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6.5-6practice - 430 Chapter 6 Applications of Definite...

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430 Chapter 6 Applications of Definite Integrals 46. (a) First, we note that y (distance from origin to AB) d a cos d d . œ± Ê œ ± Ê œ a sin a(sin cos ) ! !! !! ! ! ± Moreover, h a a cos . The graphs below suggest that œ² Ê œ œ ! ds i n c o s h a( cos ) cos a(sin cos ) ± ±± ± lim . ! Ä! sin cos 2 cos 3 ± ± ¸ (b) 0.2 0.4 0.6 0.8 1.0 f( ) 0.666222 0.664879 0.662615 0.659389 0.655145 ! ! 6.5 AREAS OF SURFACES OF REVOLUTION AND THE THEOREMS OF PAPPUS 1. (a) sec x sec x dy dy dx dx œÊ œ #% # Š‹ S 2 (tan x) 1 sec x dx Êœ ± 1 ' 0 4 1 Î È % (c) S 3.84 ¸ (b) 2. (a) 2x 4x dy dy dx dx 2 œ # S 2 x 1 4x dx ± 1 ' 0 2 # # È (c) S 53.23 ¸ (b)
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Section 6.5 Areas of Surfaces of Revolution and the Theorems of Pappus 431 3. (a) xy 1 x œÊœÊ œ ± Ê œ "" " # yd y y d y dx dx y # % Š‹ S 2 1 y dy Êœ ² 1 ' 1 2 " ±% y È (c) S 5.02 ¸ (b) 4. (a) cos y cos y dx dx dy dy œÊ œ # # S 2 (sin y) 1 cos y dy ² 1 ' 0 1 È # (c) S 14.42 ¸ (b) 5. (a) x y 3 y 3 x "Î# "Î# "Î# # ²œ Ê œ ± ˆ‰ 23 x x ± ± dy dx ˆ "Î# ±"Î# " # 13 x ± dy dx # ±"Î# # S 2 3 x 1 1 3x dx ± ²± 1 ' 1 4 É ab "Î# # ±"Î# # (c) S 63.37 ¸ (b) 6. (a) 1 y 1 y dx dx dy dy œ² Ê œ ² ±"Î# ±"Î# # # S2 y2y 1 1y d x ² ²² 1 ' 1 2 È É ±"Î# # (c) S 51.33 ¸ (b)
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432 Chapter 6 Applications of Definite Integrals 7. (a) tan y tan y dx dx dy dy œÊ œ Š‹ # # S 2 tan t dt 1 tan y dy Êœ ± 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 ab ## 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 ˆ‰ 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 ± "" # # # # # $ ± # " ' 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± œ ± È 3 5 in agreement with the integral value œ 1 È 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 x d x dy dy " # ' c1 1 2 1 Ê 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 È # # " slant height (2 1) (3 1) 5 Frustum surface area (1 3) 5 4 5 in agreement with ± ² œ ± œ È È the integral value
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Section 6.5 Areas of Surfaces of Revolution and the Theorems of Pappus 433 13. S 1 dx; dy dy dx 3 dx 9 9 9 xx 2 x x œÊ œ ± # %$ % Š‹ 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 œ² œ œ 11 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
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6.5-6practice - 430 Chapter 6 Applications of Definite...

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