Calculus 8th Edition

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

Chapter 5 - Applications of Integration - 5.3 Volumes by Cylindrical Shells - 5.3 Exercises - Page 382: 12



Work Step by Step

Using the quadratic formula $\displaystyle{y = \frac{12 \pm \sqrt{\left(-12\right)^2-4(3)(9)}}{2(3)}}\\ y=1\qquad y=3$ $\displaystyle{V=\int_{1}^{3}(2\pi y)\left(-3y^2+12y-9\right)\ dy}\\ \displaystyle{V=2\pi\int_{1}^{3}12y^2-9y-3y^3\ dy}\\ \displaystyle{V=2\pi\left[4y^3-\frac{9}{2}y^2-\frac{3}{4}y^4\right]_{1}^{3}}\\ \displaystyle{V=2\pi\left(\left(4(3)^3-\frac{9}{2}(3)^2-\frac{3}{4}(3)^4\right)-\left(4(1)^3-\frac{9}{2}(1)^2-\frac{3}{4}(1)^4\right)\right)}\\ \displaystyle{V=16\pi}$
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