Boole's rule

Boole's rule is a method of numerical quadrature from calculus, named after the mathematician George Boole. It approximates the area under a curve over a fixed interval by sampling the function at five equally spaced points and combining the five values with fixed weights, so that polynomials up to the fifth order are integrated exactly over that interval. It belongs to the Newton-Cotes family of rules, the same family that contains the trapezoid rule and Simpson's rule. When an interval is split into equal sections whose count is a multiple of four, a composite version applies the same weighting to each group of four sections and adds the results, which gives a more accurate estimate of a long integral. The form printed in Abramowitz and Stegun's Handbook of Mathematical Functions has a small typographical slip that led many later writers to call it Bode's rule by mistake. Its references include Boole's Treatise on the Calculus of Finite Differences.