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: 24


$$3.5846 $$

Work Step by Step

Given $$y=\cos x ; \quad\left[0, \frac{\pi}{2}\right] ; \quad y \text { -axis } ; \quad S_{8}$$ Since $$ V= 2\pi \int_{0}^{\pi/2}x f(x)dx= 2\pi \int_{0}^{\pi/2} x\cos xdx$$ Now, we will evaluate the integral using $M_8$ Since $\Delta x=\dfrac{b-a}{n}=\dfrac{\pi/2}{8}=\dfrac{\pi}{16}$, then by using Simpson’s rule, we get: \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_{8}&=\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) +f(x_{8}) \right] \\ &=\dfrac{\pi}{48}\left[f(0)+4f(\pi/16)+2f(2\pi/16)+4f(3\pi/16)+2f(4\pi/16) +4f(5\pi/16)+2f(6\pi/16) +4f(7\pi/16)+f(\pi/2) \right]\\ &\approx 0.570799 \end{align*} Hence $$ V=2\pi (0.570799)\approx 3.5846 $$
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