Calculus (3rd Edition)

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

Chapter 16 - Multiple Integration - 16.2 Double Integrals over More General Regions - Exercises - Page 859: 35



Work Step by Step

The domain $D$ for the given region can be expressed as: $0 \leq x \leq 1$ and $x \leq y \leq 1$ The iterated integral can be calculated as: $\iint_{D} x^2y d A=\int_0^1 \int_{x}^{1} xe^{y^3} dy dx\\=\int_0^1 \int_0^y x e^{y^3} dx dy\\=\int_0^1 (\dfrac{y^2}{2}e^{y^3}) \ dy$ Suppose that $y^3=t \implies y^2 dy=\dfrac{dt}{3}$ Now, $\int_0^1 \dfrac{y^2}{2} \times e^{y^3} \ dy=[\dfrac{e^{y^3}}{6}]_0^1\\=\dfrac{1}{6}(e)[e^1-e^0]\\=\dfrac{e-1}{6}$
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