EVEN13 - 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 . . . . . . . . . . . . . 365 Section 13.2 Double Integrals and Volume . . . . . . . . . . . . . . . . . . . 369 Section 13.3 Change of Variables: Polar Coordinates . . . . . . . . . . . . . 375 Section 13.4 Center of Mass and Moments of Inertia . . . . . . . . . . . . . 379 Section 13.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Section 13.6 Triple Integrals and Applications . . . . . . . . . . . . . . . . . 388 Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates . . . . 393 Section 13.8 Change of Variables: Jacobians . . . . . . . . . . . . . . . . . . 397 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 C H A P T E R 13 Multiple Integration Section 13.1 Iterated Integrals and Area in the Plane Solutions to Even-Numbered Exercises x 2 2. x y dy x x 1 y2 2x x2 x 1 x4 2x x2 x x2 x 2 cos y cos y 1 4. 0 y dx yx 0 y cos y x 6. x 3 x2 3y2 dy x2y y3 x3 x2 x x 3 x2x3 x3 3 x5 2 x3 2 x5 x9 1 y2 8. 1 y2 x2 y 2 dx 13 x 3 2 y 2x 1 1 y2 y2 2 1 1 3 y2 32 y2 1 y2 12 21 3 y2 1 2y 2 2 10. y sin3 x cos y dx y 1 cos x cos2 x sin x cos y dx 1 cos3 x cos y 3 y3 3 2 1 2 cos y y 1 cos3 y cos y 3 1 2 1 12. 1 2 x2 y2 dy dx 1 1 x2y 4x2 1 dx 2 1 2x2 4x3 3 16 x 3 8 3 1 1 2x2 4 3 8 dx 3 16 3 4 3 16 3 8 16 dx 3 x2 4 x 2 4 14. 40 64 x3 dy dx 4 4 y 64 64 4 x3 0 dx 2 64 9 2y 4 x3 x2 dx x3 32 4 0 2 128 9 16 3 y 3 y4 2 2 32 2048 9 2 2 2y 2 16. 0 y 10 2x2 2y2 dx dy 0 2 10x 10y 0 2x3 3 14 3 y 3 2 2 2y2x y dy 0 20y 5y2 7y4 6 83 y 3 4y3 20 0 10y 56 3 8 23 y 3 140 3 2y3 dy 2y3 dy 2 2y 3y2 y2 2 2y y2 2 18. 0 6y 3y dx dy 0 3xy 3y 2 dy 6y 3 0 8y2 4y3 dy 3 y4 0 16 2 2 cos 2 20. 0 0 r dr d 0 r2 2 2 cos 2 d 0 0 2 cos2 d 1 sin 2 2 2 0 2 4 cos 4 cos 22. 0 0 3r 2 sin dr d 0 4 r 3 sin 0 d cos4 4 4 0 cos3 sin d 0 1 4 1 2 4 1 3 16 365 366 3 Chapter 13 x2 0 Multiple Integration 3 3 24. 0 1 y2 dy dx 0 x 2 arctan y 0 dx 0 x2 2 dx 2 x3 3 3 0 9 2 26. 0 0 xye x2 y2 dx dy 0 1 ye 2 3 x2 y2 0 dy 0 1 ye 2 y2 dy 1 e 4 y2 0 1 4 3 3 3 3 3 y 28. 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 5 1 0 5 x 1 5 1 x 1 5 30. A 2 12 dy dx 2 1 1 y 0 1 y2 dx 2 1 x 1 5 dx y 2x 1 2 2 A 0 12 2 dx dy 12 2 5 1 1 dx dy 1 y2 5 4 x 0 12 2 dy 12 1 x 2 dy 3 y= 2 1 1 x−1 3 dy 0 12 12 1 y2 1 1 dy 1 2 3 4 5 x 3y 0 1 y x2 y 12 2 2 4 0 y 4 2x y 32. A 0 2 dy dx 2 y= 4 − x2 34. A 0 4 x3 2 dy dx 2x 8 7 6 5 4 3 2 1 (4, 8) y = 2x 4 0 2 x2 dx 1 y 0 4 x3 2 dx y = x 3/2 x 1234567 8 4 0 2 cos2 d x 1 2 2x 0 x3 25 x 5 2 32 5 2 dx 4 −1 2 0 1 2 cos 2 d 2 0 x2 16 8 y 2 0 1 sin 2 2 16 5 23 x 2 sin , dx 2 4 0 2 y2 2 cos d, 2 4 x2 2 cos A 0 8 y2 dx dy y dy 2 y2 4 16 8 0 A 0 dx dy 0 2 4 y2 dy 0 y2 35 y 5 3 32 5 3 4 0 cos2 2 d 2 0 2 0 1 cos 2 d 3 1 sin 2 2 16 5 y 2 sin , dy 2 cos d, 4 y2 2 cos Section 13.1 3 x 9 9x Iterated Integrals and Area in the Plane 2 y 4 2 367 36. A 0 3 0 dy dx 3 x 9 0 dy dx 9x 3 9 38. A 0 2 y2 dx dy 2 4 y2 dx dy 2 y dy 2 4 2 y 0 0 3 dx 3 y 0 9 dx 0 x dx 3 9 dx x 0 y dy 2 2 2 12 x 2 9 1 2 1 9 9 ln x 0 3 9 2 9 ln 9 ln 3 1 y2 4 2y 0 y2 4 2 2 4 2 2x 3 dy dx ln 9 A 3 9y 2x 0 x dx 0 x x2 2 2 2 0 A 0 1 y dx dy 1 9 3 y dx dy 9y 4 y y = 2x x 0 1 y dy 1 x y 3 dy 9 y 3 2 9 0 y dy 1 y dy 12 y 2 3 1 1 y=x x 1 2 3 4 9y y 6 12 y 2 1 9 ln y 0 9 1 2 ln 9 y=x 4 (3, 3) y= 9 x (9, 1) x 2 2 4 6 8 −2 4 2 2 4 0 x 2 40. 0 y f x, y dx dy, 2 0 x2 y ≤ x ≤ 2, 0 ≤ y ≤ 4 y 42. 0 f x, y dy dx, 0 ≤ y ≤ 4 4 4 0 y x2, 0 ≤ x ≤ 2 y f x, y dy dx 0 4 3 y = x2 f x, y dx dy 0 4 3 2 2 1 −1 x 1 −1 2 3 4 1 x 1 2 3 4 2 e x 2 cos x 44. 10 f x, y dy dx, 0 ≤ y ≤ e x, e 0 y 2 1≤x≤2 ln y 46. 20 1 0 f x, y dy dx, 0 ≤ y ≤ cos x, arccos y 2 y 2 3 2 ≤x≤ 2 2 e f x, y dx dy 1 e 2 f x, y dx dy 1 arccos y f x, y dx dy 3 1 2 2 −π 4 π 4 x −1 x 1 2 368 2 Chapter 13 4 4 2 Multiple Integration 2 4 4 4 4 x2 2 48. 1 y 4 3 2 dx dy 2 2 1 dy dx 2 50. 2 2 2 x2 y2 dy dx 2 4 4 x2 4 x2 dx 4 dx dy y2 y 1 x 1 2 3 4 1 x 1 −1 −1 4 x2 6 4 2 0 6 0 6 2y x 4 0 52. 0 0 dy dx y dy dx 2 x dx 2 6 6 y 6 4 x dx 6y 4 3y2 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 9 3 9 54. 0 3 0 x y 0 2 dy dx 0 3 3 y2 dy 0 x dx y3 3 3 0 3x 9 23 x 3 9 2 0 y 27 18 9 dx dy 5 4 3 2 1 −1 −2 −3 −4 −5 y= x x 123456789 2 4 y2 4 y 4 4 x 56. 20 dx dy 0 x dy dx 32 3 2 1 x 1 2 3 −1 −2 x = 4 − y2 58. 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 1 4 1 cos 4 4 Section 13.2 2 2 2 y Double Integrals and Volume 369 60. 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 2 4 4 x 62. 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 1 2y 64. 0 y sin x y dx dy sin 2 2 4 16 dx dy sin 3 3 y2 a a 0 x 0.408 66. 0 x2 y 2 dy dx a4 6 y 4 68. (a) y y 2 4 4 2 4 y 2 x2 ⇔ x x2 ⇔x 4 x2 xy y2 1 4y 3 3 2 2 0 (b) 0 x2 xy y2 4 16 0 4y 1 dx dy 3 x2 xy y2 2 1 dx dy 1 x 1 2 (c) Both orders of integration yield 1.11899. 2 2 2 1 0 sin 70. 0 x 16 x3 y3 dy dx 6.8520 72. 0 15 r dr d 45 2 32 135 8 30.7541 74. 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. 76. The integrations might be easier. See Exercise 59-62. 78. False, let f x, y x. Section 13.2 For Exercises 2 and 4, 11 ,, 22 31 ,, 22 Double Integrals and Volume xi yi 71 ,, 22 1 and the midpoints of the squares are 13 ,, 22 33 ,, 22 53 ,, 22 73 ,. 22 4 3 2 y 51 ,, 22 1 x 1 2 3 4 2. f x, y 12 xy 2 1 16 4 0 f xi, yi xi yi i 1 4 0 2 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 12 x y dy dx 2 dx 0 21.3 370 Chapter 13 1 1y xi yi 1 x 1y Multiple Integration 4. f x, y 8 x 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 1 1 ln 3 dx x1 4 ln 3 ln x 1 0 ln 3 ln 5 1.768 2 2 6. 0 0 f x, y dy dx 4 2 8 6 20 2 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 y 4 y 4 10. 0 x2y2 dx dy 1 2y 0 4 0 x3y2 3 y 72 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 1 0 1 1 0 y 1 0 1 1 y y 12. 0 y 1 ex y dx dy 0 ex y dx dy 0 1 ex y y 1 dy 0 ex y 0 dy 2 y=x+1 y = −x + 1 e 0 e2y 1 2y e 2 e 1 1 dy 1 x 1 ey 1 e 2 1 0 −1 Section 13.2 2 2 4 4 0 x Double Integrals and Volume 4 0 4 0 y 371 14. 0 sin x sin y dx dy 0 sin x sin y dy dx 2 16. 0 xey dy dx xey dx dy For the first integral, we obtain: dx 4 4 x 4 x sin x cos y 0 xey 0 0 x dx 0 xe 4 e4 x x dx x e4 x2 2 e4 4 0 sin x dx 1 0 5 y 5π 2 2π 3π 2 8 13. y 4 3 2 x π 1 x 1 2 3 4 − 3π − π − π 2 2 π 2 π 3π 2 2 4 y2 18. 0 y 1 x 4 2 dx dy 0 0 4 0 4 0 x y 1 y2 x x x 2 y 2 dy dx 4 3 1 2 1 2 x 1 1 dx 0 y= 2 x x2 dx 4 1 x 1 2 3 4 1 ln 1 4 2 4 4 y2 x2 0 1 ln 17 4 y 20. 0 y2 2 x2 4 y 2 dx dy x2 4 3 x2 20 2 y2 4 0 dy dx x2 x=− 4 − y2 x= 4 − y2 1 x 2y 2 2 13 y 3 x2 x2 32 dx 1 4 3 −2 −1 x 1 2 x2 4 2 x2 32 dx x 2 1 x4 12 x2 32 x 4 4 1 x4 2 x2 4 arcsin 6x 4 x2 24 arctan x 2 2 4 2 4 2 4 2 2 x 2 2 22. 0 0 6 2y dy dx 0 4 6y y2 0 dx 24. 0 0 y 4 dy dx 0 4x dx 2x2 0 8 8 dx 0 y 4 32 2 y=x 1 3 2 x 1 2 1 x 1 2 3 4 372 2 Chapter 13 2 0 x Multiple Integration 2 26. 0 2 x y dy dx 0 2 0 2y 1 2 2 xy x 2 dx 2 y2 2 2 0 x 2 y 2 dx 28. 0 0 4 y 2 dx dy 0 4y y3 dy y4 4 2 0 2y 2 4 3 2 1 x 6 y 2 3 0 4 y 2 y=2−x 1 1 y=x x x 1 2 1 2 30. 0 0 e x y2 dy dx 0 2e x y2 0 dx 0 2e x2 dx 4e x2 0 4 1 x 32. 0 0 1 x2 dy dx 1 3 y 5 x 34. V 0 5 0 x dy dx x 5 5 4 y=x xy 0 0 5 0 dx 0 x2 dx 3 2 1 x 1 2 3 4 5 13 x 3 125 3 r r2 x2 36. V 8 0 r 0 r2 y r2 0 r x2 y2 y 2 dy dx r2 x 2 arcsin y r2 x2 0 r2 x2 y y= r2 − x2 4 x2 x 2 dx 13 x 3 r 0 dx r 4 2 4 2 r2 0 r x r 2x r3 3 2 4 0 4 x 2 2 38. V 0 2 4 4 0 2 x2 dy dx 4 x2 dx y 40. V 0 2 0 1 1 y2 dy dx 2 y x2 3 2 y=4− x2 arctan y 0 2 0 0 dx 1 16 0 8x2 x 8 3 3 x4 dx x 5 52 0 1 x 1 2 3 4 2 x 2 dx x 1 2 2 0 16x 32 64 3 32 5 256 15 Section 13.2 5 y Double Integrals and Volume y 373 9 9 0 42. V 0 5 0 0 sin2 x dx dy dy 5 5 4 44. V 0 9 y dx dy 81 2 2 y 3 2 1 x 1 2 3 4 5 2 5 2 0 16 4 0 y 46. V 0 ln 1 x y dx dy 38.25 48. x a z V y b c1 z c x a 1 y b a b1 0 a xa a z f x, y dA R 0 c1 c 0 a x a y dy dx b x R a a y y b1 0 xy a x a x a ab 3 y2 b 1 2b 0 xa dx x a x3b 3a2 ab 6 b2 1 2b ab 1 6 abc 6 2 2 2 2y c c c xb 1 a 2 x a x a 2 dx 3a 0 ab 1 2 ab 2 10 1 10 1 10 ln y 0 x 2b 2a ab 2 ln 10 10 e x 50. 0 1 dy dx ln y 1 dx dy ln y ln y 52. 0 1 2x 2 2 y cos y dy dx 0 0 y y cos y dx dy x ln y dy dy 0 10 y cos y 2y dy 0 2 2 y 1 9 2 0 y cos y dy 2 1 (2, 2) 1 y 10 8 6 4 2 x 1 2 3 4 5 2 cos y 2 cos 2 y sin y 0 y = 1 x2 2 2 sin 2 1 x 1 2 y = ex 54. Average 1 8 4 0 2 xy dy dx 0 1 8 4 2x dx 0 x2 8 4 2 0 56. Average 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 1 2x e 2 1 2 e2 0 2e 1 2 374 Chapter 13 Multiple Integration 58. The second is integrable. The first contains sin y2 dy 60. (a) The total snowfall in the county R. (b) The average snowfall in R. which does not have an elementary antiderivation. 62. Average 1 150 60 45 50 192x 40 576y x2 5y 2 2xy 5000 dx dy 13,246.67 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 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 1 x dx b→ 1 lim 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 2x e 1 x 1 dx 0.1998. 2x e x 1 e 2 2 e 1 e 2 1 2 2 4 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 70. 0 0 20e x3 8 dy dx m 10, n 20 (a) 129.2018 (b) 129.2756 : For I, 1, M, 1 : For J, 1, N, 1 :A :C :V : End : End : Disp V 0.5G 2I 0.5H 2J sin x 1 →x 1 →y y G H→V S ection 13.3 Change of Variables: Polar Coordinates 375 4 2 72. 1 1 x3 (a) 13.956 (b) 13.9022 y3 dx dy m 6, n 4 74. V 50 Matches a. z 4 3 x 3 3 y 76. True 2 78. 1 e Thus, xy dy 1 e x 2 xy 1 e x e x 2x e 0 x e x 2x 2 dx 0 2 1 e xy dx dy e 1 2 1 2 1 0 xy dx dy e xy y 1 dy y dy 0 2 ln y 1 ln 2. Section 13.3 2. Polar coordinates 6. R 4 Change of Variables: Polar Coordinates 4. Rectangular coordinates ≤ r, 4 : 0 ≤ r ≤ 4 sin , 0 ≤ 4 8. R 4 2 r, 3 : 0 ≤ r ≤ r cos 3 , 0 ≤ 2 ≤ 10. 0 0 r 2 sin cos dr d 0 r3 sin 3 cos 0 4 0 d 12. 0 0 re r2 dr d 0 1 e 2 9 3 r2 0 2 d 64 sin2 3 2 16 3 π 2 1 e 2 4 π 2 1 0 1 1 e9 0 1 2 3 4 0 1 2 3 376 Chapter 13 Multiple Integration π 2 2 1 0 cos 2 14. 0 sin r dr d 0 2 0 sin sin 2 r2 2 1 1 0 cos d (x, y) = (0, 1) cos 2 3 0 2 d 1 6 0 1 1 1 6 cos a a2 0 x2 2 a 16. 0 x dy dx 0 0 r 2 cos dr d a3 3 2 cos d 0 a3 sin 3 2 0 a3 3 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 4 4y 0 y2 2 4 sin 2 20. 0 x 2 dx dy 0 0 2 r3 cos2 dr d 0 64 sin4 64 sin5 6 cos2 d sin3 cos 4 3 8 2 64 0 sin4 sin6 d cos sin cos 0 2 5 2 2 x 5 25 2 2 0 x2 4 5 22. 0 0 xy dy dx 5 xy dy dx 0 4 0 0 r3 sin cos dr d 625 sin cos d 4 4 0 625 2 sin 8 625 16 2 5 2 y 5 24. 2 0 e r2 2 r 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 3 9 0 x2 2 3 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 S ection 13.3 Change of Variables: Polar Coordinates 377 2 1 2 28. V 4 0 0 r2 3 r dr d 4 4 0 4 0 r4 4 3r 2 2 1 2 2 d 0 30. R ln x2 y 2 dA 0 2 1 2 ln r 2 r dr d 7 d 4 7 4 2 0 2 1 r ln r dr d r2 4 ln 4 0 2 7 4 2 0 2 1 2 ln r 1 d 2 4 3 d 4 3 4 ln 4 2 4 2 32. V 0 1 16 r 2 r dr d 0 1 3 16 r2 3 4 2 d 1 0 5 15 d 10 15 34. x 2 V y2 8 z2 2 0 2 a2 ⇒ z a a2 r 2 r dr d x2 y2 a2 r2 a2 0 (8 times the volume in the first octant) a 8 0 2 1 2 a3 d 3 22 a 3 8a3 3 r2 32 0 2 d 4 a3 3 1 1 ≤r≤ 1 4 2 8 0 0 36. 4 x2 (a) 4r 2 9 y2 9 9 36 ≤z≤ ≤z≤ 4 x2 9 9 y2 36 9 ; cos2 −1 0.7 1 4r 2 r2 (b) Perimeter r dr d 1 1 2 cos2 dr d 1 2 2 d. − 0.7 1 cos2 2 1 z cos sin 2 0 2 1 21 14 cos2 Perimeter 1 1 4 9 4r 2 1 cos2 2 cos2 sin2 d 5.21 x 1 y (c) V 2 0 36 r dr d 0.8000 2 4 2 38. A 0 2 r dr d 0 6d 12 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 1 2 4 2 4 0 4 sin 1 cos 2 2 d 1 4 2 1 sin 2 4 9 2 378 Chapter 13 4 3 cos 2 Multiple Integration 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 44. See Theorem 13.3. 46. (a) Horizontal or polar representative elements (b) Polar representative element (c) Vertical or polar 48. (a) The volume of the subregion determined by the point 5, volumes, ending with 45 10 8 12 , you obtain V 10 8 57 9 9 5 15 8 35 12 5 150 4 (b) 56 24013.5 (c) 7.48 24103.5 4 4 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 24013.5 ft3 1,344,759 pounds 179,621 gallons 50. 0 0 5e r r dr d 87.130 52. Volume base 9 4 height 3 21 6 4 2 z Answer (a) y 4 2 4 x 54. True 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 . 2 2 58. 0 0 ke x2 y2 dy dx 0 0 ke r2 r dr d 0 k e 2 2 r2 0 d 0 k d 2 k 4 For f x, y to be a probability density function, k 4 k 1 4 . 2 2 2 4 y 2 y (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 Section 13.4 Center of Mass and Moments of Inertia 379 Section 13.4 3 9 0 x 2 Center of Mass and Moments of Inertia 3 2. m 0 xy dy dx 0 3 0 xy2 2 x9 9 0 x2 dx x2 2 2 dx 33 0 1 9 x2 4 3 1 0 12 93 243 4 3 3 3 9 x2 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 54 297 8 a b 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 a2k 2 a a 0 x My 0 0 k x3 My m Mx m xy 2 dy dx 2b2 2b2 b2 3b2 b2 x y ka3b2 6 ka2b2 4 ka2b3 6 ka2b2 4 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 8. (a) m Mx y ky dy dx 0 ka3 6 a y=a−x My x Mx by symmetry y Mx m ka3 6 ka2 2 a 3 a x —CONTINUED— 380 Chapter 13 Multiple Integration 8. —CONTINUED— a a 0 x (b) m 0 a x2 x 2y 0 a a 0 x y 2 dy dx a 0 x a 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 10. The x-coordinate changes by h units horizontally and k units vertically. This is not true for variable densities. a a2 0 a2 0 a x2 12. (a) m 0 a x2 k dy dx k a2 4 a a2 0 2 a x2 (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 ka3 3 k a2 4 4a 3 Mx by symmetry y My m ka5 5 8 ka4 8a 5 4a by symmetry 3 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 4 4x 16. m 1 4 0 4x kx 2 dy dx 30k Mx 0 2 3 kxy dy dx kx2 dy dx 0 Mx 1 4 0 4x kx 2 y dy dx 24k My x y My m Mx m My 1 0 kx3 dy dx My m My m 84k 30k 24k 30k 14 5 4 5 84k 32k 3 16k 5 32k 5 32k x y y 4 3 2 y= 4 x 1 x 1 2 3 4 Section 13.4 Center of Mass and Moments of Inertia 381 L2 cos x L 18. x m 0 by symmetry 3 9 x2 20. m 0 0 cos x L k dy dx L2 kL kL 8 L2 2 L kL 2 2 ky 2 dy dx 30 3 9 x2 23,328k 35 139,968k 35 6 Mx 0 0 cos x L L2 ky dy dx kx dy dx 0 0 Mx 30 ky3 Mx m y dy dx My x y My m Mx m t 2k 2 y 139,968k 35 35 23,328k L2 2 kL 8 2 2k 12 kL 8 y = 9 − x2 6 3 x 3 6 1 y = cos π x L −6 −3 x L 2 4 a 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 a 0 kr3 sin d y=x r=a My R k x2 My m Mx m ka4 2 8 ka4 2 8 y 2 dA 0 0 kr 3 cos d 3 2a 2 12 ka3 32 2 y a 0 x 12 ka3 2 y 2a e ln x 0 ln x 0 ln x 0 24. m 1 e 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 3 Mx 1 e 2 y = ln x 1 My 1 1 2 e3 x x y My m Mx m k 1 k 6 382 26. y m Chapter 13 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 28. m 0 b hx b dy dx bh 2 bh3 12 b3h 12 b 6 h 6 6 b 6 6 h 6 30. m Ix a2 2 a Ix 0 b hx b y2 dy dx y 2 dA R 0 0 a r3 sin2 dr d a4 8 a4 8 Iy 0 x2 dy dx Iy m Ix m b3h 12 bh 2 bh3 12 bh 2 Iy R x 2 dA 0 0 r3 cos2 a4 8 a4 8 a4 8 a4 4 2 a2 dr d x y I0 x Ix y Ix Ix m a 2 32. m Ix 4 ab a 0 a 0 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 a3b 4 ab 4 1 ab 1 ab a 2 b 2 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 Iy a2 b2 Section 13.4 34. m ky a a2 0 x2 Center of Mass and Moments of Inertia kxy 1 x 383 36. y dy dx 0 a 2k m k 0 x2 x 1 xy dy dx k 2 k 4 k 2 3k 80 1 x3 0 1 x5 dx k 24 k 60 k 48 k 0 a a2 a2 x 2 dx x2 2ka3 3 4ka5 15 2ka5 15 Ix Iy I0 x a2 5 2a2 5 a 5 2a 10 40. y k 0 1 x2 x xy3 dy dx x5 0 1 x9 dx Ix Iy I0 x y k a0 a a2 x2 y3 dy dx k 0 x2 x3y dy dx Iy Iy m Ix m 9k 240 x5 0 x7 dx k a0 x 2y dy dx Iy Iy m Ix m 2ka5 5 2ka5 15 2ka3 3 4ka5 15 2ka3 3 Ix Ix k 48 k 24 k 60 k 24 1 2 2 5 38. m x2 1 0 1 x2 y2 x ky 2 4x x2 x y 2 dy dx 6 35 158 2079 158 2079 m 2 0 2 x3 4x ky dy dx 512k 21 32,768k 65 2048k 45 Ix 0 1 x2 x x2 y 2 y 2 dy dx Ix Iy I0 2 0 2 x3 4x ky3 dy dx Iy 0 x2 x2 Ix Iy Iy m Ix m 4 2 y 2 x 2 dy dx 2 0 x3 kx 2 y dy dx Iy Iy m Ix m 321,536k 585 2048k 45 32,768k 65 I0 x y 316 2079 158 2079 x 35 6 395 891 Ix x y 21 512k 21 512k 28 15 8 1365 65 2 105 15 395 891 4 4 0 42. I 0 0 kx 6 dy dx 0 2 2k x 6 dx 2 2k x 3 6 3 416k 3 a a2 x2 44. I a0 a ky y y4 4 14 a 4 2ay3 3 2a2x 2 2a2x3 3 a2 25 a 3 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 x a x2 x2 x 2 a2 a2 arcsin a2 2 ax 2 a2 3 a 2 x2 x a a a x2 dx k 14 ax 4 1 x 2x 2 8 x3 3 a3 3 2k 15 a 4 2a a4 34 a4 16 2k 7a5 15 a5 8 ka5 56 15 60 384 Chapter 13 x2 Multiple Integration 2 4 2 46. I 20 ky k 3 2 2 2 dy dx 2 k y 3 1 3 4 0 x2 2 dx 65 x 5 k 2 3 2 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 2k 32 3 32 48. x, y k2 x. 50. x, y k4 x4 y . Both x and y will decrease x, y will be the same. 52. Ix R y2 x2 R x, y dA Moment of inertia about x-axis. x, y dA Moment of inertia about y-axis. 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 r. Therefore, V 2 rA. 56. y a ,A 2 b a ab, h a 2 2 L a 2 a3b 12 a 3L 3 2L 2a a 58. y Iy 0, A a a 2 a2, h x 2 L y 2 dy dx a 2 a a2 x2 Iy 0 0 y a 2 dy dx ya a3b 12 L a 2 ab r3 sin2 0 2 0 0 dr d a4 2 sin 4 d a4 4 ya a4 4 L a2 a2 4L Section 13.5 Surface Area 385 Section 13.5 2. f x, y fx 1 3 Surface Area 2x 3y 4. f x, y R fy 2 15 10 x, y : x2 2x y2 3 3y ≤9 2, fy fx 3 2 3 14 3 fx 9 14 S 1 2, fy fx 3 3 2 3 2 9 9 S 0 y 14 dy dx 0 0 3 14 dx fy x2 2 14 14 dy dx x2 3 R 2 14 r dr d 0 0 y 9 14 1 2 y = 9 − x2 x 1 2 3 R −2 −1 1 x −1 −2 1 y = − 9 − x2 6. f x, y R fx 1 3 y2 3 y square with vertices 0, 0 , 3, 0 , 0, 3 , 3, 3 0, fy fx 3 2 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 23 y 3 y1 fy2 2 0 y 2 4y 2 ln 2y 4y 2 0 3 6 37 4 ln 6 37 8. f x, y fx 1 S 0 2 0, fy fx2 2 2 10. f x, y fx 1 y 2 9 x2 2y y2 2x, fy 1 2 fx 2 2 fy 1 2 1 4x 2 4y 2 1 21 2 12 5 y y dx dy 0 1 2 y2 y dy S 0 2 0 0 4r2 r dr d 2 y 32 2 1 5 35 2 y 2 52 0 1 1 12 1 173 12 y 4r2 2 32 0 d 17 17 1 2 0 33 2 3 8 5 2 5 2 5 1d 6 R −1 1 x 1 −1 2 y=2−x R 1 x 1 2 386 Chapter 13 xy x, y : y, fy 1 4 Multiple Integration y 12. f x, y R fx x 2 + y 2 = 16 x2 x y2 ≤ 16 2 fx 2 16 16 4 fy x2 2 1 1 y2 y2 x2 x2 dy dx −2 −2 x 2 S 4 2 0 x2 1 0 r 2 r dr d 2 17 17 3 1 14. See Exercise 13. a a2 a 2 x2 x 2 S a a2 a x2 2 a 0 y dy dx 2 0 a r dr d a2 r 2 2 a2 16. z 1 16 fy 2 4 x2 y2 fy2 1 1 4 4x 2 x2 y2 4y 2 dy dx 18. z 1 S 2 x2 fx2 2 0 2 y2 fy2 5r dr d 1 4 4x2 x2 5 y2 x2 4y2 y2 5 16 0 2 4 x 2 S 0 0 y 1 0 y 4r 2 r dr d 0 24 65 65 1 x2 + y2 = 4 1 6 y= 4 16 − x 2 −1 −1 x 1 2 x 2 4 6 20. f x, y R 1 2 2x y2 22. f x, y R 0≤ 55 fx 1 4 x2 y2 triangle with vertices 0, 0 , 2, 0 , 2, 2 fx x 2 x, y : 0 ≤ f x, y ≤ 16 x2 2x, fy fx 2 16 16 4 x fy 5 2 5 4y 2 y 2 ≤ 16 2y fy 2 S 0 y 4y 2 dy dx 0 1 21 21 12 2 1 1 4x 2 4x 2 4y 2 S 3 4y 2 dy dx 65 65 6 1 4 x2 y=x 2 2 0 1 1 4r 2 dr d R 0 y x x 2 + y 2 = 16 1 2 3 2 −2 −2 x 2 Section 13.5 24. f x, y R fx 1 1 Surface Area 387 2 32 3x cos x 26. Surface area Matches (c) z 9 x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 x 12 sin x, fy 2 0 1 x x 2 fx 1 fy 1 2 sin x 2 3 2 S 0 0 sin x dy dx 1.02185 x 3 3 y 28. f x, y R fx 1 1 2 52 5y 30. f x, y R fx 2 x2 3xy y2 x, y : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0, fy fx 1 2 x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ x 2x 1 fx 4 x y3 2 fy 1 1 y3 3y, fy fy 2 3x 1 1 2y 2x 13 x 2 y 2 dy dx 3x 3y 2 2y 3x 2y 2 S 0 1 0 y3 dx dy S 1 0 0 y2 13 x 2 1 0 y dy 3 1.1114 32. f x, y R fx 1 S cos x 2 x, y : x 2 2x sin fx 2 2 2 ( ( y2 y2 ≤ y2 fy 2 2) 2) x 2 2 2y sin x 2 1 1 4x 2 sin2 x 2 4 x2 y2 y2 4y 2sin2 x 2 y 2 dy dx y2 1 4 sin2 x 2 y 2 x2 y2 x2 , fy y 2 sin2 x 2 x2 34. f x, y R fx 1 e x sin y 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 2x 2x x, y : 0 ≤ x ≤ 4, 0 ≤ y ≤ x e x sin y, fy 2 e 2 x cos y 1 1 e e sin2 y e 2x fx 4 x fy cos2 y 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, z 1. (b) Yes. Let R be the region in part (a) and S the surface xy. given by f x, y (c) No. S 0 0 1 e 2x dy dx 38. f x, y 1 S R 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 1 388 Chapter 13 13 y 75 Multiple Integration 42 y 25 13 y 75 12 y 25 16 y 15 42 y 25 8 y 25 fy2 15 40. (a) z 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 15 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 42. False. The surface area will remain the same for any vertical translation. Section 13.6 1 1 1 1 Triple Integrals and Applications 1 3 2 3 1 1 1 1 1 1 2. 1 1 x 2y 2z2 dx dy dz 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 9 y3 0 0 y 2 9x 2 4. 0 z dz dx dy 1 2 1 2 9 0 9 y3 y2 0 9x 2 dx dy y3 xy 2 0 3x3 0 dy 2 18 9 y3 dy 0 14 y 36 9 0 729 4 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 dx 1 1 2 dx x 2 ln x 1 2 ln 4 2 y2 0 1y 2 y2 0 8. 0 0 sin y dz dx dy 0 sin y dx dy y 1 2 2 sin y dy 0 1 cos y 2 2 0 1 2 2 2 0 x2 4 y2 2 2 0 x2 10. 0 2x 2 y 2 y dz dy dx 0 4y 2x 2y 2y3 dy dx 16 2 15 3 2 0 (2y 3) 6 0 2y 3z 6 (6 0 3 0 x) 2 (6 0 x 2y) 3 12. 0 ze x 2y 2 dx dz dy 0 6 0 (x 2) ze 16 2 x 3 2y x 2y 2 dz dy dx 2 e x 2y 2 dy dx 2.118 3 2x 0 9 0 x 2 14. 0 dz dy dx S ection 13.6 12 x 2 y2 16 16 Triple Integrals and Applications 389 16. z x2 4 4 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 y2 dz dy dx x 2 1 2x 2 1 1 0 xy 1 1 1 18. 0 0 dz dy dx 0 0 xy dy dx 0 x dx 2 x2 x2 4 1 0 1 4 6 36 0 x2 36 0 x 2 y 2 6 36 0 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 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 z2 y2 24. Top plane: x y z y2 6 9 6 z 26. Elliptic cone: 4x 2 z 5 4 3 2 Side cylinder: x2 3 0 0 9 y2 6 0 x y dz dx dy 3 6 x x 3 y 3 2 1 6 1 5 y 4 0 z 4 0 y 2 z2 2 dx dy dz 28. Q x, y, z : 0 ≤ x ≤ 2, x 2 ≤ y ≤ 4, 0 ≤ z ≤ 2 2 4 2 x 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 y 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 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 104 21 xyz dx dz dy 0 0 dx dz dy 390 30. Q Chapter 13 Multiple Integration x2, 0 ≤ z ≤ 6 6 z x, y, z : 0 ≤ x ≤ 1, y ≤ 1 1 1 0 1 0 6 0 1 0 6 0 1 0 0 0 1 x2 0 1 x2 0 1 y2 1 y2 x2 6 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 y2 xyz dx dz dy xyz dx dy dz 2 x 1 1 2 y xyz dy dz dx xyz dy dx dz 5 5 0 5 0 x 1 5 15 0 3x 3y 32. m Mxz y 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 b a1 0 a[1 0 (y b) c1 0 (y b) (x a) 34. m k 0 b (y b)] c[1 0 (y b) (x a)] dz dx dy kabc 6 kab2c 24 k Mxz m 2 Mxz y k 0 y dz dx dy kab2c 24 kabc 6 b 4 Mxz m a b 0 b 0 b 0 b 0 c 36. 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 k k 0 a 0 c xz dz dy dx yz dz dy dx 0 0 k Myz m Mxz m Mxy m ka2bc2 4 kabc2 2 kab2c2 4 kabc2 2 kabc3 3 kabc2 2 a 2 b 2 2c 3 38. z will be greater than 8 5, whereas x and y will be unchanged. 40. x, y and z will all be greater than their original values. S ection 13.6 2 4 0 x2 y Triple Integrals and Applications 391 42. m 2k 0 2 0 dz dy dx 16k 3 x dz dy dx 2 2 0 4 0 4 0 x2 x2 0 y k 0 2 4 4 x 2 dx x2 y Myz Mxz Mxy x y z 44. x m k 0 2k 0 2 0 y y dz dy dx 2k 2k 0 0 z dz dy dx 0 16k 3 2k 16k 3 k 16k 3 0 3 8 3 16 k Myz m Mxz m Mxy m 0 2 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 12x 3 2 k 0 1 2 dy dx 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 ln 4 2 4 y 5 k 46. f x, y m k 1 60 15 5 0 5 0 (3 5)x 0 (3 5)x 0 (3 5)x 0 (3 5)x 20y (1 15)(60 12x 20y) dz dy dx 0 3 (1 15)(60 0 3 0 3 (1 15)(60 0 12x 20y) (1 15)(60 12x 20y) 12x 20y) 10k 25k 2 15k 2 10k 4 3 2 1 x y = 3 (5 - x) 5 Myz Mxz Mxy x y z k 0 5 x dz dy dx 1 2 3 4 5 k 0 5 y dz dy dx k 0 z dz dy dx 25k 2 10k 15k 2 10k 10k 10k 1 5 4 3 4 Myz m Mxz m Mxy m 392 Chapter 13 Multiple Integration a2 a2 a2 a2 48. (a) Ixy Ixz Ix (b) Ixy k a2 a2 z2 dz dy dx ka5 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 4 2 0 4 0 y2 4 2 0 50. (a) Ixy k 0 z3 dz dy dx 4 0 4 2 k 0 1 4 4 y 2 4 dy dx k 4 k 4 Ixz k 256 0 256y 2 256y3 3 96y4 96y5 5 k 0 16y6 16y7 7 4 2 0 y8 dy dx y9 9 2 4 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 k 0 12 x4 2 k 2 4 y 2 2 dy dx 8y3 3 Ixz y5 5 2 k 0 8y 2 y4 dy dx x 2 16y 0 dx 0 k 2 4 0 256 2 x dx 15 8192k 45 Ix Ixz Ixy 2048k , Iy 9 Iyz Ixy 8192k , Iz 21 Iyz 63,488k 315 —CONTINUED— S ection 13.7 50. —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 in Cylindrical and Spherical Coordinates 393 (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 b2 k 0 x 2z dz dy dx Ixy 118,784k , Iz 315 a2 Ix Ixz Iyz c2 a2 a2 a2 a2 a2 a2 52. Ixy c2 c2 b2 b2 z2 dz dy dx b3 12 b c2 c2 c2 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 x2 y2 ab c2 x 2 dx abc3 12 12 c abc 12 Ix Iy Iz b2 c2 c2 1 1 1 x2 4 54. 1 x2 0 kx 2 x 2 y 2 dz dy dx 56. 6 58. Because the density increases as you move away from the axis of symmetry, the moment of intertia will increase. Section 13.7 4 2 0 2 0 r Triple Integrals in Cylindrical and Spherical Coordinates 4 2 0 4 0 2. 0 rz dz dr d 0 rz2 2 2 2 0 r dr d 4r 2 r 3 dr d 1 2 2 4 1 2 2 2 4r 0 2r 2 0 4r 3 3 r4 4 2 d 0 2 3 4 d 0 6 2 3 4. 0 0 0 e 2 ddd 0 0 1 e 3 2 3 dd 0 0 0 1 1 3 2 e 8 dd 6 1 e 8 394 Chapter 13 4 4 0 cos 2 0 0 Multiple Integration 1 3 1 3 1 3 4 0 4 0 4 0 0 4 4 6. sin cos ddd cos3 sin sin 0 4 sin cos sin cos d cos cos dd 1 sin3 3 sin2 4 dd cos sin d 0 4 0 52 36 2 sin 0 5 2 sin2 36 2 52 144 8. 0 0 0 2 cos 2 ddd 8 9 2 3 0 3 0 r2 2 0 2 0 2 0 0 3 10. 0 r dz dr d r3 3r 2 2 9 d 4 r4 4 9 2 r 2 dr d 4 z 3 d 0 2 3 x 2 3 y 2 5 2 12. 0 0 2 sin ddd 117 3 117 3 468 3 2 sin 0 2 0 dd 7 z r=5 r=2 cos 0 0 d 7 x 7 y 2 2 0 0 6 0 16 r2 14. (a) 0 2 4 3 0 0 r 2 dz dr d sin2 ddd 8 3 2 2 2 0 3 2 6 2 csc 3 4 (b) sin2 ddd 8 3 2 2 2 3 2 1 0 0 1 r2 2 0 1 3 16. (a) 0 r r2 z2 dz dr d 8 2 4 2 20 16 r2 (b) 0 0 sin ddd 8 18. V 2 3 2 2 0 2 r 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 2 r 16 1 16 3 4 r 2 dr d x 7 7 y 4 4 82 3 82 3 r2 32 2 2 d 82 3 64 3 2 2 64 2 3 Section 13.7 Triple Integrals in Cylindrical and Spherical Coordinates 2 395 2 2 0 2 r 4 r2 2 2 0 12e 0 r 2 2 20. V 0 2 0 2 0 0 r dz dr d 22. 0 k r dz dr d 0 2 0 12ke 6ke 0 2 r 2 r dr d 2 r4 1 4 3 2 r2 r 2 dr d r3 3 2 r 2 0 r2 32 d 0 6ke 0 4 6k d 8 2 3 3k 1 e 4 24. x m Mxy y 0 by symmetry 26. x m z r dz dr d kz y 4k 0 0 0 1 r 2hk from Exercise 23 30 2 r0 0 2 0 h r0 0 r0 r r0 0 by symmetry 2 r0 h(r0 r) r0 zr dz dr d 4k 0 2kh r02 2 r0 0 4 2r 2r0 k r0 12 3 r02hk 2 r2 r3 dr d Mxy 1 k r02h2 12 2 r0 0 h(r0 0 r) r0 2kh r02 z 2 r0 12 h2 h 4 4k 0 z2r dz dr d 2 2h2 Mxy m k r0 12 1 k r02h3 30 z Mxy m k r02h3 30 k r02h2 12 2h 5 28. Iz Q x2 2 r0 0 2 y2 h r0 0 r0 0 x, y, z dV r r0 30. m Iz k a2h 2 2a sin 0 h 2k 0 0 r 3 dz dr d 4k 0 r 4 dz dr d r0 r0 5 4kh 0 2 r r 4 dr d 3 k a4h 2 32 ma 2 4kh 0 2 r 5 r 6 r0 d 6r0 0 r05 d 6 4kh 0 2 r05 5 15 rd 30 0 4kh 0 4kh 15 r 30 0 2 15 r kh 15 0 396 Chapter 13 4 2 0 b Multiple Integration 32. V 8 0 a 2 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 34. m 8k sin2 ddd 36. x m y k 4k 0 0 by symmetry 23 R 3 2 2ka4 0 2 0 sin2 dd Mxy 23 r 3 2 0 r 2 R 3 2 k R3 3 cos 2 r3 ddd sin k a4 0 sin2 1 2 1 k 4 d 1 sin 2 4 2a4 2 0 k a4 k a4 1 k R4 2 1 k R4 4 r4 0 0 2 sin 2 d d sin 2 d 0 2 r4 4 1 k R4 8 z Mxy m k R4 2k R3 r 4 cos 2 0 1 k R4 4 r4 r3 r4 r4 4 r3 3 3 R4 8 R3 2 2 0 r R 4 38. Iz 4k 0 sin3 2 ddd sin3 dd cos2 cos3 3 d 2 0 40. x y z sin sin cos cos sin tan cos 2 x2 y x x2 y2 z2 4k 5 R 5 2k 5 2k 5 R5 R5 2 r5 0 0 2 r5 0 sin cos 1 z y2 z2 r5 r5 4k R5 15 2 2 2 42. 1 1 1 f sin cos , sin sin , cos 2 sin ddd 44. (a) You are integrating over a cylindrical wedge. (b) You are integrating over a spherical block. . in a z 46. 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 ρi sin φi ∆ θi ∆ ρi ρi ∆φi sin sin x y S ection 13.8 Change of Variables: Jacobians 397 Section 13.8 2. x y au cu xy uv bv dv yx uv Change of Variables: Jacobians 4. x y ad cb uv uv xy uv yx uv v 2u vu 2u 2u 6. x y u v xy uv a a yx uv 11 00 1 8. x y u v u xy uv v yx uv 1 1 v 1 u v2 1 v u v2 u v2 v 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) 12. x y 1 u 2 1 u 2 xy uv 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 v 3 23 15 2u2 2 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 15 2 3 26 3 120 398 Chapter 13 1 u 2 1 u 2 xy uv 4x R Multiple Integration v 14. x y 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 ueu 2 eu 2 0 21 e 2 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 4 4 3 sin u du 1 3 cos u 1 3 cos 1 cos 4 3.5818 18. u u x x x 1 u 2 x, y u, v x y y , 2, v, 1 2 v v y x x 1 u 2 y y v 0 3π 2 y 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 8 R 1 du dv 2 cos 2v 2 2 dv 0 73 v 12 1 sin 2v 2 0 74 12 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 x x 3v y 5 2y − x = 8 (2, 5) 3x + 2y = 16 (−2, 3) 3 2 (4, 2) 2y − x = 0 xy uv 3x R 13 48 x 32 1 8 8 16 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 Section 13.8 22. u u x x x u, xy uv yx uv 1 u 4 1 4 1 4 1 Change of Variables: Jacobians 399 1, 4, v v y xy xy v u 1 4 4 3 y x=1 xy = 4 2 x=4 1 x 1 2 3 4 xy = 1 R xy dA 1 x 2y 2 v 1 1 dv du v2 u 4 1 ln 1 2 v2 1 1 du u 1 ln 17 2 4 ln 2 ln u 1 1 17 ln ln 4 2 2 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 12dv 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 x2 a2 7 sin 2 16 (b) f x, y x2 a2 A cos y2 ≤1 b2 2 y2 b2 R: Let x au and y f x, y dA bv. 1 1 1 u2 A cos 1 u2 R 2 u2 v 2 ab dv du Let u 2 r cos , v 1 r sin . r r dr d Aab 2r sin 2 r 2 0 4 2 Aab 0 0 cos 2 cos r 2 4 2 1 2 0 2 Aab 0 4 2 Aab 400 Chapter 13 Multiple Integration 28. x 4u x, y, z u, v, w v, y 4 0 1 4v 1 4 0 w, z 0 1 1 u 17 w 26. See Theorem 13.5. 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 Review Exercises for Chapter 13 2y 2. y x2 y 2 dx x3 3 2y xy 2 y 10y3 3 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 43 x 3 14 x 2 25 x 5 2 0 88 15 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 2 x 3 6 0 2x 2 6 y y2 8. 0 0 dy dx 2 2 6 y 2 y2 dy dx 0 dx dy A 0 dx dy 1 2 1 6y 2 1 1 y 6 0 3y dy 32 y 2 8 2 3 0 4 6x x2 x2 0 1 1 3 1 9 y 9 3 3 9 9 y 10. 0 2x dy dx 11 4 6x x2 x2 y 4 dy dx 0 y dx dy 8 y dx dy 4 0 A 0 2x dy dx 0 8x 2x2 dx 4x2 23 x 3 64 3 2 y2 1 1 2 5 2 12. A 0 0 dx dy 0 0 dy dx 1 x 1 dy dx 14 3 1 1 x 3 2y y y2 0 1 x 1 x 1 1 1 14. A 0 dx dy 3 dy dx 0 x dy dx 9 2 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) ex 0 y dy dx 3 ex y dy dx 35 e 5 e3 2 5 e5 e3 85 e 5 2 5 2 1 x 1 2 3 4 5 Review Exercises for Chapter 13 401 3 x 18. V 0 3 0 x y dy dx 12 y 2 x 20. Matches (c) 3 z xy 0 dx 0 2 1 y 2 3 2 3 x2 0 dx 2 x 13 x 2 3 0 27 2 1 x 1 22. 0 0 k xy dy dx 0 1 0 k xy 2 2 kx 3 dx 2 1 0 x 1 1 2 2 dx 0 24. False, 0 0 x dy dx 1 1 x dy dx kx 4 8 Since k 8 0.5 k 8 8. 0.03125 1, we have k 0.25 P 0 0 8xy dy dx 1 1 0 26. True, 0 1 1 x2 1 1 0 dx dy < y2 1 1 x2 dx dy 0 4 4 16 0 y2 2 4 2 28. 0 x2 y 2 dx dy 0 0 r 3 dr d 0 r4 4 4 2 d 0 0 64 d 32 2 R 30. V 8 0 b 2 R2 8 3 r 2r dr d R 32. tan 12 13 8 13 3 ⇒ 2 0.9828 R2 0 r2 2 32 b d The polar region is given by 0 ≤ r ≤ 4 and 0 ≤ ≤ 0.9828. Hence, arctan 3 2 4 82 R 3 4 3 R2 b2 32 0 d y r cos 0 0 r sin r dr d 288 13 b2 32 (8/ 13, 12/ 13) 4 3 2 1 x 2 + y 2 =16 y=2 x 3 θ x 1 8/ 13 4 402 Chapter 13 Multiple Integration 2 2 L h2 2 0 h2 2 0 L xL xL 34. m Mx k 0 L xL xL 2 2 dy dx y dy dx 0 2 kh 2 L 2 0 x L x dx L2 2 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 2 dx L2 3x2 L2 x3 L2 xL 2 2 L kh2 8 kh2 8 L 4 0 2x3 L3 x4 2L3 x dy dx x4 dx L4 x5 5L4 L 0 4x 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 kh 2 x 2 x3 3L x4 4L2 L 0 kh 2 5L2 12 5khL2 24 5khL2 24 17kh2L 80 12 7khL 12 7khL 51h 140 2 2 4 0 4 0 x 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 I0 m R x, y dA 16,384k 315 2 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 y2 4 7 128 k 15 x y Iy m Ix m 128 21 38. f x, y R fx 1 S 16 x x, y : 0 ≤ x ≤ 2, 0 ≤ y ≤ x 1, fy fx 2 2 0 2 2y fy 2 y 2 2 4y2 2 4y2 dx dy 0 22 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 R eview Exercises for Chapter 13 403 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 4,540 sq. ft. R 50 x 50 y 2 4 4 x2 x2 (x2 0 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 5 25 0 x2 0 25 x2 y2 44. 0 1 1 x2 y2 z2 2 2 0 5 0 2 dz dy dx 0 2 0 2 0 2 1 2 sin ddd 5 arctan 0 sin dd 2 5 0 arctan 5 cos 0 d 2 5 arctan 5 2 4 0 x2 0 4 x2 y2 46. 0 xyz dz dy dx 4 3 2 2 sin 0 16 0 r2 2 2 sin 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 a 0 cr sin 2 a 50. m Mxz Mxy x y z 2k 0 2 0 cr sin a 0 2 a 0 r dz dr d 2kc 0 0 r 2 sin dr d 2 a 2 kca3 3 dr d 2 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 kca4 8 2kca3 3 kc2 0 0 1 24 kc a 4 sin2 0 d 3a 16 3 ca 32 kc2a4 16 2kca3 3 404 Chapter 13 Multiple Integration 52. m 500 3 500 3 2 0 2 3 0 3 0 3 0 2 0 4 25 r2 r dz d dr 1 25 3 r2 32 500 3 3 3 0 2 r 25 0 r2 18 4r d dr 125 3 500 3 14 3 162 2r 2 0 500 3 2 64 3 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 81 1 4 162 2 a 2 54. Iz k 0 0 0 sin2 2 sin ddd 4k a6 9 56. x 2 y2 z2 a2 x2 Q 2 1 0 r2 1 y2 d V z2 z2 a2 a2 1 1 y 2 z2 y 2 z2 a2 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 paraboloid and above the semi-circle 4 x2 in the xy-plane. y Iz a a 1 1 x2 a2 y 2 dx dy dz 8 a 15 x, y u, v xy uv 2u 2v yx uv 2u 2v 8uv y 60. 62. x x, y u, v u, y xy uv v ⇒u u xy vu x, v 1 1 u xy 0 1 u 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 ...
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