Chapter 8 - Techniques of Integration - 8.9 Numerical Integration - Exercises - Page 457: 13

$$S_{4} \approx 5.2522$$

Given$$\int_{0}^{4} \sqrt{x} d x, \quad N=4 $$ Since $\Delta x=\dfrac{b-a}{N}=1$ , 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}{3}\left[f(0)+4f(1)+2f(2)+4f(3)+ f(4) \right]\\ &\approx 5.2522 \end{align*}