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ISM_T11_C06_E - Chapter 6 Practice Exercises(d A =2/< 7/>29...

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Chapter 6 Practice Exercises 413 (d) : A+=2/< 7/>29. V R (x) r (x) dx œ ' a b 1 c d # # 4 dx œ 1 ' 1 2 ˆ ‰ ˆ 7 4 x # # # 16 1 2x x dx œ 49 4 1 1 ' 1 2 a b $ ' 16 x x œ 49 x 4 5 1 1 # # " 16 2 1 1 œ 49 4 4 5 3 5 1 1 ˆ ˆ " " " # 16 œ 49 4 4 160 5 1 1 ˆ " " " (40 1 32) œ œ œ 49 16 49 71 103 4 160 4 10 20 1 1 1 1 1 9. (a) : .3=5 7/>29. V x 1 dx (x 1) dx x œ œ œ 1 1 1 ' ' 1 1 5 5 Š È # & # " x 5 1 4 8 œ œ œ 1 1 1 ˆ ˆ ˆ 25 24 # # # " (b) : A+=2/< 7/>29. R(y) 5, r(y) y 1 V R (y) r (y) dy 25 y 1 dy œ œ Ê œ œ # # # # # ' ' c 2 d 2 1 1 c d a b 25 y 2y 1 dy 24 y 2y dy 24y y 2 24 2 8 œ œ œ œ 1 1 1 1 ' ' 2 2 2 2 a b a b ˆ % # % # $ # # y 5 3 5 3 2 32 2 32 3 (45 6 5) œ œ œ 1 ˆ 2 32 1088 5 3 15 15 " 1 1 (c) : .3=5 7/>29. R(y) 5 y 1 4 y œ œ a b # # V R (y) dy 4 y dy Ê œ œ ' ' c 2 d 2 1 1 # # # a b 16 8y y dy œ 1 ' 2 2 a b # % 16y 2 32 œ œ 1 1 ˆ 8y y 3 5 3 5 64 32 # # 64 1 (15 10 3) œ œ œ 1 ˆ 2 64 512 3 5 15 15 " 1 1 10. (a) : =2/66 7/>29. V 2 dy 2 y y dy œ œ ' ' c 0 d 4 1 1 ˆ Š Š shell shell radius height 4 y 2 y dy 2 2 œ œ œ 1 1 1 ' 0 4 Š ˆ # % ! y y y 4 3 16 3 4 64 64 64 œ œ 2 32 1 3 1 1 # (b) : =2/66 7/>29. V 2 dx 2 x 2 x x dx 2 2x x dx 2 x œ œ œ œ ' ' ' a 0 0 b 4 4 1 1 1 1 ˆ ˆ ˆ Š È shell shell radius height 5 3 4 x $Î# # &Î# % ! 2 32 œ œ 1 ˆ 4 64 128 5 3 15 1 (c) : =2/66 7/>29. V 2 dx 2 (4 x) 2 x x dx 2 8x 4x 2x x dx œ œ œ ' ' ' a 0 0 b 4 4 1 1 1 ˆ ˆ ˆ Š È shell shell radius height "Î# $Î# # 2 x 2x x 2 8 32 32 64 1 œ œ œ 1 1 1 ˆ ˆ 16 4 x 16 4 64 4 4 2 3 5 3 3 5 3 3 5 3 $Î# # &Î# % ! 64 1 œ œ 1 ˆ 4 64 5 5 1 (d) : =2/66 7/>29. V 2 dy 2 (4 y) y dy 2 4y y y dy œ œ œ ' ' ' c 0 0 d 4 4 1 1 1 ˆ Š Š Š shell shell radius height 4 4 y y # #
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414 Chapter 6 Applications of Definite Integrals 2 4y 2y dy 2 2y y 2 32 64 16 32 2 1 œ œ œ œ œ 1 1 1 1 ' 0 4 Š ˆ ˆ # # $ % ! y y 4 3 16 3 3 3 2 2 8 32 1 11. : .3=5 7/>29. R(x) tan x, a 0, b V tan x dx sec x 1 dx [tan x x] œ œ œ Ê œ œ œ œ 1 1 1 1 3 3 3 3 1 1 1 ' ' 0 0 3 3 # # Î$ ! a b Š È 12. : .3=5 7/>29. V (2 sin x) dx 4 4 sin x sin x dx 4 4 sin x dx œ œ œ 1 1 1 ' ' ' 0 0 0 # # # a b ˆ 1 cos 2x 4x 4 cos x 4 4 0 (0 4 0 0) 8 (9 16) œ œ œ œ 1 1 1 1 1 ˆ ˆ x sin 2x 9 4 # # # # ! 1 1 1 1 13. (a) : .3=5 7/>29. V x 2x dx x 4x 4x dx x x 16 œ œ œ œ 1 1 1 1 ' ' 0 0 2 2 a b a b ˆ # % $ # % $ # # ! x 4 32 32 5 3 5 3 (6 15 10) œ œ 16 16 15 15 1 1 (b) : A+=2/< 7/>29. V 1 x 2x dx dx x dx œ " œ " œ # œ # œ ' ' ' 0 0 0 2 2 2 1 1 1 1 1 1 1 a b a b # # # % " & & & # ! # ) a b x 1 (c) : =2/66 7/>29. V 2 dx 2 (2 x) x 2x dx 2 (2 x) 2x x dx œ œ œ ' ' ' a 0 0 b 2 2 1 1 1 ˆ Š c d a b a b shell shell radius height # # 2 4x 2x 2x x dx 2 x 4x 4x dx 2 x 2x 2 4 8 œ œ œ œ 1 1 1 1 ' ' 0 0 2 2 a b a b ˆ # # $ $ # $ # # ! x 4 32 4 3 3 (36 32) œ œ 2 8 3 3 1 1 (d) : A+=2/< 7/>29. V 2 x 2x dx 2 dx 4 4 x 2x x 2x dx 8 œ œ 1 1 1 1 ' ' ' 0 0 0 2 2 2 c d a b a b a b # # # # # # 4 4x 8x x 4x 4x dx 8 x 4x 8x 4 dx 8 œ œ 1 1 1 1 ' ' 0 0 2 2 a b a b # % $ # % $ x 4x 4x 8 16 16 8 8 (32 40) 8 œ œ œ œ œ 1 1 1 1 1 ˆ x 32 72 40 32 5 5 5 5 5 5 % # # !
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