Suppose that a region $ \Re $ has area $ A $ and lies above the x-axis. When $ \Re $ is rotated about the x-axis, it sweeps out a solid with volume $ V_1 $. When $ \Re $ is rotated about the line $ y = -k $ (where $ k $ is a positive number), it sweeps out a solid with volume $ V_2 $. Express $ V_2 $ in terms of $ V_1 $, $ k $, and $ A $.

$V_{2}=V_{1}+2 k \pi A$

Applications of Integration

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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