How do you find the volume of the solid generated by revolving the region bounded by the graphs x=y2,x=4, about the line x=6?

1 Answer
Jim H
Mar 31, 2017

Please see below.

Explanation:

The graph of the region is shown below. A representative slice of thickness dy has been taken at height y. The radii of rotation are shown as dashed and dotted red lines.

enter image source here

When revolved about the line x=6 the resulting slice has volume

(πR2−πr2)⋅thickness where R is the greater radius and r the lesser.

In this problem

R=6−y2 and
r=6−4=2 and
thickness=dy.

y varies from −2 to 2. (Or go from 0 to 2 and double the result.)

So the volume of the solid of revolution is

V=π∫2−2((6−y2)2−(2)2)dy=π∫2−2(32−12y2+y4)dy

=384π5

or about 241.3.

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