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We arrived at Formula 6.3.2, $ V = \displaystyle \int_a^b 2 \pi x f(x) 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

$$ V = \pi b^2 d - \pi a^2 c - \int_c^d \pi [g(y)]^2 dy $$

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

$$ V = \int_a^b 2 \pi x f(x) dx $$

$V=$ $\int_{a}^{b} 2 \pi x f(x) d x$

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