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: 28


$3 \pi$

Work Step by Step

Consider, $A=\oint_{C} \dfrac{x}{2} dy - \dfrac{y}{2} dx$ We have $ dx=1- \cos t dt $ and $dy= \sin t dt$ Now $Area=\int_{2 \pi}^{0} \dfrac{1}{2} (t-\sin t) (\sin t dt) -\dfrac{1}{2} (1-\cos t) (1-\cos t \ dt) \\=\dfrac{1}{2} \int_{2 \pi}^{0} t \sin t -\sin^2 t -1-\cos^2 t + 2 \cos t d t \\=\dfrac{1}{2} [\int t \sin t \ dt]_0^{2 \pi} -[t+\sin t]_{2 \pi}^0 \\=\dfrac{1}{2} [ -t \cos t +\int \cos t ]_{2 \pi}^{0} +2 \pi \\=3 \pi$
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