Chapter 14.8, Problem 56ES

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 result of Example 3 to find the volume of the solid generated when the region bounded by the x-axis and the semicircle $y=\sqrt{{\text{a}}^{\text{2}}-x^2}$ is revolved about

(a) the line $y=-a$

(b) the line $y=x-a.$