Simpson's Rule.[Wikipedia]

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:

The method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. Kepler used similar formulas over 100 years prior and in German the method is sometimes called Keplersche Fassregel for this reason. Kepler would, of course, have been unaware of the connection to calculus.
Simpson's rule is a staple of scientific data analysis and engineering. It is widely used, for example, by Naval architects to calculate the capacity of a ship or lifeboat.^[1]

Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P (x) (in red).