Business Calc Homework w answers_Part_62

Business Calc Homework w answers_Part_62 - 306 Section 7.3...

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43. continued (c) A shell has radius } 8 5 } 2 y . The volume is E 1 0 2 p 1 } 8 5 } 2 y 2 12( y 2 2 y 3 ) dy 5 24 p E 1 0 1 y 4 2 } 1 5 3 } y 3 1 } 8 5 } y 2 2 dy 5 24 p 3 } 1 5 } y 5 2 } 1 2 3 0 } y 4 1 } 1 8 5 } y 3 4 5 2 p . (d) A shell has radius y 1 } 2 5 } . The volume is E 1 0 2 p 1 y 1 } 2 5 } 2 12( y 2 2 y 3 ) dy 5 24 p E 1 0 1 2 y 4 1 } 5 3 } y 3 1 } 2 5 } y 2 2 dx 5 24 p 3 2 } 1 5 } y 5 1 } 2 3 0 } y 4 1 } 1 2 5 } y 3 4 5 2 p . 44. A shell has height } y 2 2 } 2 1 } y 4 4 } 2 } y 2 2 } 2 5 y 2 2 } y 4 4 } . (a) A shell has radius y . The volume is E 2 0 2 p ( y ) 1 y 2 2 } y 4 4 } 2 dy 5 2 p 3 } 1 4 } y 4 2 } 2 1 4 } y 6 4 5 } 8 3 p } . (b) A shell has radius 2 2 y . The volume is E 2 0 2 p (2 2 y ) 1 y 2 2 } y 4 4 } 2 dy 5 2 p E 2 0 1 } y 4 5 } 2 } y 2 4 } 2 y 3 1 2 y 2 2 dy 5 2 p 3 } 2 1 4 } y 6 2 } 1 1 0 } y 5 2 } 1 4 } y 4 1 } 2 3 } y 3 4 5 } 8 5 p } . (c) A shell has radius 5 2 y . The volume is E 2 0 2 p (5 2 y ) 1 y 2 2 } y 4 4 } 2 dy 5 2 p E 2 0 1 } y 4 5 } 2 } 5 4 y 4 } 2 y 3 1 5 y 2 2 dy 5 2 p 3 } 2 1 4 } y 6 2 } 1 4 } y 5 2 } 1 4 } y 4 1 } 5 3 } y 3 4 5 8 p . (d) A shell has radius y 1 } 5 8 } . The volume is E 2 0 2 p 1 y 1 } 5 8 } 21 y 2 2 } y 4 4 } 2 dy 5 2 p E 2 0 1 2 } y 4 5 } 2 } 5 3 y 2 4 } 1 y 3 1 } 5 8 y 2 } 2 dy 5 2 p 3 2 } 2 1 4 } y 6 2 } 3 1 2 } y 5 1 } 1 4 } y 4 1 } 2 5 4 } y 3 4 5 4 p . 45. [ 2 1, 3] by [ 2 1.4, 9.1] The functions intersect at (2, 8). (a) Use washer cross sections: a washer has inner radius r 5 x 3 , outer radius R 5 4 x , and area A ( x ) 5 p ( R 2 2 r 2 ) 5 p (16 x 2 2 x 6 ). The volume is E 2 0 p (16 x 2 2 x 6 ) dx 5 p 3 } 1 3 6 } x 3 2 } 1 7 } x 7 4 5 } 51 2 2 1 p } . (b) Use cylindrical shells: a shell has a radius 8 2 y and height y 1/3 2 } 4 y } . The volume is E 8 0 2 p (8 2 y ) 1 y 1/3 2 } 4 y } 2 dy 5 2 p E 8 0 1 8 y 1/3 2 2 y 2 y 4/3 1 } y 4 2 } 2 dy 5 2 p 3 6 y 4/3 2 y 2 2 } 3 7 } y 7/3 1 } 1 1 2 } y 3 4 5 } 83 2 2 1 p } . 46. [ 2 0.5, 1.5] by [ 2 0.5, 1.5] The functions intersect at (0, 0) and (1, 1). (a) Use cylindrical shells: a shell has radius x and height 2 x 2 x 2 2 x 5 x 2 x 2 . The volume is E 1 0 2 p ( x )( x 2 x 2 ) dx 5 2 p 3 } 1 3 } x 3 2 } 1 4 } x 4 4 5 } p 6 } . (b) Use cylindrical shells: a shell has radius 1 2 x and height 2 x 2 x 2 2 x 5 x 2 x 2 . The volume is E 1 0 2 p (1 2 x )( x 2 x 2 ) dx 5 2 p E 1 0 ( x 3 2 2 x 2 1 x ) dx 5 2 p 3 } 1 4 } x 4 2 } 2 3 } x 3 1 } 1 2 } x 2 4 5 } p 6 } . 1 0 1 0 8 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 306 Section 7.3
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47. [ 2 0.5, 2.5] by [ 2 0.5, 2.5] The intersection points are 1 } 1 4 } , 1 2 , 1 } 1 4 } , 2 2 , and (1, 1). (a) A washer has inner radius r 5 } 1 4 } , outer radius R 5 } y 1 2 } , and area p ( R 2 2 r 2 ) 5 p 1 } y 1 4 } 2 } 1 1 6 } 2 . The volume is E 2 1 p 1 } y 1 4 } 2 } 1 1 6 } 2 dy 5 p 3 2 } 3 1 y 3 } 2 } 1 1 6 } y 4 5 } 1 4 1 8 p } . (b) A shell has radius x and height } ˇ 1 x w } 2 1. The volume is E 1 1/4 2 p ( x ) 1 } ˇ 1 x w } 2 1 2 dx 5 2 p 3 } 2 3 } x 3/2 2 } 1 2 } x 2 4 5 } 1 4 1 8 p } . 48. (a) For 0 , x # p , x f ( x ) 5 } x (si x n x ) } 5 sin x . For x 5 0, x f ( x ) 5 0 ? 1 5 sin 0 5 sin x . So x f ( x ) 5 sin x for 0 # x # p .
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