Alex has four books of different sizes which he wishes to place on a bookshelf. Unfortunately, the bookshelf has a conspicuous h
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Consider a circle with radius centered at some point
on the
-axis. This circle has equation
Revolve the region bounded by this circle across the -axis to get a torus. Using the shell method, the volume of the resulting torus is
where .
So the volume is
Substitute
and the integral becomes
Notice that is an odd function, so the integral over
is 0. This leaves us with
Write
so the volume is