Chapter 7.1, Problem 73E

We arrived at Formula 6.3.2, $V={\displaystyle {\int }_a^{b}2\pi xf\left(x\right)}dx$, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least

for the case where f is one-to-one and therefore has an inverse function g. Use the figure to show that

$V=\pi b^{2}d-\pi a^{2}c-{{\displaystyle {\int }_c^d\pi \left[g\left(y\right)\right]}}^{2}dy$

Make the substitution y = f(x) and then use integration by parts on the resulting integral to prove that

$V={\displaystyle {\int }_a^{b}2\pi xf\left(x\right)}dx$