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: 21

Answer

\begin{aligned} (a)V &=\int_{2\pi}^{3\pi } 2\pi x\sin(x) d x\\ (b)V&= 98.69604 \end{aligned}

Work Step by Step

Given $$y=\sin x, y=0, x=2 \pi, x=3 \pi ; \quad \text { about the } y \text {-axis }$$ The volume of the solid when the region rotate about y -axis given by \begin{aligned} V&= \int_a^b 2\pi r(x) h(x)dx \end{aligned} where $r(x)$ is the radius of shell and $h(x)$ is the height. Here \begin{aligned} r(x)&= x,\\ h(x)&= \sin(x) \end{aligned} Hence \begin{aligned} V &=\int_{2\pi}^{3\pi } 2\pi x\sin(x) d x\\ &=2 \pi \left( -x\cos(x)+\int_{2\pi}^{3\pi 2} \cos(x)dx \right)\\ &=2 \pi \left( -x\cos(x)+ \sin(x)\right)\bigg|_{2\pi}^{3\pi }\\ &= 10\pi ^2\approx 98.69604 \end{aligned}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.