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: 29

Answer

(a) $0$ b) $(h-x) \times ( \ Area \ enclosed \ inside \ the \ curve \ C)$

Work Step by Step

The tangential form for Green Theorem is given as: Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) \ dx \ dy ....(1)$ (a) $\oint_C f(x) dx +g(y) dy= \iint_{R} (0-0) dx dy $ or, $\oint_C f(x) dx +g(y) dy=0$ (b) $\oint_C k(y) \ dx +h(x) \ dy= \iint_{R} (\dfrac{h(x) }{\partial x}-\dfrac{k (y)}{\partial y}) \ dx \ dy \\=(h-x) \iint_{R} \ dx \ dy \\=(h-x) \times ( \ Area \ enclosed \ inside \ the \ curve \ C)$
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