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: 30


$8 \pi$

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: $V=\pi \int_{m}^{n} (R^2_{outside}-R^2_{inside}) \ dy$ where, $R_{outside}=x^2+2-2=x^2 $ and $ R_{inside}=6-2=4$ Now, $V=\pi \int_2^6 [R(y)^2] \ dy \\ = \pi \int_2^6 [y-2] \ dy \\=\pi [\dfrac{y^2}{2}-2y ]_2^6 \\=\pi[(\dfrac{6^2}{2}-\dfrac{2^2}{2})-2(6-2)] \\=8 \pi$
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