Chapter 16.5, Problem 56E

(a)

To determine

Show that the vector function $r\left(x,\theta \right)=xi+f\left(x\right)\cos \theta j+f\left(x\right)\sin \theta k$ is the parameterization of the surface $S$ obtained by rotating the smooth curve $y=f\left(x\right),a\le x\le b,f\left(x\right)\ge 0$ about the $x\text{-axis}$.

(b)

To determine

Show that the surface area of the surface $S$ of revolution is $\text{A}={\displaystyle \underset{a}{\overset{b}{\int }}2\pi f\left(x\right)\sqrt{1+{\left[f^{\prime }\left(x\right)\right]}^2}dx}$.