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: 46



Work Step by Step

The domain $D$ for given region can be expressed as: $-1 \leq y \leq 1$ and $1-y^2 \leq y \leq y^2-1$ The iterated integral can be calculated as: $\iint_{D} f(x,y) d A=\int_{-1}^1 \int_{-1+y^2}^{-y^2+1} dz dy\\=\int_{-1}^1 (-2y^2+2) \ dy \\=[-\dfrac{2y^3}{3}+2y]_{-1}^1 \\=\dfrac{-2}{3}+2-\dfrac{2}{3}-2 \\=\dfrac{8}{3}$ Thus, the volume of a given region is: $2 \times \dfrac{8}{3}=\dfrac{16}{3}$
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