Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 16: Integrals and Vector Fields - Section 16.4 - Green's Theorem in the Plane - Exercises 16.4 - Page 979: 32


The integral is $0$ irrespective of the curve $C$.

Work Step by Step

The tangential form for Green's Theorem can be computed as: The counterclockwise Circulation is : $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $ $\implies \oint_C x^3 \ dx -y^3 \ dy= \iint_{R} (\dfrac{\partial (-y^3) }{\partial x}-\dfrac{\partial (x^3) }{\partial y}) \ dx \ dy $ $\implies \oint_C x^3 \ dx -y^3 \ dy= \iint_{R} [0-0] \ dx \ dy$ $\implies \oint_C x^3 \ dx -y^3 \ dy=\oint_C x^3 \ dx -y^3 \ dy=0$ This means that the integral is $0$ irrespective of the curve $C$.
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