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: 19



Work Step by Step

Given $$ \int_{1}^{4} \ln x d x, \quad N=8 $$ Since $\Delta x=\dfrac{b-a}{N}=\dfrac{3}{8}$ , 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_{8}&=\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}{8}\left[f(1)+4f(11/8 )+2f(14/8)+4f(17/8)+2f(20/8) +4f(23/8)+2f(26/8)+4f(29/8)+f(4) \right]\\ &\approx 2.5416 \end{align*}
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