Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 5 - Applications of Integration - 5.2 Volumes - 5.2 Exercises - Page 374: 9


$\displaystyle{V=\frac{64\pi }{15}}\\ $

Work Step by Step

$y^2=2y\\ y^2-2y=0\\ y(y-2)=0\\ y=0 \qquad y=2$ $\displaystyle{A\left(y\right)=\pi \left(2y\right)^2-\pi \left(y^2\right)^2}\\ \displaystyle{A\left(y\right)=\pi \left(4y^2-y^4\right)}\\$ $\displaystyle{V=\int_0^2A\left(y\right)\ dy}\\ \displaystyle{V=\int_0^2\pi \left(4y^2-y^4\right)\ dy}\\ \displaystyle{V=\pi \int_0^24y^2-y^4\ dy}\\ \displaystyle{V=\pi\left[\frac{4}{3}y^3-\frac{1}{5}y^5\right]_0^2}\\ \displaystyle{V=\pi\left(\left(\frac{4}{3}2^3-\frac{1}{5}2^5\right)-\left(\frac{4}{3}0^3-\frac{1}{5}0^5\right)\right)}\\ \displaystyle{V=\frac{64\pi }{15}}\\ $
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