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: 52

Answer

$\dfrac{11 \pi }{21}$

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: $V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$ Now, $V=2\pi \int_{0}^{1} x(x-x^{12}) \ dx \\ = 2\pi \int_0^1 [x^2-x^{13}] \ dx \\= 2 \pi [ \dfrac{x^{3}}{3}-\dfrac{x^{14}}{14}]_0^1\\=2 \pi ( \dfrac{1}{3}-\dfrac{1}{14}) \\= \dfrac{11 \pi }{21}$
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