Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Section 5.2 - The Definite Integral - 5.2 Exercises - Page 389: 12


$\int_{0}^{\pi}x~sin^2~x~dx \approx 2.4674$

Work Step by Step

$\Delta x = \frac{b-a}{n} = \frac{\pi-0}{4} = \frac{\pi}{4}$ We can find the midpoints of the four subintervals: $x_1 = \frac{\pi}{8}$ $x_2 = \frac{3\pi}{8}$ $x_3 = \frac{5\pi}{8}$ $x_4 = \frac{7\pi}{8}$ $\int_{0}^{\pi}x~sin^2~x~dx \approx \sum_{i=1}^{4} f(x_i)~\Delta x$ $\int_{0}^{\pi}x~sin^2~x~dx \approx \frac{\pi}{4}\cdot (\frac{\pi}{8}~sin^2~\frac{\pi}{8}+\frac{3\pi}{8}~sin^2~\frac{3\pi}{8}+\frac{5\pi}{8}~sin^2~\frac{5\pi}{8}+\frac{7\pi}{8}~sin^2~\frac{7\pi}{8})$ $\int_{0}^{\pi}x~sin^2~x~dx \approx \frac{\pi}{4}\cdot (3.14159)$ $\int_{0}^{\pi}x~sin^2~x~dx \approx 2.4674$
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