Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 15: Multiple Integrals - Section 15.4 - Double Integrals in Polar Form - Exercises 15.4 - Page 894: 44

Answer

$\int^{\beta}_{\alpha} \frac{1}{2}r^2 d\theta $

Work Step by Step

We calculate the area using polar coordinates as follows: $A= \int^{\beta}_{\alpha} \int^{f(\theta)}_{0} rdrd\theta $ =$\int^{\beta}_{\alpha} [\frac{r^2}{2}]^{f(\theta)}_0 d\theta $ =$\frac{1}{2}\int^{\beta}_{\alpha} f^2(\theta) d\theta $ Substitute $r=f(\theta)$: =$\int^{\beta}_{\alpha} \frac{1}{2}r^2 d\theta $
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