# Cotes formulas

2010 Mathematics Subject Classification: Primary: 65D32 [MSN][ZBL]

Formulas for the approximate computation of definite integrals, given the values of the integrand at finitely many equidistant points, i.e. quadrature formulas with equidistant interpolation points. Cotes' formulas are $$\label{eq:1} \int_0^1 f(x) \, dx \approx \sum_{k=0}^n a_k^{(n)} f\left({ \frac{k}{n} }\right)\,,\ \ \ n=1,2,\ldots$$

The numbers $a_k^{(n)}$ are known as Cotes' coefficients; they are determined from the condition that formula \eqref{eq:1} be exact if $f(x)$ is a polynomial of degree at most $n$.

The formulas were proposed by R. Cotes (1722) and considered in a more general form by I. Newton. See Newton–Cotes quadrature formula.