Numerical integration
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]
People who studied about numerical integration
Numerical Integration Media
Numerical integration is used to calculate a numerical approximation for the value S, the area under the curve defined by f(x).
Antique method to find the Geometric mean
The area of a segment of a parabola 
Illustration of the rectangle rule.
Illustration of the trapezoidal rule.
Illustration of Simpson's rule.
References
- ↑ Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.
- ↑ Weisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianQuadrature.html
- ↑ Weisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Newton-CotesFormulas.html
- ↑ Weisstein, Eric W. "Euler-Maclaurin Integration Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html
- ↑ Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.
Numerical integration software
- MATLAB, INTLAB
- GNU Octave, Scilab - open source versions of MATLAB
- Wolfram Mathematica - made by Wolfram Research