Simpson’s Rule (popular as Simpson’s 1/3^{rd} rule) is a numerical method that approximates the value of a definite integral (integral with limits) by using quadratic functions. This method is named in honour of the English mathematician Thomas Simpson (born 1710−died 1761).

Simpson’s Rule is based on the fact that with three given points, we can find the equation of a quadratic with the help of those points. To obtain an approximation of the definite integral b∫af(x)dx using Simpson’s Rule, we partition the interval [a,b] into an even number n of subintervals, each of width Δx=b−an.

It is said that the more the number of sub-intervals, the better results it will give (otherwise ideally, sub-intervals should be at least 8, if nothing otherwise is given).

On each pair of consecutive sub-intervals [xi−1,xi], [xi,xi+1], we consider a quadratic function y=ax2+bx+c such that it passes through the points (xi−1, f(xi−1)), (xi, f(xi)), (xi+1,f(xi+1)).

- First you will have to enter a function.
- Then you are required to enter and set a lower and an upper limit to it.
- Finally, you enter the number of rectangles.