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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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Integration Techniques

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Integration Techniques

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