Chapter 14.8, Problem 55ES

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

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

Perform the following steps to prove the Theorem of Pappus:

(a) Introduce an xy-coordinate system so that L is along the y-axis and the region R is in the first quadrant. Partition R into rectangular subregions in the usual way and let $R_k$ be a typical subregion of R with center $\left(x_k^{*},y_k^{*}\right)$ and area $\Delta A_k=\Delta x_k\Delta y_k$ . Show that the volume generated by $R_k$ as it revolves about L is

$2\pi x_k^{*}\Delta x_k\Delta y_k=2\pi x_k^{*}\Delta A_k$

(b) Show that the volume generated by R as it revolves about L is

$V={\displaystyle \underset{R}{\iint }2\pi xdA=2\pi \cdot \bar{x}\cdot [\text{area of }R}]$