Question

Complex Analysis

A torus is a doughnut shape with a minor radius r (the radius of one vertical slice) and a major radius R(the distance from the origin to the center of a slice). Prove that the volume of a torus is equal to27 Rr2 by following the steps below.

\text { Compute the integral } \int_{0}^{a} \sqrt{a^{2}-x^{2}} d x \text { by rewriting as } \int_{0}^{a} \sqrt{a^{2}-x^{2}} d x=\int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} d y d x \text { and }

changing to polar coordinates. Describe the region whose area we find by taking this integral.