Answer
Using Pappus's Theorem, we show that the volume of the torus is $V = 2{\pi ^2}a{b^2}$.
Work Step by Step
We have ${\cal D}$ defined by a disk of radius $b$ centered at $\left( {0,a} \right)$. So, the area of ${\cal D}$ is $A = \pi {b^2}$.
Since the centroid of a disk occurs at its center, so $\left( {\bar x,\bar y} \right) = \left( {0,a} \right)$.
Using Pappus's Theorem in Exercise 60 we obtain the volume of the torus by revolving ${\cal D}$ about the $x$-axis:
$V = 2\pi A\bar y = 2\pi \left( {\pi {b^2}} \right)\left( a \right)$
So, the volume of the torus: $V = 2{\pi ^2}a{b^2}$.