Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 8 - Further Applications of Integration - 8.1 Arc Length - 8.1 Exercises: 25

Answer

$\approx 15.498085$

Work Step by Step

$y = x \sin x$ then $dy/dx = x \cos x + (\sin x)(1)$ and $1+(dy/dx)^{2} = 1+(x \cos x + \sin x)^{2}$ Let $f(x) = \sqrt{1+(dy/dx)^{2}} = \sqrt{1+(x \cos x + \sin x)^{2}}$ Then $L = \int ^{2\pi}_{0} f(x) dx$. Since $n = 10, \Delta x = \frac{2\pi - 0}{10} = \frac{\pi}{5}$ $L \approx S_{10} = \frac{\pi/5}{3}[f(0) + 4f(\pi/5) + 2f(2\pi/5) + 4f(3\pi/5) + 2f(4\pi/5) + 4f(5\pi/5) + 2f(6\pi/5) + 4f(7\pi/5) + 2f(8\pi/5) + 4f(9\pi/5) + f(2\pi) ] \approx 15.498085$
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