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 '' ' ' 00 0 0 32 3 3 ab ’“ ±œ ± œœ # # ! y 33 16 $ 2. (x y 2xy dy dx xy dx ' 02 0 30 3 ## ! ±# ± xy 2 4x 2x dx 2x 0 œ± œ ± œ ' 0 3 $ ! 2x 3 $ 3. (x y 1) dx dy yx x dy ' 11 1 01 0 ²² œ ² ² x 2 # " ±" (2y 2) dy y 2y 1 œ² œ ² œ ' 1 0 cd # ! ±" 4. (sin x cos y) dx dy ( cos x) (cos y)x dy ' 1 22 0 ²œ ± ² 1 ! ( cos y 2) dy sin y 2y 2 œ ² œ ' 1 1 2 1 # 1 1 5. (x sin y) dy dx x cos y dx ' 0 x x ! (x x cos x) dx (cos x x sin x) œ ± ² ' 0 1 x 2 # 1 ! 2 1 # # 6. y dy dx dx sin x dx ' ' 0 0 sin x sin x 1 y 2 # ! " # # (1 cos 2x) dx x sin 2x œ ± œ "" " ! 44 2 4 ' 0 1 ± 1 1
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942 Chapter 15 Multiple Integrals 7. e dx dy e dy ye e dy '' ' ' 10 1 1 ln 8 ln y ln 8 ln 8 xy y y ln y œœ ± cd a b ! (y 1)e e 8(ln 8 1) 8 e œ± ± œ ± ± ² yy ln 8 1 8 l n 81 6e ² 8. dx dy y y dy ' 1y 1 2y 2 # œ ± ab’“ # # # " 3 $ # 2 œ ± ± ± œ±œ ˆ‰ 87 3 5 33 3 6 "" ## 9. 3y e dx dy 3y e dy ' 00 0 11 0 y y # # $# xy xy œ 3y e 3y dy e y e 2 œ ± œ ± ' 0 1 Š‹ $ " ! $$ 10. e dy dx x e dx ' 1 4x 4 yx x 0 È ÈÈ 2 # ÎÎ œ ± È È (e 1) x dx (e 1) x 7(e 1) œ ± œ ± 2 23 # $Î# % " ' 1 4 È ± 11. dy dx x ln y dx (ln 2) x dx ln 2 ' ' 1x 1 1 22 x 2 2 2x x x3 y œ # 12. dy dx (ln 2 ln 1) dx (ln 2) dx (ln 2) ' ' 1 1 2 2 1 xy x œ œ " # 13. x y dy dx x y dx x (1 x) dx x x dx ' ' ' 0 0 0 x 1 1 1 x 0 " ab ’“ # # #$ ±± ²œ ± ² œ ± ² y ( ) ( ) 3 $ 0 œ ±± œ±±±±± œ ˆ xx 34 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 1 1 œ ± œ ± ± ± œ ± 1 1 ! " ! 2 15. v u dv du v u du u(1 u) du ' ' 0 0 u 1 1 u 0 " È ±œ ± œ ± ± v1 2 u u 2 ±² # uu u d u u u œ ±² ± ² œ ± ² ± ² œ±²±²œ±²œ± ' 0 1 " """ " " # # # # "Î# $Î# $Î# &Î# " ! u u uu2 2 2 2 2 2 6 3 5 635 5 1 0 $
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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 '' ' ' 10 1 1 2l n t 2 2 ln t 0 ss tt 24 œœ ± œ ± ± ² cd ’“ ## # " ( 2 l n 212 l n 22 ) 1 œ± ± ² ± ± ² œ ˆ‰ "" 44 17. 2 dp dv 2 p dv 2 2v dv ' ' 2v 2 2 0v 0 0 v v ± ± 8 œ # 0 2 18. 8t dt ds 4t ds ' 00 0 11 s 1 1s 0 È È # # œ # 41 s d s 4s œ ± œ ' 0 1 ab # " ! s8 33 $ 19. 3 cos t du dt (3 cos t)u ' Î Î ÎÎ 30 3 3s e c t 3 sec t 0 œ 3 dt 2 ' Î Î 1 1 3 3 1 20. dv du du ' 01 0 34 2 u 3 42 u 1 u 2 u4 vv ±± # œ ± (3 u) du 3u u # œ ± œ ! ' 0 3 # $ ! 21. dx dy 20 y ) 2 Ð Î 22. dy dx 0x 2
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944 Chapter 15 Multiple Integrals 23. dy dx '' 0x 1x # 24. dx dy 01 y 11 y È 25. dx dy 1l n y e1 26. dy dx 10 2l n x 27. 16x dx dy 00 99 y 1 2 È 28. y dy dx 44 x È
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Section 15.1 Double Integrals 945 29. 3y dy dx '' 10 11 x È # 30. 6x dx dy 20 24 y È # 31. dy dx dx dy sin y dy 2 ' 0x 00 0 y 1 1 sin y sin y yy œœ œ 32. 2y sin xy dy dx 2y sin xy dx dy 22 2y ## œ 2y cos xy dy 2y cos y 2y dy œ± ² y 0 cd a b # sin y y 4 sin 4 ² œ ± # ! 33. x e dx dy x e dy dx xe dx ' 0y 0 1x 1 xy xy xy x 0 xe x dx e œ ± œ ' 0 1 xx ab ’“ # " !
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ISM_T11_C15_A - CHAPTER 15 MULTIPLE INTEGRALS 15.1 DOUBLE...

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