Chapter 14.8, Problem 58ES

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 to find the volume of the solid that is generated when the region enclosed by $y=x^2\text{ and }y=8-x^2$ is revolved about the x-axis.