Calculus (3rd Edition)

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

Chapter 6 - Applications of the Integral - 6.4 The Method of Cylindrical Shells - Exercises - Page 313: 61


$\dfrac{4\pi a^2 b}{3}$

Work Step by Step

The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by: $V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(y) \ dy$ Now, $V=2\pi \int_{-b}^{b} (\sqrt{a^2-\dfrac{a^2y^2}{b^2}})^2 \ dy\\= \pi \int_{-b}^{b} (a^2-\dfrac{a^2y^2}{b^2}) \ dy \\= \pi [a^2y -\dfrac{a^2 y^3}{3b^2}]_{-b}^{b} \\=\pi (a^2b - \dfrac{a^2 b^3}{3b^2 }+a^2b -\dfrac{a^2 b^3}{3b^2}) \\= \pi [2a^2 b -\dfrac{a^2 b}{3}-\dfrac{2 a^2 b}{3}] \\=\dfrac{4\pi a^2 b}{3}$
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