How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y=1/x, y=0, x=1, and x=2 about the y-axis?

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Answer 1 · 2016-10-11T15:03:29

Please see the explanation section, below.

Here is the region (blue) with a thin slice taken parallel to the axis of rotation. To set up for cylindrical shells. (The slice has black edges.)

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The volume of a representative shell is #2pirh*"thickness"#

In this case, we have

radius #r = x# (the dashed black line),

height #h = 1/x# and

#"thickness" = dx#.

#x# varies from #1# to #2#, so the volume of the solid is:

#int_1^2 2pi x(1/x)dx =2 piint_1^2 dx = 2pi[x]_1^2 = 2pi#