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: 31


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

Work Step by Step

The tangential form for Green's Theorem is given as: Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$ $\implies \oint_C4x^3 \ dx +x^4 \ dy= \iint_{R} (\dfrac{\partial (x^4) }{\partial x}-\dfrac{\partial (4x^3 y) }{\partial y}) dx dy= \iint_{R} [ 4x^3-4x^3 ] \ dx \ dy$ and, $\oint_C4x^3 \ dx +x^4 \ dy=0$ This means that the integral is $0$, irrespective of the curve $C$.
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