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 15.2 - Double Integrals over General Regions - Exercises 15.2 - Page 883: 72



Work Step by Step

Use the definition for improper integrals $I= \int_{0}^{\infty} \lim\limits_{b \to \infty} \int_0^b xe^{-x-2y} dx \space dy$ or, $= \int_{0}^{\infty} e^{-2y} \lim\limits_{b \to \infty} [-x e^{-x} -e^{-x}]_0^b dy$ or, $=\int_{0}^{\infty} e^{-2y} \lim\limits_{b \to \infty} [-b e^{-b} -e^{-b}+1] dy $ Thus, we have $I=\dfrac{1}{2} \lim\limits_{b \to \infty} (-e^{-2b}+1)=\dfrac{1}{2}$
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