6.2-6.3 - 406 Chapter 6 Applications of Definite Integrals...

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406 Chapter 6 Applications of Definite Integrals 6.2 VOLUME BY CYLINDRICAL SHELLS 1. For the sketch given, a 0, b 2; œœ V 2 dx 2 x 1 dx 2 x dx 2 2 œ œ ±œ ±œ± œ ± '' ' a0 0 b2 2 11 1 1 1 ˆ‰ Š Š‹ shell shell radius height 4 4 16 16 xx x x 4 1 6 #$ # % ## # ! 236 2. For the sketch given, a 0, b 2; V 2 dx 2 x 2 dx 2 2x dx 2 x 2 (4 1) 6 œ œ ²œ ²œ² œ ² œ ' 0 2 1 1 1 1 Š Š shell shell radius height 4 4 16 x % # # ! 3. For the sketch given, c 0, d 2; È V 2 dy 2 y y dy 2 y dy 2 2 œ œ œ ' c0 0 d2 2 1 1 1 ab shell shell radius height 4 y ÈÈ # ! % È 4. For the sketch given, c 0, d 3; È V 2 dy 2 y 3 3 y dy 2 2 ² ² œ œ œ ' 0 d3 3 1 1 cd shell shell radius height 4 y 3 9 ! # % È 1 5. For the sketch given, a 0, b 3; È V 2 dx 2 x x 1 dx; ± b3 Š È shell shell radius height È # u x 1 du 2x dx; x 0 u 1, x 3 u 4 ’“ È œ±Ê œ œÊœ œ Êœ # V u du u 4 1 (8 1) Äœ œ œ ' 1 4 "Î# $Î# $Î# % " ±‘ ˆ ˆ 22 2 1 4 33 3 3 1 6. For the sketch given, a 0, b 3; V 2 dx 2 x dx; Š shell shell radius height 9x x9 È $ ± u x 9 du 3x dx 3 du 9x dx; x 0 u 9, x 3 u 36 c d Ê œ œÊœ œÊœ $# # V 3u du 6 2u 12 36 9 36 œ œ ² œ 1 1 ' 9 36 ²"Î# "Î# $' * 7. a 0, b 2; V 2 dx 2 x x dx ² ² ± shell shell radius height 2 x 2 x dx 3x dx x 8 œ œ 00 1 1 $ # # ! 3 8. a 0, b 1; V 2 dx 2 x 2x dx ² b1 ˆ shell shell radius height 2 x 2 dx x œ œ 1 1 ' 0 1 1 0 ' 3x # # " !
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Section 6.2 Volume by Cylindrical Shells 407 9. a 0, b 1; œœ V 2 dx 2 x (2 x) x dx ± ± '' a0 b1 11 ˆ‰ Š‹ cd shell shell radius height # 2 2 xx x d x2x œ± ± œ ± ± ' 0 1 ab ’“ #$ # " ! xx 34 $ % 21 2 ± œ œ œ ˆ "" ± ± 3 4 12 12 6 1 243 1 0 5 10. a 0, b 1; V 2 dx 2 x 2 x ± ± shell shell radius height ## 2 x2 2 x d x 4 x x d x œ ± 00 a b 44 œ ± œ 1 42 4 # % # " ! 11. a 0, b 1; V 2 dx 2 x x (2x 1) dx ± ± ± È shell shell radius height 2 x 2x 2 x x x ² œ ± ² ' 0 1 ± $Î# # &Î# $ # " # " ! 22 53 ² œ œ ˆ 2 2 12 20 15 7 5 3 30 15 ² # 1 12. a , b 4; œ" œ V 2 dx 2 x x dx a1 b4 ˆ shell shell radius height 2 3 ±"Î# 3 x dx 3 x 2 4 œ ± " 1 ' 1 4 "Î# $Î# $Î# % " ±‘ ˆ 2 3 2( 8 1 ) 1 4 œ 13. (a) xf(x) xf(x) ; since sin 0 0 we have x, 0 x x, x 0 sin x, 0 x 0, x 0 œÊ œ œ ³Ÿ œ œ sin x x 1 1 xf(x) xf(x) sin x, 0 x sin x, 0 x sin x, x 0 œ Ÿ Ÿ œ œ 1 1 (b) V 2 dx 2 x f(x) dx and x f(x) sin x, 0 x by part (a) œ Ÿ Ÿ b 1 shell shell radius height 1 †† V 2 sin x dx 2 [ cos x] 2 ( cos cos 0) 4 Êœ ² œ 1 1 1 ' 0 1 1 ! 14. (a) xg(x) xg(x) ; since tan 0 0 we have 0 x x 0, x 0 tan x, 0 x /4 0, x 0 œ œ œ œ tan x x4 # 1 # 1 xg(x) xg(x) tan x, 0 x /4 tan x, 0 x /4 tan x, x œ Ÿ Ÿ œ œ # # # 1 1
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408 Chapter 6 Applications of Definite Integrals (b) V 2 dx 2 x g(x) dx and x g(x) tan x, 0 x /4 by part (a) œœ œ Ÿ Ÿ '' a0 b4 11 1 ˆ‰ Š‹ shell shell radius height 1 Î †† # V 2 tan x dx 2 sec x 1 dx 2 [tan x x] 2 1 Êœ œ ± œ ± œ ±œ 00 44 ÎÎ ## Î% ! ± # ab 1 1 4 4 # 15. c 0, d 2; V 2 dy 2 y y ( y) dy ± ± c0 d2 ± È shell shell radius height 2 y y dy 2 œ² œ ² ' 0 2 ’“ $Î# # # !
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6.2-6.3 - 406 Chapter 6 Applications of Definite Integrals...

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