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?

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Answer 1 · 2017-04-11T19:14:35

Please see below.

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))#