## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 8 - Techniques of Integration - 8.9 Numerical Integration - Exercises - Page 457: 17

#### Answer

$$0.746855$$

#### Work Step by Step

Given$$\int_{0}^{1} e^{-x^{2}} d x, \quad N=4$$ Since $\Delta x=\dfrac{b-a}{N}=\dfrac{1}{4}$ , 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_{4}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+ f(x_{4}) \right] \\ &=\dfrac{1}{12}\left[f(0)+4f(1/4)+2f(2/4)+4f(3/4)+ f(1) \right]\\ &\approx 0.746855 \end{align*}

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