Chapter 7 - Applications of Integration - 7.3 Exercises: 11

$\frac{128\sqrt 2 \pi}{15}$

Work Step by Step

Setup the integration using shell method about the y-axis $2\pi \int_2^4 x(\sqrt {x-2})dx$ $2\pi \int_2^4 (u+2)(u^{\frac{1}{2}})du$ , Use u-substitution $2\pi \int_2^4 (u^{\frac{3}{2}} +2u^{\frac{1}{2}})du$ $2\pi (\frac{2}{5}u^{\frac{5}{2}} +\frac{4}{3} u^{\frac{3}{2}})]_0^2$, use change of variables $2\pi (\frac{8\sqrt 2}{5} + \frac{8\sqrt 2}{3})$ $\frac{128\sqrt 2 \pi}{15}$

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