Chapter 4.2, Problem 42E

Simpson’s Rule.

To use Simpson’s Rule to approximate the area under a graph of a function f, interval $\left[a,b\right]$ is subdivided into n equal subintervals, where n is an even number. As show in the following graph a parabola is fitted to the first, second, and third points, another parabola is fitted to the third, fourth, and fifth points, and so on.

The area under the graph of f over $\left[a,b\right]$ is approximated by

$\begin{array}{l}\frac{b-a}{3n}\left(f\left(x_1\right)+4f\left(x_2\right)+2f\left(x_3\right)\right)+\cdot \cdot \cdot \\ +2f\left(x_{n-1}\right)+4f\left(x_n\right)+f\left(x_{n+1}\right)\end{array}$ where $x_n=a\text{ and }{x}_{n+1}=b$.

Use Simpson’s Rule and the interval subdivision of Exercises 17 (a) to approximate the area under the graph of $f\left(x\right)=\frac{1}{x^2}$ over $\left[1.7\right]$.