ISM_T11_C15_A - CHAPTER 15 MULTIPLE INTEGRALS 15.1 DOUBLE...

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CHAPTER 15 MULTIPLE INTEGRALS 15.1 DOUBLE INTEGRALS 1. 4 y dy dx 4y dx dx 16 ' ' ' ' 0 0 0 0 3 2 3 3 a b œ œ œ # # ! y 3 3 16 2. (x y 2xy dy dx xy dx ' ' ' 0 2 0 3 0 3 a b # # ! # œ x y 2 4x 2x dx 2x 0 œ œ œ ' 0 3 a b # # $ ! 2x 3 3. (x y 1) dx dy yx x dy ' ' ' 1 1 1 0 1 0 œ x 2 " " (2y 2) dy y 2y 1 œ œ œ ' 1 0 c d # ! " 4. (sin x cos y) dx dy ( cos x) (cos y)x dy ' ' ' 2 2 0 œ c d 1 ! ( cos y 2) dy sin y 2y 2 œ œ œ ' 2 1 1 1 c d # 1 1 5. (x sin y) dy dx x cos y dx ' ' ' 0 0 0 x x œ c d ! (x x cos x) dx (cos x x sin x) œ œ ' 0 x 2 1 ! 2 œ 1 # 6. y dy dx dx sin x dx ' ' ' ' 0 0 0 0 sin x sin x œ œ y 2 ! " # # (1 cos 2x) dx x sin 2x œ œ œ " " " ! 4 4 2 4 ' 0 1 1
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942 Chapter 15 Multiple Integrals 7. e dx dy e dy ye e dy ' ' ' ' 1 0 1 1 ln 8 ln y ln 8 ln 8 x y x y y y ln y œ œ c d a b ! (y 1)e e 8(ln 8 1) 8 e œ œ c d y y ln 8 1 8 ln 8 16 e œ 8. dx dy y y dy ' ' ' 1 y 1 2 y 2 œ œ a b # # # " y y 3 2 œ œ œ ˆ ˆ 8 7 3 5 3 3 3 6 " " # # 9. 3y e dx dy 3y e dy ' ' ' 0 0 0 1 1 0 y y $ # xy xy œ c d 3y e 3y dy e y e 2 œ œ œ ' 0 1 Š # # $ " ! y y 10. e dy dx x e dx ' ' ' 1 0 1 4 x 4 y x y x x 0 3 3 2 # œ È È (e 1) x dx (e 1) x 7(e 1) œ œ œ 3 3 2 2 3 # $Î# % " ' 1 4 È ˆ ‰ 11. dy dx x ln y dx (ln 2) x dx ln 2 ' ' ' ' 1 x 1 1 2 2x 2 2 2x x x 3 y œ œ œ c d # 12. dy dx (ln 2 ln 1) dx (ln 2) dx (ln 2) ' ' ' ' 1 1 1 1 2 2 2 2 1 xy x œ œ œ " # 13. x y dy dx x y dx x (1 x) dx x x dx ' ' ' ' ' 0 0 0 0 0 1 1 x 1 1 1 x 0 a b # # # # # $ œ œ œ y (1 x) (1 x) 3 3 3 0 0 0 œ œ œ ˆ ˆ x x 3 4 1 3 4 1 6 (1 x) # # " ! " " " " 14. y cos xy dx dy sin xy dy sin y dy cos y ( 1 1) ' ' ' ' 0 0 0 0 1 1 1 œ œ œ œ œ c d 1 1 1 1 ! " " " ! 1 1 2 15. v u dv du v u du u(1 u) du ' ' ' ' 0 0 0 0 1 1 u 1 1 u 0 ˆ È È È œ œ v 1 2u u 2 # u u u du u u œ œ œ œ œ ' 0 1 Š " " " " " " # # # # # # "Î# $Î# $Î# &Î# " ! u u u u 2 2 2 2 2 2 6 3 5 6 3 5 5 10
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Section 15.1 Double Integrals 943 16. e ln t ds dt e ln t dt (t ln t ln t) dt ln t t ln t t ' ' ' ' 1 0 1 1 2 ln t 2 2 ln t 0 s s t t 2 4 œ œ œ c d # " (2 ln 2 1 2 ln 2 2) 1 œ œ ˆ " " 4 4 17. 2 dp dv 2 p dv 2 2v dv ' ' ' ' 2 v 2 2 0 v 0 0 v v œ œ c d 2 v 8 œ œ c d # 0 2 18. 8t dt ds 4t ds ' ' ' 0 0 0 1 1 s 1 1 s 0 œ c d # 4 1 s ds 4 s œ œ œ ' 0 1 a b # " ! s 8 3 3 19. 3 cos t du dt (3 cos t)u ' ' ' 3 0 3 3 sec t 3 sec t 0 œ c d 3 dt 2 œ œ ' 3 3 1 20. dv du du ' ' ' 0 1 0 3 4 2u 3 4 2u 1 4 2u 2u 4 v v œ (3 u) du 3u u œ # œ œ ! ' 0 3 c d # $ ! 21. dx dy ' ' 2 0 4 4 y) 2 22. dy dx ' ' 2 0 0 x 2
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944 Chapter 15 Multiple Integrals 23. dy dx ' ' 0 x 1 x 24. dx dy ' ' 0 1 y 1 1 y 25. dx dy ' ' 1 ln y e 1 26. dy dx ' ' 1 0 2 ln x 27. 16x dx dy ' ' 0 0 9 9 y 1 2 28. y dy dx ' ' 0 0 4 4 x
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Section 15.1 Double Integrals 945 29. 3y dy dx ' ' 1 0 1 1 x 30. 6x dx dy ' ' 2 0 2 4 y 31. dy dx dx dy sin y dy 2 ' ' ' ' ' 0 x 0 0 0 y sin y sin y y y œ œ œ 32. 2y sin xy dy dx 2y sin xy dx dy ' ' ' ' 0 x 0 0 2 2 2 y # # œ 2y cos xy dy 2y cos y 2y dy œ œ ' ' 0 0 2 2 y 0 c d a b # sin y y 4 sin 4 œ œ c d # # # !
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