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 312: 56


$\dfrac{4 \pi (R^2-r^2)^{3/2}}{3}$

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: $V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(y) \ dy$ $V=2\pi \int_{r}^{R} (x) [2\sqrt {R^2-x^2}] \ dx$ Let us apply the substitution method: $a=R^2-x^2 \implies da=-2x dx$ Now, $V= 2 \pi \int_0^{R^2-r^2} (a^{1/2}) \ da \\= 2\pi [\dfrac{2a^{3/2}}{3}]_0^{R^2-r^2} \\=\dfrac{4 \pi (R^2-r^2)^{3/2}}{3}$
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