Numerical integration

Numerical integration is the term used for a number of methods to find an approximation for an integral. 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 (named after Gauss), the Newton-Cotes formula (named after Isaac Newton), and the Euler-Maclaurin formula (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".