Simpson Rule

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

for points that are equally spaced. For unequally spaced points, see Cartwright.

Simpson's rule also corresponds to the three-point Newton-Cotes quadrature rule.

The method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. Johannes Kepler used similar formulas over 100 years prior. For this reason the method is sometimes called Kepler's rule, or Keplersche Fassregel in German.

Simpson's rule can be derived in various ways.

One derivation replaces the integrand ${\displaystyle f(x)}$ by the quadratic polynomial (i.e. parabola)${\displaystyle P(x)}$ which takes the same values as ${\displaystyle f(x)}$ at the end points a and b and the midpoint m = (a + b) / 2. One can use Lagrange polynomial interpolation to find an expression for this polynomial,

...
Wikipedia