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Assignment 2 Solutions

# Assignment 2 Solutions - Math 138 Assignment 2 Solutions...

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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 0 y x x=4y 2 -y 3 4 0 Using cylindrical shells, the volume is given by: V = 2 π 4 0 ( y (4 y 2 - y 3 )) dy = 3 π 4 0 (4 y 3 - y 4 ) dy = 2 π y 4 - 1 5 y 5 4 0 = 2 π 256 - 1024 5 - 0 = 512 5 π. The volume of the solid is 512 5 π . 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).

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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 π 3 0 y (4 - ( y - 1) 2 - (3 - y )) dy = 2 π 3 0 ( - y 3 + 3 y 2 ) dy = 2 π - 1 4 y 4 + y 3 3 0 = 2 π - 81 4 + 27 - 0 = 27 2 π. The volume is 27 2 π .
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Assignment 2 Solutions - Math 138 Assignment 2 Solutions...

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