Simpson's Rule

Simpson's rules are numerical methods for approximating definite integrals, named after Thomas Simpson; the most basic, Simpson's one-third rule, approximates the integral of a function over an interval using the values of the function at the endpoints and the midpoint. The approximation is exact when the function is a polynomial of degree three or lower, and applying the rule to many equal subdivisions of the interval produces the composite Simpson's rule; a related Simpson's three-eighths rule uses one additional evaluation point for a lower error bound.