ISM_T11_C06_B - Section 6.2 Volume by Cylindrical Shells 3....

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Section 6.2 Volume by Cylindrical Shells 371 3. For the sketch given, c 0, d 2; œœ È V 2 dy 2 y y dy 2 y dy 2 2 œ œ œ '' ' c0 0 d2 2 11 1 1 1 ˆ‰ Š‹ 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 ±± 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; ² a0 b3 Š È shell shell radius height È # u x 1 du 2x dx; x 0 u 1, x 3 u 4 ’“ È œ ² Êœ œ Ê œ œ Ê œ # V u du u 4 1 (8 1) Äœ œ œ ± 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 œ ² Ê œ œ Ê œ œ Ê œ $# # V 3u du 6 2u 12 36 9 36 œ œ ± œ 1 1 ' 9 36 ²"Î# "Î# $' * ± 7. a 0, b 2; V 2 dx 2 x x dx b2 ± 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 ± b1 ˆ shell shell radius height 2 x 2 dx x œ œ 1 1 ' 0 1 1 0 ' 3x # # " ! 9. a 0, b 1; V 2 dx 2 x (2 x) x dx shell shell radius height # 2 2 xx x d x2x œ œ ' 0 1 # " ! xx 34 $ % 21 2 œ œ ˆ "" ²² 3 4 12 12 6 1 243 1 0 5
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372 Chapter 6 Applications of Definite Integrals 10. a 0, b 1; œœ V 2 dx 2 x 2 x x dx ±± '' a0 b1 11 ˆ‰ Š‹ cd ab shell shell radius height ## 2 x2 2 x d x 4 x x d x œ ± œ ± 00 a b #$ 44 œ ± œ ± œ 1 ’“ xx 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 œ ±² œ ' 0 1 ± $Î# # &Î# $ # " # " ! 22 53 œ ˆ 2 2 12 20 15 7 5 3 30 15 " # 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 ± 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 ³ Ÿ œ ³ Ÿ 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 ³ 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 œ ± œ 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 ³ Ÿ œ ³ Ÿ 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 ³ Ÿ œ œ # # # 1 1 (b) V 2 dx 2 x g(x) dx and x g(x) tan x, 0 x /4 by part (a) œ Ÿ Ÿ 1 shell shell radius height 1 Î # V 2 tan x dx 2 sec x 1 dx 2 [tan x x] 2 1 œ ± œ ± œ ± œ ÎÎ Î% ! ± # 1 1 4 4 #
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Section 6.2 Volume by Cylindrical Shells 373 15. c 0, d 2; œœ V 2 dy 2 y y ( y) dy ±± '' c0 d2 11 ˆ‰ ± Š‹ È shell shell radius height 2 y y dy 2 œ ² œ ² ' 0 2 ’“ $Î# # # ! 2y y 53 &Î# $ 22 2 1 6 œ ² œ ² œ ² 1 ”• Š Š È 8 5 35 3 82 2 & " $ ÈÈ 32 5 œ ² 16 15 1 È 16. c 0, d 2; V 2 dy 2 y y ( y) dy cd shell shell radius height # 2 y y dy 2 16 œ ² œ ² œ ² 1 ' 0 2 ab $# # ! " yy 43 4 3 2 % $ 16 1 54 0 63 1 17. c 0, d 2; V 2 dy 2 y 2y y dy ± shell shell radius height # 2 2y y dy 2 2 œ ± œ ± œ ± 1 ' 0 2 ab ’“ #$ # !
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This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

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ISM_T11_C06_B - Section 6.2 Volume by Cylindrical Shells 3....

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