Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 16 - Multiple Integration - 16.5 Applications of Multiple Integrals - Exercises - Page 893: 61

Answer

Using Pappus's Theorem, we show that the volume of the torus is $V = 2{\pi ^2}a{b^2}$.

Work Step by Step

We have ${\cal D}$ defined by a disk of radius $b$ centered at $\left( {0,a} \right)$. So, the area of ${\cal D}$ is $A = \pi {b^2}$. Since the centroid of a disk occurs at its center, so $\left( {\bar x,\bar y} \right) = \left( {0,a} \right)$. Using Pappus's Theorem in Exercise 60 we obtain the volume of the torus by revolving ${\cal D}$ about the $x$-axis: $V = 2\pi A\bar y = 2\pi \left( {\pi {b^2}} \right)\left( a \right)$ So, the volume of the torus: $V = 2{\pi ^2}a{b^2}$.
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