Chapter 8.2, Problem 39E

Formula 4 is valid only when f(x) ≥ 0. Show that when f(x) is not necessarily positive, the formula for surface area becomes

$S={\displaystyle {\int }_a^{b}2\pi |f\left(x\right)|\sqrt{1+{\left[f^{\prime }\left(x\right)\right]}^2}}dx$