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 306: 59


$4 \pi \sqrt 3$

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, $V=\pi \int_{-\sqrt 3}^{\sqrt 3} [(\sqrt {4-y^2})^2-1] \ dy \\ = \pi \int_{-\sqrt 3}^{\sqrt 3} (3-y^2) \ dy \\=\pi (3y-\dfrac{y^3}{3}]t_{-\sqrt 3}^{\sqrt 3}\\=\pi (6 \sqrt 3-2 \sqrt 3) \\= 4 \pi \sqrt 3$
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