A2W08-138Sol

A2W08-138Sol - Math 138 Assignment 2 Solutions Fall 2008 1....

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Unformatted text preview: Math 138 Assignment 2 Solutions Fall 2008 1. Determine the volume obtained by rotating the region defined by the given curves about the specified line. Include a sketch of the region with a typical cylindrical shell, and sketch the solid. a) x = 4 y 2- y 3 , x = 0; about the x-axis. Solution The points of intersection are (0 , 0) and (0 , 4). The region is shown below. y x x=4y 2-y 3 4 y x x=4y 2-y 3 4 Using cylindrical shells, the volume is given by: V = 2 Z 4 ( y (4 y 2- y 3 )) dy = 3 Z 4 (4 y 3- y 4 ) dy = 2 y 4- 1 5 y 5 4 = 2 256- 1024 5- = 512 5 . The volume of the solid is 512 5 . b) x + y = 3, x = 4- ( y- 1) 2 ; about the x-axis. Solution For intersection: 3- y = 4- ( y 2- 2 y + 1) 0 = 3 y- y 2 0 = y (3- y ) The points of intersection are (3 , 0) and (0 , 3). Math 138 Assignment 2 Solutions Fall 2008 y x x+y=3 x=4-(y-1) 2 3 3 y x x+y=3 x=4-(y-1) 2 3 3 Using cylindrical shells, the volume is given by: V = 2 Z 3 y (4- ( y- 1) 2- (3- y )) dy = 2 Z 3 (- y 3 + 3 y 2 ) dy = 2 - 1 4 y 4 + y 3 3 = 2 - 81...
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A2W08-138Sol - Math 138 Assignment 2 Solutions Fall 2008 1....

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