EVNREV13 - 400 Chapter 13 Multiple Integration 28. x 4u x,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 Problem Solving for Chapter 13 405 Problem Solving for Chapter 13 1 d c 2 5 4 2 2. z fx 1 S ax by b c Plane 4. A: 0 r 16 10 r2 r dr d 160 r2 r dr d 160 1333 960 523 960 4.36 ft3 1.71 ft3 a ,f cy fx2 fy2 1 R B 0 9 r 16 1 a2 c2 c 2 a2 c2 b2 c2 The distribution is not uniform. Less water in region of greater area. In one hour, the entire lawn receives 2 0 10 0 b2 dA c2 dA R a 2 b c 2 r 16 r2 r dr d 160 125 12 32.72 ft3. a2 b2 c c2 AR 2 2 0 2 4 0 8 r2 6. (a) V 0 2 2 2 r dz dr d 2 0 2 sec 8 42 3 ddd 5 5 (b) V sin 8 42 3 8. Volume 5 6 5 54 84 m3 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 (PS #9) 12. Essay 14. The greater the angle between the given plane and the xyplane, the greater the surface area. Hence: z2 < z1 < z4 < z3 ...
View Full Document

This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

Ask a homework question - tutors are online