ISM_T11_C06_D - 394 2(a Chapter 6 Applications of Definite...

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394 Chapter 6 Applications of Definite Integrals 2. (a) 2x 4x dy dy dx dx œÊ œ Š‹ # S 2 x 1 4x dx Êœ ± 1 ' 0 2 # # È (c) S 53.23 ¸ (b) 3. (a) xy 1 x œÊœÊ œ ² Ê œ "" " # yd y y d y dx dx y # % S 2 1 y dy ± 1 ' 1 2 " ±% y È (c) S 5.02 ¸ (b) 4. (a) cos y cos y dx dx dy dy œ # # S 2 (sin y) 1 cos y dy ± 1 ' 0 1 È # (c) S 14.42 ¸ (b) 5. (a) x y 3 y 3 x "Î# "Î# "Î# # ±œ Ê œ ² ˆ‰ 23 x x ² ² dy dx ˆ "Î# ±"Î# " # 13 x ² dy dx # ±"Î# # S 2 3 x 1 1 3x dx ² ±² 1 ' 1 4 É ab "Î# # ±"Î# # (c) S 63.37 ¸ (b)
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Section 6.5 Areas of Surfaces of Revolution and the Theorems of Pappus 395 6. (a) 1 y 1 y dx dx dy dy œ± Ê œ ± ±"Î# ±"Î# # # Š‹ ˆ‰ S2 y2y 1 1y d x Êœ ± ±± 1 ' 1 2 È É ab ±"Î# # (c) S 51.33 ¸ (b) 7. (a) tan y tan y dx dx dy dy œÊ œ # # S 2 tan t dt 1 tan y dy ± 1 '' 00 3y 1 Î È # 2 tan t dt sec y dy œ 1 1 Î (c) S 2.08 ¸ (b) 8. (a) x 1 x 1 dy dy dx dx œ² Ê œ ² È # # # S 2 t 1 dt 1 x 1 dx ² ± ² 1 11 5x È È È ## 2 t 1 dt x dx 1 È È # (c) S 8.55 ¸ (b) 9. y ; S 2 y 1 dx S 2 1 dx x dx œÊ œ œ ± ± œ x x dy dy dx dx 4 5 # # " " # ' a0 0 b4 4 Ê É 1 È 4 5; Geometry formula: base circumference 2 (2), slant height 4 2 2 5 œœ œ œ ± œ 1 È 5 x % ! ’“ È È È # Lateral surface area (4 ) 2 5 4 5 in agreement with the integral value œ " # ÈÈ 10. y x 2y 2; S 2 x 1 dy 2 2y 1 2 dy 4 5 y dy 2 5 y œÊœ Ê œ œ ± œ ± œ œ xd x d x dy dy # # # # # ! ' c0 0 d2 2 1 1 Ê È cd 2 5 4 8 5; Geometry formula: base circumference 2 (4), slant height 4 2 2 5 œ œ ± œ 1 È È Lateral surface area (8 ) 2 5 8 5 in agreement with the integral value œ " # 11. ; S 2 y 1 dx 2 1 dx (x 1) dx x dy dy (x 1) dx dx 55 x ± œ ± œ ± œ ± "" # # # # # $ ² # " ' a1 1 b3 3 Ê É # 3 1 (4 2) 3 5; Geometry formula: r 1, r 2, œ ± ² ± œ œ±œ 9 3 # # # # " " "# ± È 1 slant height (2 1) (3 1) 5 Frustum surface area (r r ) slant height (1 2) 5 ± ² œ ± È 3 5 in agreement with the integral value œ 1 È
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396 Chapter 6 Applications of Definite Integrals 12. y x 2y 1 2; S 2 x 1 dy 2 (2y 1) 1 4 dy 2 5 (2y 1) dy œ± Ê œ ²Ê œ œ ± œ ² ± œ ² xd x d x dy dy ## " # '' ' c1 1 d2 2 11 1 Ê Š‹ È È 2 5 y y 2 5 [(4 2) (1 1)] 4 5; Geometry formula: r 1, r 3, œ ² œ ²²² œ œ œ 1 ÈÈ È cd # # " "# slant height (2 1) (3 1) 5 Frustum surface area (1 3) 5 4 5 in agreement with œ² ± ² œÊ œ ± œ È È the integral value 13. S 1 dx; dy dy dx 3 dx 9 9 9 xx 2 x x œ ± # %$ % É # ' 0 2 1 u 1 du x dx du dx; œ± Ê œ Ê œ x4 x 994 9 $ " x 0 u 1, x 2 u œÊœ œÊ œ 25 9 S 2 u du u Äœ œ 1 ' 1 25 9 Î "Î# $Î# " # #&Î* " 43 2 1 ± 1 œ œ 1 3 27 3 27 81 125 125 27 98 ˆ‰ ˆ ± 14. x dy dy dx dx 4x œ "" # ±"Î# # S 2 x 1 dx Êœ ± ' 34 15 4 Î Î 1 È É " 4x 2 x dx 2 x œ± œ ± ' 15 4 Î Î É ’“ $Î# "&Î% $Î% 4 2 1 ² ± œ ² 41 5 3 44 344 4 4 32 ˆ ˆ $Î# $Î# $ (8 1) œ 42 8 33 15. dy (2 2x) dy (1 x) dx dx 2x x 2x x 2x x 1x œœ Ê œ ±± # # # S 2 2x x 1 dx ² ± ' 05 15 Þ Þ 1 È É # ± ± (1 x) 2x x # # 2 2x dx 1 ' Þ Þ È # ±²±² ± È È 2x x 1 2x x 2x x # 2d x 2 [ x ]2 œ 1 ' Þ Þ "Þ& !Þ& 16.
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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_D - 394 2(a Chapter 6 Applications of Definite...

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