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  • 5 years ago

Consider the following graph: http://i55.tinypic.com/288nvpz.gif a=4 and b=5 Refer to the figure and find the volume V generated by rotating the region R1 about the line AB.

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  1. dumbcow
    • 5 years ago
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    R1 is region bounded by y = 4x^5, y=0, and x=1 Rotating around vertical axis x=1 means we will look at horizontal cross-sections which are circular with radius that is x-distance from 4x^5 and x=1 Also we will integrate w/ respect to y since each horizontal cross-section has height of dy get x in terms of y y=4x^5 --> x = (y/4)^1/5 radius = 1 - (y/4)^1/5 V = pi*integral 0-4 R^2 dy R^2 = (1-(y/4)^1/5)^2 = 1-2(y/4)^1/5 +(y/4)^2/5 \[V = \pi \int\limits_{0}^{4}(y/4)^{2/5} -2(y/4)^{1/5}+1 dy\] u=y/4 ---> du =dy/4 -->4du = dy \[V = 4\pi \int\limits_{0}^{4}u ^{2/5} - 2u ^{1/5}+1 du \] \[V=4\pi |_{0}^{4} \left( 5/7(y/4) ^{7/5}- 5/3(y/4) ^{6/5} +y/4 \right)\] \[V =4\pi/21\]

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