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 311: 25


$\dfrac{256 \pi}{15}$

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, $V=2\pi \int_{0}^{4} y \sqrt {4-y} \ dy\\= -2 \pi \int_0^4 (4-t) \sqrt t \ dt \\=-2 \pi [ \dfrac{8t^{3/2}}{3}-\dfrac{2y^{5/2}}{5}]_4^0 \\= -2\pi [\dfrac{-64}{3}+\dfrac{64}{5}] \\=\dfrac{256 \pi}{15}$
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