Chapter 14.8, Problem 57ES

These exercises reference the Theorem of Pappus: If R is a bounded plane region and L is a line that lies in the plane of R such that R is entirely on one side of L, then the volume of the solid formed by revolving R about L is given by

$\text{volume}=\left(\text{area of }R\right)\cdot \left(\begin{array}{l}\text{distance}\text{traveled}\\ \text{by}\text{the}\text{centroid}\end{array}\right)$

Use the Theorem of Pappus and the fact that the area of an ellipse with semiaxes a and b is $\pi ab$ to find the volume of the elliptical torus generated by revolving the ellipse

$\frac{{\left(x-k\right)}^2}{a^2}+\frac{y^2}{b^2}=1$

about the y-axis. Assume that $k>a.$