1 2 3 x 1 −1 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 Section 13.2 2 4 0 x2 2 4 0 x2 Double Integrals and Volume 141 39. V 0 2 x 12 y 2 x2 y dy dx 4 0 x2 41. V 4 0 2 x2 x2 4 0 2 y 2 dy dx x2 1 4 3 32 cos4 3 x2 32 xy 0 2 dx 12 x dx 2 4 4 0 dx, x 2 sin x4 0 2 16 cos2 32 3 3 16 d 1 4 3 y x2 32 2x 13 x 6 2 0 16 3 4 16 8 y 4 2 y= 4 − x2 1 x 2 + y2 = 4 1 −1 x 1 −1 x 1 2 2 4 0 x2 2 0.5x 0 1 43. V 4 0 4 x2 y 2 dy dx 8 45. V 0 1 2 x2 y2 dy dx 1.2315 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 ≤ 1 12 12 2x 1 arccos y 49. 0 y2 e x 2 dx dy 0 12 0 e x 2 dy dx 51. 0 0 sin x 1 2 cos x sin2 x dx dy 2xe 0 x 2 dx 0 12 0 2 sin x 1 sin2 x dy dx e e 1 y x2 0 14 0 1 1 0.221 y sin2 x 12 sin x cos x dx 2 e 14 1 2 2 1 3 sin2 x 32 0 1 22 3 1 y = 2x 1 2 y = cos x 1 2 1 x 1 2 1 π 2 π x 142 Chapter 13 1 8 4 0 2 Multiple Integration 1 8 4 53. Average x dy dx 0 2x dx 0 x2 8 4 2 0 55. Average 1 4 1 4 2 0 2 0 2 x2 0 y 2 dx dy 2 x3 3 xy 2 0 dy 2 0 1 4 8 3 2 0 8 3 2y 2 dy 18 y 43 23 y 3 57. See the definition on page 946. 59. The value of R f x, y dA would be kB. 61. Average 1 1250 1 1250 325 300 325 250 100x 0.6y 0.4 dx dy 200 100y 0.4 300 x1.6 1.6 250 dy 200 128,844.1 1250 325 y 0.4 dy 300 103.0753 y1.4 1.4 325 25,645.24 300 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 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 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 2500m3. (15, 5, 6) (25, 5, 4) 7 z (15, 15, 7) (5, 5, 3) (5, 15, 2) 20 y 30 x (25, 15, 3) 1 2 6 2 69. 0 0 sin (a) 1.78435 (b) 1.7879 x y dy dx m 4, n 8 71. 4 0 y cos (a) 11.0571 (b) 11.0414 x dx dy m 4, n 8 S ection 13.3 Change of Variables: Polar Coordinates 143 73. V 125 (4, 0, 16) 16 z 75. False (4, 4, 16) 1 1 0 y2 Matches d. V 8 0 1 x2 y 2 dx dy (4, 0, 0) 5 x (0, 4, 0) 5 y (4, 4, 0) 1 1 x 1 1 t 77. Average 0 f x dx 0 1 0 t 1 e t dt dx 0 1 x 2 et dt dx 1 2 e t dx dt 0 1 0 0 2 te t dt 1 1 1 2 x 2 1 t2 e 2 1 e 2 e 1 Section 13.3 Change of Variables: Polar Coordinates 3. Polar coordinates ≤ 1. Rectangular coordinates 5. R r, : 0 ≤ r ≤ 8, 0 ≤ 7. R r, :0≤r≤3 3 sin , 0 ≤ ≤2 Cardioid 2 6 2 6 9. 0 0 3r 2 sin dr d 0 2 r 3 sin 0 d π 2 216 sin d 0 2 0 4 216 cos 0 0 2 3 2 11. 0 2 9 r 2 r dr d 0 1 9 3 2 0 3 r2 32 2 d π 2 55 3 55 6 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 π 2 sin 2 d 1 2 1 cos 2 sin 1 2 12 sin 8 2 0 0 1 2 1 8 3 32 2 sin 9 8 cos 2 144 a Chapter 13 a2 0 y2 Multiple Integration 2 a 15. 0 y dx dy 0 0 r 2 sin dr d a3 3 2 sin d 0 a3 3 2 cos 0 a3 3 3 9 0 x2 2 3 17. 0 x2 y2 32 dy dx 0 0 r 4 dr d 243 5 2 d 0 243 10 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 π 2 2 0 2 3 2 x 2 2 0 8 x2 4 2 0 2 21. 0 0 x2 y 2 dy dx 2 x2 y 2 dy dx 0 4 0 r 2 dr d 16 2 d 3 0 1 2 3 42 3 2 4 0 x2 2 2 2 2 23. 0 x y dy dx 0 0 2 r cos 8 3 r sin r dr d 0 0 cos 2 sin 16 3 π 2 r2 dr d cos 0 sin d 8 sin 3 cos 0 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 ( 1 , 2 1 2 ( ( 2, 2) r dr d 1 3 d 2 32 4 4 0 32 64 0 1 2 2 1 27. V 0 0 2 0 r cos 1 2 1 r sin r dr d 1 8 2 r3 sin 2 dr d 0 sin 2 d 0 1 cos 2 16 2 0 1 8 2 5 29. V 0 0 r 2 dr d 250 3 2 4 cos 2 31. V 2 0 0 2 16 r2 r 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 1 0 sin 1 cos2 d 128 3 cos cos3 3 4 Section 13.3 2 4 2 4 Change of Variables: Polar Coordinates 145 33. V 0 a 16 r 2 r dr d 0 1 3 16 3. r2 3 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 83 2 4 23 2 2.4332 23 2 2 4 35. Total Volume V 0 2 0 25e r2 4 r 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 e c2 4 50 e 0 c2 4 1 d ⇒ 30.84052 ⇒e c2 4 100 1 0.90183 0.10333 0.41331 0.6429 2c c2 4 c2 c ⇒ diameter 6 cos 1.2858 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 1 0 cos 39. 0 r dr d 1 2 1 2 2 1 0 2 2 cos cos2 1 d cos 2 2 3 1 0 2 cos d 1 2 2 sin 1 2 1 sin 2 2 2 0 3 2 3 2 sin 3 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 3 0 43. Let R be a region bounded by the graphs of r r g2 , and the lines a and b. g1 and 45. r-simple regions have fixed bounds for . -simple regions have fixed bounds for r. When using polar coordinates to evaluate a double integral over R, R can be partitioned into small polar sectors. 146 Chapter 13 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. 2 5 49. 4 0 r1 r 3 sin dr d 2 56.051 5 Note: This integral equals 4 sin d 0 r1 r3 dr 51. Volume base 8 height 12 300 16 z 53. False r 1 where R is the circular sector Let f r, 0 ≤ r ≤ 6 and 0 ≤ ≤ . Then, r R Answer (c) 1 dA > 0 but r 1 0 for all r. 6 x 4 46 y 2 2 2 55. (a) I 2 (b) Therefore, I 7 49 49 x2 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. 2 7 2 7 57. 7 4000e x2 0.01 x 2 y2 dy dx 0 0 4000e 2 0.01r 2 r dr d 0 200,000e 400,000 1 y 0.01r 2 0 d 486,788 200,000 e 0.49 1 e 0.49 4 y 59. (a) 2 2 y 3 f dx dy 3x 4 3 x 3x 4 4 5 y= 3x y=x (b) 2 3 3 2 4 csc f dy dx 2 f dy dx 4 3x f dy dx 3 ( ( 1 2 (4, 4) 4 ,4 3 ( x 1 (c) 4 2 csc f r dr d (2, 2) 2 ,2 3 3 ( 4 5 61. A r22 2 r12 2 r1 2 r2 r2 r1 rr Section 13.4 4 3 Center of Mass and Moments of Inertia 4 0 1. m 0 0 xy dy dx xy2 2 3 4 dx 0 0 9 x dx 2 9x2 4 4 36 0 2 2 2 2 3. m 0 0 r cos r sin r dr d 0 2 0 cos sin 4 cos sin d 0 r 3 dr d 4 sin2 2 2 2 0 Section 13.4 Center of Mass and Moments of Inertia a b 147 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 ka2 b2 kab x y x, y kab2 2 kab 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 k bh 2 b by symmetry 2 b2 2hx b x y x, y 2 a 3 b 2 7. (a) m x Mx y y= h 2hx b y=− 2 h (x − b ) b b 2h x bb ky dy dx 0 0 b2 0 ky dy dx x b kbh2 12 y x, y Mx m bh , 23 kbh2 12 kbh2 6 kbh 2 kbh2 6 h 3 —CONTINUED— 148 Chapter 13 Multiple Integration 7. —CONTINUED— b2 2hx b b 2h x bb (b) m 0 b2 0 2hx b ky dy dx b2 0 b 2h x 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 (c) m My m Mx m b2 0 kb2 2 h 12 kbh2 6 b 2 h 2 b 2h x bb kbh3 12 kbh2 6 2hx b kx dy dx 0 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 My m Mx m 11 3 kb h 96 7kb3h 48 kb2h 4 kh2b2 12 kb2h 4 7 b 12 h 3 a 2 a 2 2 9. (a) The x-coordinate changes by 5: x, y (b) The x-coordinate changes by 5: x, y a 5 b 5, b 2 11. (a) x m Mx 0 by symmetry a2k 2 a a2 x2 2b 5, 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 25 kb 2 25 2 kb 4 125 kb 3 yk dy dx a0 2a3k 3 4a 3 Mx My 5 0 y Mx m a 2a3k 3 a2 x2 2 a2k ka kx 2 dy dx My m Mx m 5b (b) m a0 a a2 x2 y y dy dx a4k 16 24 a5k 15 120 0 3 x y 2 a2 15a 75 3 a 10 b 2 Mx a0 a a2 x2 ka y y 2 dy dx 32 My a0 kx a My m Mx m 0 a 15 5 16 32 3 y y dy dx x y Section 13.4 Center of Mass and Moments of Inertia 149 4 x 13. m 0 4 0 kxy dy dx x 32k 3 256k 21 32k 3 8 7 15. x m 0 by symmetry 1 11 x2 k dy dx 10 1 11 x2 Mx 0 4 0 x kxy 2 dy dx k 2 k 2 8 2 4 My 0 0 kx My m Mx m 32k 1 256k 21 2y dy dx Mx 10 ky dy dx Mx m y x y y 3 3 32k 3 32k y k 2 8 2 k 2 y= 1 1 + x2 y= 2 x −1 x 1 1 x 1 2 3 4 −1 17. y m 0 by symmetry 4 16 y2 19. x 8192k 15 524,288k 105 64 7 m L by symmetry 2 L 0 sin x L kx dx dy 40 4 16 y2 ky dy dx 0 L sin x L kL 4 4kL 9 My 40 kx 2 dx dy My m 524,288k 105 15 8192k Mx 0 0 ky 2 dy dx Mx m 4kL 9 4 kL 16 9 x y 8 y y x = 16 − y 2 2 4 x 4 −4 −8 8 y = sin π x L 1 x L 2 L 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 2 150 Chapter 13 Multiple Integration 2 e 0 e 0 e 0 x 23. m 0 2 x ky dy dx ky 2 dy dx 0 2 x k 1 4 k 1 9 k1 4e4 ke 4 e e 4 25. 6 y m 0 by symmetry 6 2 cos 3 Mx My 0 k dA R 60 kr dr d k 3 kxy dy dx My m Mx m y 5e 8 1 4 My R kx dA 6 2 cos 3 x y k e4 5 8e4 k e6 1 9e6 e4 2 e4 4 e6 9 e6 5 1 1 e2 0.46 60 kr 2 cos dr d x π 2 1.17k 4e4 k e4 1 0.45 My m 1.17k 3 k 1.12 2 π θ= 6 r = 2 cos 3θ 1 y = e −x 0 1 x 1 2 π θ =−6 27. m Ix bh b 0 b h 29. m y 2 dy dx 0 h a2 2 a bh3 3 b3h 3 Ix R y 2 dA 0 2 0 a r3 sin2 dr d a4 4 a4 4 Iy 0 0 x 2dy dx Iy m Ix m a2 4 Iy R x 2 dA 0 0 r3 cos2 a4 4 a4 4 a4 4 a4 2 1 a2 dr d x y b3h 3 bh3 3 1 bh 1 bh b2 3 h2 3 b 3 h 3 3 b 3 3 h 3 I0 x Ix y Iy Ix m a 2 31. m Ix 33. 2 a ky a b y2 R dA 0 2 0 a r3 sin2 dr d a4 16 a4 16 m 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 Iy R x2 Ix y Iy dA 0 0 r3 a4 16 a4 16 a 16 4 cos2 a4 8 4 a2 dr d Ix Iy k 0 a 0 b y3 dy dx I0 x k 0 0 x 2y ydy dx Iy Iy m Ix m 3kab4 Ix m a 2 I0 x y Ix Section 13.4 35. m kx 2 4 0 4 0 4 0 x2 x2 x 2 Center of Mass and Moments of Inertia kxy 4 x 151 37. x dy dx 0 2 k 4k 32k 3 16k 3 m 0 4 0 x kxy dy dx 32k 3 16k 512k 5 Ix k 0 2 xy 2 dy dx Ix 0 4 0 x kxy3 dy dx Iy I0 x k 0 x3 dy dx Iy Iy m Ix m 16k 16k 3 4k 32k 3 4k Iy 0 0 kx3 y dy dx 592k 5 512k 5 16k 1 Ix I0 4 3 8 3 2 3 4 6 23 3 26 3 x Ix Iy Iy m Ix m 3 32k 3 32k 3 2 48 5 4 15 5 6 2 y y 39. m kx 1 0 1 x 2 x kx dy dx x 3k 20 3k 56 k 18 Ix 0 1 kxy 2 dy dx x 2 x Iy 0 x 2 kx3 dy dx 55k 504 k 18 3k 56 I0 x Ix Iy Iy m Ix m 20 3k 20 3k 30 9 70 14 y b b2 0 x2 b 41. I 2k b b x a 2 dy dx b 2k b x a 2 b2 x2 dx b 2k b x 2 b2 b4 8 x 2 dx a2b2 2 2a b x b2 4a2 x2 dx a2 b b2 x 2 dx 2k 0 k b2 2 b 4 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 4 2 0 42,752k 315 152 Chapter 13 Multiple Integration a a2 x2 a a2 0 a2 0 x2 x2 a 45. I 0 0 a ka k 4 k 4 k 4 yy a 2 dy dx 0 ka y dy dx 0 3 k a 4 y 4 0 a2 x2 dx a4 0 a 4a3y 6a2 y2 4ay3 y4 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 47. x, y ky. y will increase 49. x, y kxy. Both x and y will increase 51. Let 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 53. See the definition on page 968 55. y L ,A 2 b L bL, h L 2 2 L 2 dy dx 3L 57. y 2L .A 3 b2 bL ,h 2 L L 3 2L 3 3L 2 Iy 0 b 0 0 y y Iy hA Iy L3b 12 L 3 2 0 2Lx b b2 y 2L 3 dy dx L2 3 L 2 dx 0 2 3 2 3 y 0 b2 0 dx 2L x b ya y L3b 12 L 2 bL L 27 2Lx b 2L 3 2L 3 3 dx L3b 36 2 L3x 3 27 ya 2L 3 b 2Lx 8L b L3b 36 L2b 6 L 2 4 b2 0 S ection 13.5 Surface Area 153 Section 13.5 1. f x, y R fx 1 2 Surface Area 2y 3. f x, y R fx fy 2 2x 8 2x 2y y2 ≤ 4 triangle with vertices 0, 0 , 2, 0 , 0, 2 2, fy fx 2 0 2 x x, y : x 2 2, fy 1 fx 2 2 4 4 y 2 3 2 2 fy x2 2 3 2 2 S 0 3 dy dx 3 2x y 3 0 2 x dx S 2 x2 3 dy dx 0 0 3r dr d 12 x2 2 2 6 0 y = 4 − x2 R −1 1 2 x y = −x + 2 R 1 −1 1 y = − 4 − x2 x 1 2 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 7. f x, y R fx 1 3 4x 2 ln 2x 4x 2 0 3 6 37 4 ln 6 37 2 x3 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 9x 1 x 1 2 3 4 S 0 9x dy dx 3 4 dx 4 4 27 9. f x, y R fx 1 S 0 0 4 31 31 27 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 1 154 Chapter 13 x2 y2 Multiple Integration y 11. f x, y R 0≤ fx 1 1 x, y : 0 ≤ f x, y ≤ 1 x2 x x2 fx 2 1 x 2 + y2 = 1 y 2 ≤ 1, x 2 y2 , fy fy x2 2 y2 ≤ 1 x y x2 1 2 1 y2 x2 x 2 1 y2 y 2 x 2 y2 2 2 1 1 S 1 x2 2 dy dx 0 0 2 r dr d 13. f x, y R fx 1 b a2 x, y : x 2 a2 fx 2 x2 y2 a y y 2 ≤ b2, b < a y2 fy 2 x x2 , fy 1 a x2 a2 a2 y2 y x2 x2 x2 b x 2 + y 2 ≤ b2 y2 y2 2 a2 b 0 y2 x2 y2 a2 a x2 2 aa −b b −b a x y2 a2 b2 b 2 x 2 S b b2 x2 a2 dy dx 0 a r dr d a2 r 2 15. z 1 24 fx 8 3x 2 3 2x 0 2y fy 12 2 y 16 12 14 14 dy dx 48 14 S 0 8 4 x 4 8 12 16 17. z 1 25 fx 3 2 x2 fy 9 9 3 0 x 2 y2 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 x S 2 3 2 x2 −1 −2 1 2 25 y2 20 2 0 5 25 x2 r2 19. f x, y R 1 1 2y y triangle with vertices 0, 0 , 1, 0 , 1, 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 Section 13.5 x2 y2 x2 y2 Surface Area 155 21. f x, y R 0≤4 fx 1 2 4 23. f x, y R y2 ≤ 4 fx 1 1 4x 2 4x 2 4y 2 S 1 0 4 x, y : 0 ≤ f x, y x2 2x, fy fx 2 4 4 2 x 2 x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 2x, fy fx 1 2 y 2, x 2 2y fy 2 2y fy 1 2 1 4x 2 4y 2 1.8616 4x2 4y2 dy dx S 2 2 0 x2 1 4y 2 dy dx 17 17 6 1 0 1 0 y 4r 2 r dr d x 2 + y2 = 4 1 −1 −1 x 1 25. Surface area > 4 Matches (e) z 10 6 24. 27. f x, y R fx 1 1 ex x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 ex, fy fx 1 2 0 fy 1 2 1 e2x S 0 1 5 x 5 y e2x dy dx 0 1 0 e2x 2.0035 29. f x, y R fx S 1 x3 3xy y3 1, 1 , 3x 9 y2 3y 1, 2 31. f x, y 1 , 1, 3y 2 e e x x sin y e 1 1 x square with vertices 1, 1 , 3x 2 1 1 x fx 1 sin y, fy fy2 4 4 x2 cos y e e 2x 2x 3y 1 1 3x 2 y , fy y 2 fx2 2 sin2 y e 2x cos2 y 1 9 x2 x 2 dy dx S 2 x2 1 e 2x dy dx 33. f x, y R fx 1 4 exy x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ 10 yexy, fy fx 2 10 xexy fy 1 2 1 y 2e2xy y 2 dy dx x 2e2xy 1 e2xy x 2 y2 S 0 0 e2xy x 2 156 Chapter 13 Multiple Integration x 12 fy2 dA dy dx 1 35. See the definition on page 972. 37. f x, y S R 1 1 1 x 0 x 2; fx fx2 1 1 x x2 x 2 x2 , fy 0 16 0 1 16 0 1 dx 16 1 x2 12 0 16 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 ft3 (b) z 1 S 20 fx 1 100 1 100 xy 100 2 1 502 15 250x x3 30 50 0 fy 2 1 x2 y2 1002 x2 x2 1002 y 2 dy dx 1002 x2 100 y2 50 0 2 0 0 502 1002 50 1002 0 r 2 r dr d 2081.53 ft2 y 41. (a) V R f x, y 24 20 8 R 2 625 25 x2 y2 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 1 cm3 fx 2 r 2 r dr d 25 4 x 4 8 12 16 20 24 4 0 r2 32 4 d 8 0 3 812 (b) A R 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 13.6 Triple Integrals and Applications 157 Section 13.6 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 x 0 xy 1 x xy 3. 0 0 x dz dy dx 0 1 0 0 x xz 0 dy dx 1 x 2y dy dx 0 0 x 2y 2 2 x 1 dx 0 0 x4 dx 2 x5 10 1 0 1 10 4 1 0 x 4 1 x 4 1 5. 1 0 2ze x2 dy dx dz 1 4 0 2ze ze 1 x2 x2 y 0 dx dz 1 4 0 2zxe e 1 x2 dx dz 1 e 1 1 dz 0 1 z1 dz z2 2 4 1 15 1 2 1 e 4 2 0 1 0 x 4 0 4 0 2 1 x 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 2 4 4 x2 x 2 x2 2 4 4 x2 9. 0 0 x dz dy dx 0 x 2 x3 dy dx 128 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 ln4 0 cos y 0 dx 0 x 2 ln 4 1 cos 4 x 2 dx 2.44167 4 4 0 x 4 0 x y 3 9 9 x2 x2 9 0 x 2 y 2 13. 0 dz dy dx 15. 3 dz dy dx 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 a a2 0 x2 0 a2 x2 y2 a a2 0 x2 19. 8 0 dz dy dx 8 0 a 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 3 158 2 Chapter 13 4 0 x2 4 0 x2 Multiple Integration 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 23. Plane: 3x z 3 6y 4z 12 25. Top cylinder: y 2 Side plane: x z z2 y 1 1 2 4 x 1 3 y x 1 y 3 0 (12 0 4z) 3 (12 0 4z 3x) 6 dy dx dz 1 0 x 0 0 1 y2 dz dy dx 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 29. Q x, y, z : x2 4 y2 ≤ 9, 0 ≤ z ≤ 4 5 z 3 3 3 3 4 9 9 9 9 9 9 9 9 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 4 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 Section 13.6 6 4 0 (2x 3) 2 0 ( y 2) (x 3) Triple Integrals and Applications 159 31. m k 0 dz dy dx 8k 6 4 0 (2x 3) 2 0 ( y 2) (x 3) Myz k 0 x dz dy dx 12k x Myz m 12k 8k 3 2 4 4 0 4 4 0 x 4 4 b b 0 b 0 b 0 b 0 b 33. m k 0 x dz dy dx 4x 0 4 4 0 4 4 0 x k 0 0 x4 x dy dx 35. m k 0 b 0 b xy dz dy dx kb5 4 kb6 6 kb6 6 kb6 8 4k x 2 dx 128k 3 4 4 Myz k 0 0 k 0 0 b b x 2y dz dy dx Mxy k 0 xz dz dy dx x 128k 3 4 2 x 2 dy dx Mxz Mxy x y z k 0 b 0 b xy 2 dz dy dx 2k 0 16x 1 8x 2 x3 dx 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 z Mxy m Myz m Mxz m Mxy m 37. x will be greater than 2, whereas y and z will be unchanged. 39. y will be greater than 0, whereas x and z will be unchanged. 1 k r 2h 3 y 4k 0 0 r 0 r 0 h r2 x2 x2 y2 r 41. m x Mxy 0 by symmetry r r2 x2 h z dz dy dx 3kh2 r2 4kh2 3r 2 r2 x2 y 2 dy dx r2 0 x2 32 dx k r 2h2 4 z Mxy m k r 2h2 4 k r 2h 3 3h 4 160 Chapter 13 128k 3 y 4 4k 0 4 0 42 0 2 x2 2 4 Multiple Integration 43. m x z Mxy 0 by symmetry x2 42 x2 0 4 y2 42 x2 y2 z dz dy dx 13 y 3 42 0 x2 2k 0 42 x2 y 2 dy dx 2k 0 16y x 2y dx 4k 3 4 4 0 2 x2 32 dx 1024k 3 64 k z Mxy m cos4 0 d let x 4 sin by Wallis’s Formula 64k 1 3 128k 3 2 45. f x, y m k 5 y 12 20 0 20 0 (3 5)x 0 (3 5) 0 (3 5)x 0 12 x 12 (3 5)x 12 (5 12)y 20 y dz dy dx 0 (5 12)y 200k 16 12 8 y = − 3 x + 12 5 Myz Mxz Mxy x y z k 0 20 0 12 x dz dy dx (5 12)y 1000k 4 x 4 8 12 16 20 k 0 20 0 (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 a a 0 a a a a 47. (a) Ix k 0 0 y2 13 y 3 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 a Iz a 0 a 0 a 2ka5 by symmetry 3 y2 z2 xyz dx dy dz y2z3 2 a k 0 0 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 dz 0 ka4 8 a2z 0 2z3 dz ka8 8 Iz ka8 by symmetry 8 Section 13.6 4 4 0 4 0 x 4 4 Triple Integrals and Applications 161 49. (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 4 x 4 0 3 dy dx 1024k 3 8k Iz k 0 4 x3 y x2 dx 43 x 3 4 14 x 4 y 2 dz dy dx 4 k 0 0 4 x 2y 8x 2 0 y3 4 64 4 x dx x dx k 0 4 x 2y 2 2 32 8x y4 4 4 4x 2 x 0 dx k 8k 0 x3 dx 8k 32x 4x 2 43 x 3 14 x 4 4 0 2048k 3 L2 a a a2 a 2 x2 L2 a a 51. Ixy k L2 x 2 z2 dz dx dy L2 k L2 22 a 3 1 x 2x 2 8 x2 a2 x 2 dx dy x a a 2 3 2k 3 Since m k L2 L2 a2 x a2 2 a4 4 a4 16 x2 a2 arcsin a4 Lk 4 x a a2 x2 a2 a4 arcsin dy a 2 L2 dy a2Lk, Ixy ma2 4. —CONTINUED— 162 Chapter 13 Multiple Integration 51. —CONTINUED— L2 a a a2 a2 x2 L2 a Ixz k L2 L2 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 dy a a k a2 L2 y 2 dy 2k a2 L3 3 8 1 mL2 12 Iyz k L2 L2 x2 x 2 dz dx dy 1 x 2x 2 28 ma 4 2 2k L2 a x 2 a2 x a a x 2 dx dy ka4 4 L2 2k L a2 mL 12 2 a2 x2 a4 arcsin L2 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 ma2 4 mL2 12 ma2 4 ma2 4 L2 1 1 1 x 53. 1 10 x2 y2 x2 y2 z2 dz dy dx 55. See the definition, page 978. See Theorem 13.4, page 979. 57. (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. Section 13.7 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 2dz 8 2 2 cos2 0 4 0 r2 2 0 2 2 cos2 2 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 cos2 d 0 2 0 4 cos8 sin d 52 45 2 4 0 cos 2 5. 0 0 sin ddd 1 3 2 0 0 4 cos3 sin dd 1 12 2 4 cos4 0 0 d 8 4 z 0 0 2 7. 0 rer d dr dz e4 3 S ection 13.7 2 3 0 e 0 r 2 Triple Integrals in Cylindrical and Spherical Coordinates 163 2 3 9. 0 r dz dr d 0 2 0 2 0 0 re 1 e 2 1 1 2 e 9 r2 dr d 3 r2 0 1 3 z d 9 2 e d 2 3 x 1 3 y 4 2 2 6 4 2 0 0 1 11. sin ddd 64 3 64 3 2 0 2 2 sin 6 dd 4 z 2 cos 0 2 6 d x 4 4 y 32 3 3 64 3 3 2 2 0 4 d 0 13. (a) 0 2 r2 r 2 cos dz dr d arctan 1 2 0 4 sec 3 0 0 0 2 2 arctan 1 2 cot 0 csc 3 0 (b) sin2 cos d d d sin2 cos d d d 0 2 a 0 4 2 0 a a a2 r2 15. (a) 0 2a cos r 2 cos dz dr d 3 0 a sec 0 (b) 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 a cos a2 0 r2 2 2 0 2 9 0 r cos 2r sin 19. V 2 0 0 a cos r dz dr d 21. m 0 2 kr r dz dr d kr 2 9 0 0 2 2 0 0 r a2 12 a 3 1 0 r 2 dr d a cos 32 0 r cos r4 cos 4 4 cos 2r sin r4 sin 2 8 sin 2 dr d 2 2 0 r2 d 0 2 k 3r 3 k 24 0 d 0 2a3 3 2a3 3 sin3 cos 4 d cos3 3 d 0 k 24 k 48 4 sin 8 8 8 cos 0 2a3 3 9 48k 164 Chapter 13 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 25. x m k x2 y 4k 0 0 y2 kr 0 by symmetry 2 r0 h(r0 0 r) r0 r 2 dz dr d 4h r0 4h r0 r0r 0 2 0 0 r 2 dr d r0 d 6 2 1 r 2h 30 3 1 k r03h 6 2 r0 0 h(r0 0 r) r0 Mxy 4k 0 r 2 z dz dr d 4h r03 r0 6 1 k r03 h2 30 z Mxy m k r03h2 30 k r03h 6 h 5 2 r0 0 2 h(r0 0 r0 r) r0 27. Iz 4k 0 r3 dz dr d 29. m Iz k b2h 2 b a 2 a2h h k h b2 a2 4kh r0 r0r3 0 0 r 4 dr d 4k 0 0 b r3 dz dr d 4kh r05 r0 20 1 k r04h 10 4kh 2 kh 0 0 2 a r3 dr d b4 a4 h 2 a4 d 1 Since the mass of the core is m kV k 3 r02h from 2h. Exercise 23, we have k 3m r0 Thus, k b4 k b2 1 m a2 2 Iz 1 k r04h 10 1 10 3m r02h r04h a2 b2 2 b2 a2 h 3 mr 2 10 0 2 4 sin 2 2 2 0 2 a 3 31. V 0 0 0 sin d d d 16 2 33. m 8k 0 0 2 sin ddd 2ka4 0 2 0 sin dd k a4 0 sin cos d 2 k a4 k a4 0 Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates 2 2 0 2 4 0 2 2 cos 4 4 0 165 35. m x Mxy 2 k r3 3 y 4k 0 0 2 0 2 0 0 2 37. Iz 4k 4 k 5 cos5 cos5 4 sin3 ddd dd sin 2 4 0 by symmetry 2 2 r 3 sin3 cos2 cos sin ddd 2 k 5 1 d 14 kr 2 kr 4 4 sin 2 d d sin 2 d 0 2 k 5 k 192 1 cos6 6 1 cos8 8 1 k r 4 cos 2 8 z Mxy m k r4 4 2k r3 3 2 0 1 k r4 4 3r 8 2 g2 g1 h2 r cos , r sin 39. x y z r cos r sin z x2 y2 tan z r2 y x z 41. 1 f r cos , r sin , z r dz dr d h1 r cos , r sin 43. (a) r z r0: right circular cylinder about z-axis 0: (b) 0: 0: 0: sphere of radius cone 0 plane parallel to z-axis plane parallel to z-axis z0: plane parallel to xy-plane a a2 0 x2 0 a2 x2 y2 0 a2 x2 y 2 z2 45. 16 0 a a2 0 2 a 0 2 a 0 2 a 0 2 0 dw dz dy dx x2 0 a2 r2 a2 x2 y2 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 0 166 Chapter 13 Multiple Integration Section 13.8 1. x y 1 u 2 1 u 2 xy uv v yx uv v Change of Variables: Jacobians 3. x y u u xy uv v2 v yx uv 11 1 2v 1 2v 1 2 1 2 1 2 1 2 1 2 5. x y u cos u sin xy uv v sin v cos yx uv cos2 sin2 1 7. x y eu sin v eu cos v xy uv yx uv 2v eu sin v eu sin v eu cos v eu cos v e2u 9. x y v u 3u 3v y 3 x 3 x 3 1 u 2 1 u 2 xy uv 4 x2 R x, y 0, 0 3, 0 2, 3 u, v 0, 0 1, 0 0, 1 1 v (0, 1) 2v 2y 9 x 2y3 3 (1, 0) u 1 11. x y v v yx uv y 2 dA 1 1 1 1 1 1 2 1 1 1 2 4 1 u 4 1 2 v 1 2 2 1 2 1 u 4 1 v 2 1 dv du 2 1 du 3 2 u3 3 u 3 1 1 u2 1 v 2 dv du 1 2 u2 8 3 13. x y u u xy uv yx R 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 Section 13.8 Change of Variables: Jacobians 167 15. R e R: y x xy 2 dA 2x, y 1 ,y x 4 x y ,v x v1 u3 v1 u1 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 uv ⇒ u x v y v 1 2 1 2 xy 1 1 R x 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 ⇒ yx 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 y x ⇒ 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 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 v 0 4 6 y 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 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 v 0 5 2 y x−y=0 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 v 0 dv 2 52 3 v 33 5 2 0 100 9 168 21. u Chapter 13 x y, v yx uv x R y Multiple Integration x y, x 1 2 a u 1 u 2 v ,y 1 u 2 v xy uv y dA 0 u u 1 dv du 2 y a u u du 0 25 u 5 a 2 0 25 a 5 2 v=u a a x+y=a x 2a x −a a v = −u 23. au 2 a2 x2 a2 y2 b2 bv 2 b2 1, x 1 1 au, y bv (a) x2 a2 y2 b2 y 1 u2 v2 v 1 1 u2 v2 b R x S u 1 a (b) x, y u, v x u ab y v yx uv 00 ab (c) A S ab dS ab 1 2 ab 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 w u uv 2 u2vw u uv2 1 w uv2w 27. x u1 x, y, z u, v, w w, z u u1 w uw 29. x sin x, y, z ,, cos , y sin cos sin sin cos cos cos 2 2 2 2 2 sin sin , z sin sin sin cos 0 cos cos cos 2 cos sin sin cos cos2 sin sin2 sin cos2 sin2 sin2 cos2 sin2 sin2 sin sin cos2 cos2 cos cos 2 sin2 sin2 sin3 sin cos2 sin sin sin sin2 ...
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