Simpsons rule is a numerical integration method which makes use of parabolas. This provides a closer approximation to numerical integrals when compared to Left, Right Riemann sums and mid point rule which use rectangles and Trapezoidal rule which measures the area of trapezoids for approximation. This rule is given its name after the English mathematician Thomas Simpson who popularized this method of numerical integration.As it is done for other numerical integration methods, the interval [a,b] is divided into n equal partitions of width = Δx. But in order to apply this rule, n has to be even. The area in any two successive partition is approximated to the area under the parabola, which passes through the three end points which define the partition. In the diagram shown below the area in the partitions [xi-1 , xn+1] are approximated to the area of the region under the parabola shown which passes through the points P1, P2 and P3.

The Simpson's Rule for approximation of a definite integral is derived by using the general equation for one such parabola and evaluating the area under the parabola applying fundamental theorem of Calculus. Then the sum of all such areas is found using the pattern observed from the area of one sub interval and is known as Simpson's Rule for numerical integration.
Let us look how Simpson's rule is stated, its proof and also solve few example problems.

Simpson's rule used for approximating a definite integral is as follows:

$\int_{a}^{b}f(x)dx$ ≈ S = $\frac{\Delta x}{3}$$[f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+........+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})]$

where 'n' is even and Δx = $\frac{b-a}{n}$.
The rule can be remembered observing pattern found in the sum as the function values are alternatively multiplied by 4 and 2 except the function values at the boundary points which are taken as such.

Simpson's rule approximation is more efficient when compared to mid point rule or Trapezoidal Rule. The Es error involved in using Simpson's rule is given by
E ≤ $\frac{K(b-a)^{5}}{180n^{4}}$ and K is so chosen that | f 4 (x) | ≤ K where f 4 (x) denotes the fourth derivative of f(x).