Simpson's Rule

To calculate the value of an integral one can approximate the function using a polynomial defined on subintervals. The following formula corresponds to passing a quadratic function through three points and integrating.

For an even number of intervals and an odd number of points, 2N+1, with a = x_o, a+x = x₁, a+2x = x₂, ..., a+2Nx = b we get Simpson's rule.

Within each pair of intervals the interpolant is continuous with continuous derivatives, but only the function is continuous from one pair to another.