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: 30


$2 \times ( \ Area \ of \ the \ Square)$

Work Step by Step

The tangential form for Green Theorem's can be calculated as: Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $ $\implies \oint_C xy^2 dx +(x^2y +2x) \ dy= \iint_{R} (\dfrac{\partial (x^2y+2x) }{\partial x}-\dfrac{\partial (xy^2) }{\partial y}) dx dy =\iint_{R} 2xy+2-2xy \ dx \ dy$ $\implies \oint_C xy^2 dx +(x^2y +2x) \ dy=2 \times ( \ Area \ of \ the \ Square)$
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