Chapter 10.3, Problem 64ES

(a) In the discussion associated with Exercises 75-80 of Section 10.1, formulas were given for the area of the surface of revolution that is generated by revolving a parametric curve about the $x\text{-axis or }y\text{-axis}\text{.}$ Use those formulas to derive the following formulas for the areas of the surfaces of revolution that are generated by revolving the portion of the polar curve $r=f\left(\theta \right)\text{from}\theta \text{=}\alpha \text{to}\theta =\beta$ about the polar axis and about the line $\theta =\pi /2:$

$\begin{array}{l}s={\displaystyle {\int }_{\alpha }^{\beta }2\pi r\sin \theta \sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}}d\theta \text{About}\theta =0\\ s={\displaystyle {\int }_{\alpha }^{\beta }2\pi r\cos \theta \sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}}d\theta \text{About}\theta =\pi /2\end{array}$

(b) State conditions under which these formulas hold.