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: 52



Work Step by Step

The iterated integral can be calculated as: $\ Area =\iint_{D} f(x,y) d A=\int_0^1 \int_{0}^{x^2} \ dy \ dx\\=\int_0^1 [x^2-0] \ dx \\=\int_0^1 x^2 \ dx \\=[\dfrac{x^3}{3}]_0^1 \\=\dfrac{1}{3}$ The average co-ordinate of $y$ is: $\overline{f}=\dfrac{1}{Area} \iint_{D} f(x,y) d A\\=\dfrac{1}{(1/3)}\int_0^1 \int_{0}^{x^2} y \ dy \ dx\\=3 \int_0^1 \dfrac{x^4}{2}\\=(3) \times [\dfrac{x^5}{10}]_0^1 \\=(3) \times [\dfrac{(1)^5}{10}]\\=\dfrac{3}{10}$
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