Chapter 10.4, Problem 55E

(a) Use Formula 10.2 to show that the area of the surface generated by rotating the polar curve

$r=f\left(\theta \right)a\le \theta \le b$

(where $f^{\prime }$ is continuous and $0\le a<b\le \pi$) about the polar axis is

$S={\displaystyle \int \begin{array}{l}b\\ a\end{array}}2\pi r\sin \theta \sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta$

(b) Use the formula in part (a) to find the surface area generated by rotating lemniscate $r^2=\cos 2\theta$ about the polar axis.