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: 8



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 $f=\ln x+y ; g =-x^2$ 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_{1}^3\int_{1}^{4} (-2x-1) \ dy \ dx \\=\int_{1}^3[y(2x-1)]_1^4 \ dx\\=\int_1^3 3(-2x-1) \ dx \\=3[-x^2-x)]_1^3\\=3(-3^2-3)-3(-1-1) \\=-30$
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