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t 4t 3t, 1 t 2 4 3 5 y e ds 3t e 5dtr i j i ja b c c c c d d dt dt x 4tr r a b' 'C 1 22 2 c a b 15 t e dt e e e e e c c c b c'c c1 2 1 16t 16t 64 16 16 6415 15 15 15 32 32 32 32 22 2 a b Copyright 2010 Pearson Education Inc. (a) t t 4t, 0 t 1 4 17 x 2y ds t 2 4t 17 dtr i j i ja b b b b bd d dt dt r r a b' 'C 0 1 17 9t dt 3 17 t dt 2 17 t 2 17 ' '0 0 1 1 2 3 1 ! (b) C : t t, 0 t 1 1 C : t t, 0 t 1 1" r i i r i j ja b a b b d d d d dt dt dt dt2 r r r r x 2y ds x 2y ds x 2y ds t 2 0 dt 1 2 t dt' ' ' ' 'C C C 0 0 1 2 a b a bb b b b b b b 1 2 t dt 1 2t dt t 1 2t b b b b b c ' '0 0 1 2 < a b2 1 2 1 3 3 3 3 3 3 2 3 1 2 3 2 5 5 5 5 1 ! ! b 21. (a) t t t, 0 t 4 x ds t dt t dt t 4 5r i j i ja b b b " " # ! d d dt dt 2 2 2 4 5 5 5 5 2 4 r r ' ' 'C 0 0 4 4 (b) t t t, 0 t 2 2t 1 4t x ds t 1 4t dtr i j i ja b b b b b2 d d dt dt 2 2r r ' 'C 0 2 1 4t b a b1 12 12 2 3 2 2 17 17 ! c " 20. (t) a cos t a sin t, 0 t 2 ( a sin t) (a cos t) a sin t a cos t a r j k j k b c b b a b a b k k 1 d d dt dt r r # x z 0 a sin t f(x y z) ds a sin t dt a sin t dt a sin t, 0 t a sin t, t 2 c b c b c b c k k k k k k k k#1 1 1 ' ' 'C 0 21 1 1 a cos t a cos t a ( 1) a a a ( 1) 4a c c c c c c cc d c d c d c d# ! #1 1 1 19. (t) t t t, 0 a t b 3 r i j k i j k b b b b d d t t t 1 dt dt x y z t t t t x y zr r b b b b b b b b # f(x y z) ds 3 dt 3 ln t 3 ln, since 0 a b ' 'C a b b a k k1 b t a 18. C : (t) t, 0 t 1 1 x y z 0 0 t t" # r k k b c b c cd d dt dt r r f(x y z) ds t (1) dt c c c' 'C 0 1 " a b # " ! "t 3 3 $ C : (t) t, 0 t 1 1 x y z 0 t 1 t 1# r j k j b b c b c cd d dt dt r r f(x y z) ds t 1 (1) dt t t 1 c c c c' 'C 0 1 # < 2 2 3 3 3 $# " ! " C : (t) t, 0 t 1 1 x y z t 1 1 t$ # r i j k i b b b c b c d d dt dt r r f(x y z) ds (t)(1) dt f(x y z) ds f ds f ds f ds b b c b c b' ' ' ' ' 'C 0 C C C C 1 $ " # $ t 2 3 3 # " ! " " " " # c " 6 17.
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C : (t) t t, 0 t 1 2t 1 4t x y z t t 0 t t 2t" #r i j i j b b b b c b c b d d dt dt r r k k since t 0 f(x y z) ds 2t 1 4t dt 4t (5) 5 5 1 b " b c c' 'C 0 1 " a b # " " " "# $#$# " !6 6 6 6 C : (t) t, 0 t 1 1 x y z 1 1 t 2 t# r i j k k b b b c b c cd d dt dt r r f(x y z) ds 2 t (1) dt 2t t 2 therefore f(x y z) ds c c c ' ' 'C 0 C 1 # a b < # $" "" !3 3 3 5 f(x y z) ds f(x y z) ds 5 b b' 'C C" # 5 3 6 # 16. 940 Chapter 16 Integration in Vector Fields 15. (t) t t t, 1 t 3 r i j k i j k b b _ b b d d dt dt x y z t t t 3t 3 3 3r r #b b b b f(x y z) ds 3 dt lim 1 1 c c b ' 'C 1 _ < 3 3t t b 1 # _ " " b _ Copyright 2010 Pearson Education Inc.
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(t) ( 2 3 ) t( 3 2 ) (1 t) (2 3t) (3 2t), 0 t 1 3 2r i j k i j k i j k i j k b b b c c c c b c b c c c cd dt r 1 9 4 14 x y z (1 t) (2 3t) (3 2t) 6 6t f(x y z) ds b b b b c b c b c c d dt r 'C (6 6t) 14 dt 6 14 t 6 14 3 14 c c '0 1 t 2 # " ! " # 14. (t) (4 cos t) (4 sin t) 3t, 2 t 2 ( 4 sin t) (4 cos t) 3r i j k i j k b b c c b b1 1 d dt r 16 sin t 16 cos t 9 5 x y 16 cos t 16 sin t 4 f(x y z) ds (4)(5) dt b b b b d dt r # ' 'C 2 2 c 1 1 20t 80 c d# c# 1 1 1 13. (t) 2t t (2 2t), 0 t 1 2 2 4 1 4 3 xy y zr i j k i j k b b c b c b b b bd d dt dt r r (2t)t t (2 2t) f(x y z) ds 2t t 2 3 dt 3 t t 2t 3 2 b b c c b c b c b ' 'C 0 1 a b < # $ #" " # " ! 2 2 13 3 3 12. (t) t (1 t), 0 t 1 2 x t, y 1 t, and z 1 x y z 2r i j k i j b c b c c c b cd d dt dt r r t (1 t) 1 2 2t 2 f(x y z) ds (2t 2) 2 dt 2 t 2t 2 c c b c c c c c' 'C 0 1 c d# " ! 11. t t 1 t, 0 t 1 2 x t and y 1 t x y t ( t) 1r i j i j ja b b c c c b b " c a b d d dt dt r r f x y z ds f t 1 t 0 dt (1) 2 dt 2 t 2 c ' ' 'C 0 0 1 1 a b a b d dt r " ! 10. 2 cos t 2 sin t x 2 cos t and z 2 sin t x z 4 (h)r i k b b a b a b # 9. t 1 2t y t 1 and z 2t y 1 (f)r j k c b c c a b# z 4 # 8. t 2 2t y t and z 2 2t z 2 2y (b)r j k b c c c a b 7. t t t x t, y t, and z t (d)r i j k b b 6. 2 cos t 2 sin t x 2 cos t and y 2 sin t x y 4 (g)r i j b b a b a b # 4. t t x t and y 1 t y 1 x (c)r i j b " c c c a b 2. CHAPTER 16 INTEGRATION IN VECTOR FIELDS 16.1 LINE INTEGRALS 1.