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 882: 18

Answer

a) $ \int_{-1}^{2} \int_{x^2}^{x+2} f(x,y) dy dx $ (b) $ \int_{0}^{1} \int_{-\sqrt y}^{\sqrt y} f(x,y) dx dy+ \int_{1}^{4} \int_{y-2}^{\sqrt y} f(x,y) dx dy$

Work Step by Step

(a) For vertical cross-sections, the region $R$ can be defined as: $y= x^2; y=x+2; \implies x^2 -x-2=0$ or, $x=-1, 2$ $\iint_{R} dA= \int_{-1}^{2} \int_{x^2}^{x+2} f(x,y) dy dx $ (b) For horizontal cross-sections, the region $R$ can be defined as: $\iint_{R} dA=\int_{R_1} dA+\iint_{R_1} dA$ or, $= \int_{0}^{1} \int_{-\sqrt y}^{\sqrt y} f(x,y) dx dy+ \int_{1}^{4} \int_{y-2}^{\sqrt y} f(x,y) dx dy$
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