How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=4-x^2# and #y=0# rotated about the y-axis?

1 Answer
Jim H
Oct 14, 2015

See the explanation, below.

Explanation:

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

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

In this case, we have radius #r = x#, height #h = 4-x^2# and #"thickness" = dx#. #x# varies from #0# to #2#, so the volume of the solid is:

#int_0^2 pi x(4-x^2)dx = piint_0^2 (4x-x^3) dx = 4pi#

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