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: 16

Answer

$$ 0.947$$

Work Step by Step

Given$$ \int_{0}^{1} \cos \left(x^{2}\right) d x, \quad N=6$$ Since $\Delta x=\dfrac{b-a}{N}=\dfrac{1}{6}$ , 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_{6}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5) +f(x_{6}) \right] \\ &=\dfrac{1}{18}\left[f(0)+4f(1/6)+2f(2/6)+4f(3/6)+2f(4/6) +4f(5/6)+f(1) \right]\\ &\approx 0.947 \end{align*}
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