Chapter 6.5, Problem 28ES

Let $y=f\left(x\right)$ be a smooth curve on the interval $\left[a,b\right]$ and assume that $f\left(x\right)\ge 0$ for $a\le x\le b$ . By the Extreme-Value Theorem $\left(\mathrm{4.4.2}\right)$ , the function $f$ has a maximum value $K$ and a minimum value $k$ on $\left[a,b\right]$ . Prove: If $L$ is the arc length of the curve $y=f\left(x\right)$ between $x=a$ and $x=b$ , and if $S$ is the area of the surface that is generated by revolving this curve about the $x\text{-axis}$ , then

$2\pi kL\le S\le 2\pi KL$