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: 5

Answer

$8 \pi$

Work Step by Step

Given $$y=x^2, \quad 0 \leqslant x \leqslant 2, \quad y=4, \quad x=0$$ 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)&= 4-x^2 \end{aligned} Hence \begin{aligned} V &=\int_0^2 2 \pi x\left(4-x^2\right) d x\\ &=2 \pi \int_0^2\left(4 x-x^3\right) d x \\ &=2 \pi\left[2 x^2-\frac{1}{4} x^4\right]_0^2\\ &=2 \pi(8-4) \\ &=8 \pi \end{aligned}
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