Calculus (3rd Edition)

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

Chapter 18 - Fundamental Theorems of Vector Analysis - 18.1 Green's Theorem - Exercises - Page 983: 6



Work Step by Step

Green's Theorem states that: $\int_C fdx+g dy=\iint_D (\dfrac{\partial g}{\partial x}-\dfrac{\partial f}{\partial y})dA$ Here, we have $f=x+y ; g =x^2-y$ Now, $\dfrac{\partial g}{\partial x} =2x$ and $\dfrac{\partial f}{\partial y}=1$ Therefore, $ \iint_D (\dfrac{\partial g}{\partial x}-\dfrac{\partial f}{\partial y})dA=\int_{0}^1\int_{x^2}^{\sqrt x} (2x-1) \ dy \ dx \\=\int_{0}^1(2x-1) (x^2-\sqrt x) \ dx\\=\int_0^1 -2x^3+x^2+2x^{3/2}-x^{1/2} \ dx \\=-\dfrac{1}{30}$
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