Numerical integration

Numerical integration is the term used for a number of methods to find an approximation for an integral.Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation. 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 quadratureWeisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianQuadrature.html (named after Gauss), the Newton-Cotes formulaWeisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Newton-CotesFormulas.html (named after Isaac Newton), and the Euler-Maclaurin formulaWeisstein, Eric W. "Euler-Maclaurin Integration Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html (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".Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.

People who studied about numerical integration

 Masaaki Sugihara
 Masao Iri
 Masatake Mori

 References 

Numerical integration software
 MATLAB, INTLAB
 GNU Octave, Scilab - open source versions of MATLAB
 Wolfram Mathematica - made by Wolfram Research

Category:Calculus
Category:Numerical analysis