Numerical integration

Numerical integration, also called numerical quadrature, is a broad family of algorithms for calculating the numerical value of a definite integral when an exact antiderivative is unavailable or inconvenient. The integrand may be known only at sampled points, may have no elementary antiderivative as with the exponential of minus x squared, or may have an antiderivative harder to evaluate than a direct approximation. Most methods evaluate the integrand at a finite set of integration points and combine the values as a weighted sum called a quadrature rule, from step-function rules and the trapezoidal rule to interpolating rules, Gaussian quadrature and adaptive algorithms. Integrals over many dimensions, sometimes called cubature, are handled by Monte Carlo and quasi-Monte Carlo methods that avoid the exponential growth in evaluations with dimension. The term quadrature descends from the ancient problem of constructing a square of equal area, and the phrase numerical integration first appears in print in 1915.