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: 36



Work Step by Step

The domain $D$ for given region can be expressed as: $0 \leq x \leq 1$ and $x^{2/3} \leq y \leq 1$ The iterated integral can be calculated as: $\iint_{D} f(x,y) d A=\int_0^1 \int_{x^{2/3}}^{1} xe^{y^4} dy dx\\=\int_0^1 \int_{0}^{y^{3/2}} xe^{y^4} dy \\=\int_0^1 (\dfrac{y^3e^{y^4}}{2}) \ dy$ Suppose that $y^4=t \implies 4y^3 dy=dt$ Now, $\int_0^1 \dfrac{y^3}{2} \times e^{y^4} \ dy=[\dfrac{e^{y^4}}{8}]_0^1\\=\dfrac{e}{8}-\dfrac{1}{8}\\=\dfrac{e-1}{8}$
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