Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 6 - Applications of the Integral - 6.3 Volumes of Revolution - Exercises - Page 305: 52


$32 \pi$

Work Step by Step

The Washer method to compute the volume of revolution: When the function $f(x)$ is continuous and $f(x) \geq g(x) \geq 0$ on the interval $[m,n]$, then the volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the x-axis is given by: $V=\pi \int_{m}^{n} (R^2_{outside}-R^2_{inside}) \ dy$ Now, $V=\pi \int_{0}^{4} [(8-x)^2-(8-2 \sqrt x)^2] \ dy \\ = \pi \int_{0}^{4} [(64-16x+x^2-(64-32x^{1/2}+4x) \ dx \\=\pi [-10x^2+\dfrac{x^3}{3}+\dfrac{32 x^{3/2}}{3/2}]_0^4 \\= 32 \pi$
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.