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 - Practice Exercises - Page 1028: 9



Work Step by Step

We have $\dfrac{\partial f_2}{\partial x}=8y \times \sin (y) \\ \dfrac{\partial f_1}{\partial y}=8x \times \cos (y) $ Now, $\oint_C F \cdot dr=\iint_S (\dfrac{\partial f_2}{\partial x}-\dfrac{\partial f_1}{\partial x}) dA $ $$\int_{0}^{\pi/2}\int_{0}^{\pi/2} (8y \times siny-8 \space x cos \space y) \space dy \space dx \\=\int_{0}^{\pi/2} (\pi^2 \sin x -8x) dx\\=(\pi^2 \cos x -4x^2)_{0}^{\pi/2} \\=0$$
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