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: 23


$-16 \pi$

Work Step by Step

The tangential form for Green Theorem -- Counterclockwise Circulation $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $ Now, $\oint_C F \cdot n ds= \iint_{R} \dfrac{\partial (y+2x)}{\partial x}-\dfrac{\partial (6y+x)}{\partial y} dx dy $ or, $= -4 \iint_{R} dx dy$ or, $= -4 \times$ Area of circle with radius 2 or, $= -4 \times \pi \times (2)^2$ or, $=-16 \pi$
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