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Pre-Calc Homework Solutions 305

Pre-Calc Homework Solutions 305 - Section 7.3 37 40 305 2 2...

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37. [ 2 2, 2] by [ 2 1, 2] The curves intersect at ( 6 1, 1). (a) A cross section has radius r 5 1 2 x 2 and area A ( x ) 5 p r 2 5 p (1 2 x 2 ) 2 5 p (1 2 2 x 2 1 x 4 ). The volume is E 1 2 1 p (1 2 2 x 2 1 x 4 ) dx 5 p 3 x 2 } 2 3 } x 3 1 } 1 5 } x 5 4 5 } 1 1 6 5 p } . (b) Use cylindrical shells: a shell has radius 2 2 y and height 2 ˇ y w . The volume is E 1 0 2 p (2 2 y )(2 ˇ y w ) dy 5 4 p E 1 0 (2 ˇ y w 2 y 3/2 ) dy 5 4 p 3 } 4 3 } y 3/2 2 } 2 5 } y 5/2 4 5 } 5 1 6 5 p } . (c) Use cylindrical shells: a shell has radius y 1 1 and height 2 ˇ y w . The volume is E 1 0 2 p ( y 1 1)(2 ˇ y w ) dy 5 4 p E 1 0 ( y 3/2 1 ˇ y w ) dy 5 4 p 3 } 2 5 } y 5/2 1 } 2 3 } y 3/2 4 5 } 6 1 4 5 p } . 38. (a) A cross section has radius r 5 h 1 1 2 } b x } 2 and area A ( x ) 5 p r 2 5 p h 2 1 1 2 } b x } 2 2 . The volume is E b 0 p h 2 1 1 2 } b x } 2 2 dx 5 p h 2 3 2 } b 3 } 1 1 2 } b x } 2 3 4 5 } p 3 } bh 2 . (b) Use cylindrical shells: a shell has radius x and height h 1 1 2 } b x } 2 . The volume is E b 0 2 p ( x ) h 1 1 2 } b x } 2 dx 5 2 p h E b 0 1 x 2 } x b 2 } 2 dx 5 2 p h 3 } 1 2 } x 2 2 } 3 x b 3 } 4 5 } p 3 } b 2 h .
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