# Numerical integration

family of algorithms for finding the definite integral of a function

Numerical integration is the term used for a number of methods to find an approximation for an integral.[1] Numerical integration has also been called quadrature. Very often, it is not possible to solve integration analytically, for example when the data consists of a number of distinct measurements, or when the antiderivative is not known, and it is difficult, impractical or impossible to find it. In such cases, the integral can be written as a mathematical function defined over the interval in question, plus a function giving the error.

One way to find a numerical integral is using interpolation. Very often these interpolating functions are polynomials.

Various formulas have been studied for many years and become famous. For example, there is the Gaussian quadrature[2] (named after Gauss), the Newton-Cotes formula[3] (named after Isaac Newton), and the Euler-Maclaurin formula[4] (named after Leonhard Euler).

## Numerical errors

Numerical errors can occur in any kind of numerical computation including numerical integration. Errors in numerical integration are considered in another area called "validated numerics".[5]

## References

1. Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.
2. Weisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianQuadrature.html
3. Weisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Newton-CotesFormulas.html
4. Weisstein, Eric W. "Euler-Maclaurin Integration Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html
5. Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.