Simpson's Rule is a method for approximating the integral of some function [math]f\left(x\right)[/math] over a given interval [math]\left[a,b\right][/math].[br][br]This is accomplished by dividing the interval into [math]n[/math] equal segments. Taking the endpoints of each sub-interval as well as their respective midpoint, a quadratic function [math]f_{i}\left(x\right)[/math] can be determined to pass through all three points evaluated in [math]f\left(x\right)[/math] where [math]1\leq{i}\leq{n}[/math]. Each quadratic is then integrated and evaluated over its respective interval. As [math]n\to\infty[/math], the sum of these definite integrals tends to the actual value of the original integral.[br][br][math]\int_{a}^{b}{f\left(x \right )}{\mathrm{d}x}=\lim_{n\to\infty}\left[{\sum_{i=1}^{n}{\int_{a+\left(i-1 \right )\left(\frac{b-a}{n} \right )}^{a+i\left(\frac{b-a}{n} \right )}{f_{i}\left(x \right )}{\mathrm{d}x}}} \right ][/math]