thomasET_226348_ism38

thomasET_226348_ism38 - Chapter 6 Practice Exercises 3....

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Chapter 6 Practice Exercises 449 3. A(x) (diameter) (2 sin x 2 cos x) œœ ± 11 44 ## 4 sin x 2 sin x cos x cos x œ± ² 1 4 ab (1 sin 2x); a , b œ œ 1 5 V A(x) dx (1 sin 2x) dx Êœ œ ± '' a4 b5 4 1 1 1 Î Î x œ² 1 ±‘ cos 2x # % Î% 1 1 ± ±œ ’“ Š‹ Š 5 cos cos 5 # 4. A(x) (edge) 6 x 0 6 x 36 24 6 x 36x 4 6 x x ; œœ± ± œ ± œ ± ² ± ² # $Î# # #% # Œ² ÈÈ È È È a 0, b 6 V A(x) dx 36 24 6 x 36x 4 6 x x dx Ê œ œ ± ²± ² a0 b6 È $Î# # 36x 24 6 x 18x 4 6 x 216 16 6 6 6 18 6 6 6 6 ²œ ± ² ± ² È È †† 22 x 8 6 35 3 5 3 $Î# # &Î# # # ' ! $ $ 216 576 648 72 360 œ±²±² œ œ 1728 1728 1800 1728 72 55 5 5 ± 5. A(x) (diameter) 2 x 4x x ; a 0, b 4 V A(x) dx ± œ ± ² œ œ Ê œ 1 4 4 1 6 xx # &Î# # Š È # % ' a b 4x x dx 2x x 32 32 32 ² œ ± ² œ ± ² 1 41 6 4 7 5 1 6 4 7 5 x2 x 8 2 ' 0 4 ˆ‰ &Î# # (Î# % ! %& 1 (35 40 14) ² ² œ 32 8 2 8 72 47 5 3 5 3 5 1 6. A(x) (edge) sin 2 x 2 x ± ± " # # # ± 1 34 3 È 4 x 4 3 x; a 0, b 1 œ œ È 3 4 È È # V A(x) dx 4 3 x dx 2 3 x œ œ b1 # " ! 23 œ È 7. (a) : .3=5 7/>29. V R (x) dx 3x dx 9x dx œ ' a1 1 1 1 ) # cd * " ±" (b) : =2/66 7/>29. V 2 dx 2 x 3x dx 2 3 x dx 2 3 œ œ œ ' 0 1 1 1 1 shell shell radius height 6 x " ! ' Note: The lower limit of integration is 0 rather than 1. ± (c) : =2/66 7/>29. V 2 dx 2 (1 x) 3x dx 2 2 ± œ ± œ ± ± ± ± œ 1 1 ± ˆ shell shell radius height 5 2 5 5 5 3x x 3 3 12 % " ±" "" &' 1 (d) : A+=2/< 7/>29. R(x) 3, r(x) 3 3x 3 1 x V R (x) r (x) dx 9 9 1 x dx ± œ ± Ê œ ± œ ± ± %% # # % # c d 9 1 1 2x x dx 2x 9 18 ± ² œ ± œ ± œ ± œ œ 1 1 a b %) " ±" " 2x x 2 2 13 26 59 5 9 5 5 &*
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450 Chapter 6 Applications of Definite Integrals 8. (a) : A+=2/< 7/>29. R(x) , r(x) V R (x) r (x) dx dx x œœ Ê œ ± œ ± œ ± ± 44 1 6 x xx 5 4 $ $ "" ## ± & # " '' a1 b2 11 1 cd ’“ ˆ‰ ˆ‰ ± ( 2 10 64 5) œ ±±±± œ±±²²œ ±±²²œ ±‘ ˆ ˆ‰ ˆ ±" " " " " # 16 16 16 57 532 5 4 10 5 4 20 0 (b) : =2/66 7/>29. V 2 x dx 2 4x 2 1 4 2 œ± œ ± ± œ ± ± ± ± ± œ œ 1 1 ' 1 2 ± ‘ˆ 4x 4 5 5 x4 4 4 $ # # ±" # " 1 (c) : =2/66 7/>29. V 2 dx 2 (2 x) dx 2 1 dx ± ± œ ± ± ² 1 ' 1 2 ˆ ˆ Š‹ shell shell radius height x x x 48 4 x $$ # " 2 x 2(1221 ) 441 œ ± ²±² œ ±²±² ±±²±² œ x 3 4 4 # # # " " # 1 (d) : A+=2/< 7/>29. V R (x) ' a b 1 4 d x ± 1 ' 1 2 ˆ 74 x # # # $ 16 1 2x x dx ± ² 49 4 1 1 ' 1 2 ab ±$ ±' 1 6xx ²± 49 x 45 1 1 ±# # " & 16 2 1 1 ² ± ± ² ± 49 5 35 1 1 ˆ " # 16 ±² 49 4 4 160 5 1 1 """ (40 1 32) ± ²œ±œ 49 16 49 71 103 4 160 4 10 20 1 9. (a) : .3=5 7/>29. V x 1 dx (x 1) dx x œ ± œ ± 1 55 È #& # " x # 51 4 8 ± ± œ ± œ 1 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 ² Ê œ ± œ ± ² # # # c2 d2 a b 25 y 2y 1 dy 24 y 2y dy 24y y 2 24 2 8 ± ± œ ± ± œ ± ± œ ± ± 1 1 22 a b %# $ # ±# y 53 23 2 2 & †† 32 3 (45 6 5) ± ± œ 1 2 32 1088 5 3 15 15 " (c) : .3=5 7/>29. R(y) 5 y 1 4 y œ± ² œ± V R (y) dy 4 y dy Êœ œ ± # 16 8y y dy ² 1 ' 2 2 #% 16y 2 32 ² ² 8y y 64 32 $ & # ±# 64 1 (15 10 3) ² ² œ 1 2 64 512 3 5 15 15 " 10. (a) : =2/66 7/>29. V 2 dy 2 y y dy ± c0 d4 Š shell shell radius height 4 y # 2 y dy 2 2 œ ± œ ± 1 ' 0 4 # % !
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This note was uploaded on 04/17/2008 for the course MA 113 taught by Professor Massman during the Spring '08 term at Rose-Hulman.

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thomasET_226348_ism38 - Chapter 6 Practice Exercises 3....

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