Chapter 6.5, Problem 30ES

Let $y=f\left(x\right)$ be a smooth curve on $\left[a,b\right]$ and assume that $f\left(x\right)\ge 0$ for $a\le x\le b$ . Let $A$ be the area under the curve $y=f\left(x\right)$ between $x=a$ and $x=b$ , and let $S$ be the are a of the surface obtained when this section of curve is revolved about the $x\text{-axis}$ .

(a) Prove that $2\pi A\le S$ .

(b) Fat what functions $f$ is $2\pi A=S$ ?