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: 62


$2\pi (1-c)$

Work Step by Step

Let us consider that $a= y=f(x)$. Now, we have: $\dfrac{dy}{dx}=-xy$ and $da=-x f(x) \ dx $ or, $-da=x f(x) dx$ $V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(x) \ dx$ Now, $V=2\pi \int_{0}^{a} x f(x) \ dx \\= 2\pi \int_{1}^{c} (-1) \ da \\= 2\pi \int_c^1 da \\= 2\pi [a]_1^c \\= 2\pi (1-c)$
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