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: 51


$\dfrac{121 \pi }{525}$

Work Step by Step

The disk 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 x-axis is given by: $V= \pi \int_{m}^{n} R^2 \ dx$ Now, $V=\pi \int_{0}^{1} (x-x^{12})^2 \ dx\\= \pi \int_0^1 [x^2-2x^{13}+x^{24}] \ dx \\= \pi [ \dfrac{x^{3}}{3}-\dfrac{2x^{14}}{14}+\dfrac{x^{25}}{25}]_0^1\\=\pi ( \dfrac{1}{3}-\dfrac{2}{14}+\dfrac{1}{25}) \\= \dfrac{121 \pi }{525}$
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