A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from th
e center of the circle. Find the volume of the torus.
1 answer:
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

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