Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.9 Numerical Integration - Exercises - Page 457: 21


$$ 0.347 $$

Work Step by Step

Given $$ \int_{0}^{\pi / 4} \tan \theta d \theta, \quad N=10 $$ Since $\Delta x=\dfrac{b-a}{N}=\dfrac{\pi}{40}$ , then by using Simpson’s rule \begin{align*} S_{n}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3).....+4f(x_{n-1})+f(x_n)\right]\\ S_{10}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5) +2f(x_6)+4f(x_7) +2f(x_8)+4f(x_9) +f(x_{10}) \right] \\ &=\dfrac{\pi}{120}\left[f(0)+4f(\pi/40)+2f(2\pi/40)+4f(3\pi/40)+2f(4\pi/40) +4f(5\pi/40)+2f(6\pi/40)+4f(7\pi/40)+2f(8\pi/40) +4f(9\pi/40)+f(\pi/4) \right]\\ &\approx 0.347 \end{align*}
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