Question

Set up an integral for the volume a solid torus (the donut-shaped solid shown in the figure) with radii br and aR. (Let a = 6 and b = 4.)


By interpreting the integral as an area, find the volume V of the torus.

Answer 1

The Theorem of Pappus will come in handy here.

Rotate the circle (x-R)^2 + y^2 = r^2
around the y-axis. The volume of the torus is the area of the circle times the distance traveled by its center.

Take a stab at it, and don't be afraid to google torus volume to motivate your solution. Reverse engineering works in math, too!

Answer 2

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