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: 42

Answer

$\frac{\pi}{4}$

Work Step by Step

$\int^{\infty}_0 \int^{\infty}_0 \frac{1}{(1+x^2+y^2)^2} dxdy $ =$\int^{\pi/2}_0 \int^{\infty}_0 \frac{r}{(1+r^2)^2}drd\theta $ =$\frac{\pi}{2} \lim\limits_{b \to \infty} \int^b_0 \frac{r}{((1+r^2)^2)}dr $ =$\frac{\pi}{4} \lim\limits_{b \to \infty}[1-\frac{1}{1+r^2}]_0^b $ =$\frac{\pi}{4} \lim\limits_{b \to \infty}(1-\frac{1}{1+b^2})$ =$\frac{\pi}{4}$
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