How do you find the volume of the solid generated by revolving the region bounded by the graphs xy=6, y=2, y=6, x=6, about the line x=6?

1 Answer
Apr 11, 2017

Please see below.

Explanation:

Find the volume of the solid generated by revolving the region bounded by xy=6, y=2, y=6, and x=6, about the line x=6.

Here is a graph. The region is in blue. A representative slice (black line segments) has been taken at a y value. The thickness is dy.

The axis of revolution is the line x=6 and is indicated with a red circular arrow.

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Because the thickness is the differential of y (rather than x), we'll want the curve in terms of y.
xy=6 if and only if x=6/y

When rotated, the radius is the x on the right minus the x on the left. (greater minus lesser values of x)

r = (6-6/y)

The volume of the representative disk is

pi r^2 dy = (6-6/y)^2dy

= pi(36-72/y+36/y^2)dy

y varies from 2 to 6, so the volume of the solid is

V = pi int_2^6 (36-72/y+36/y^2)dy

= pi(156 - 72ln(3))