Discussion of Numerical Integration...

The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially when applied to one-dimensional integrals. Two-and-higher dimensional integration is sometimes known as cubature, although the meaning of quadrature is used for higher dimensional integration as well.

Numerical integration is intrinsically a much more accurate procedure than numerical differentiation.

The basic problem considered by numerical integration is computing an approximation to a definite integral,

If f(x) is a smooth well-defined function, integrated over a small number of dimensions, and the limits of integration are bounded, there are many excellent methods to approximate to the integral with arbitrary precision.

The need for a numerical integration arises for two reasons: First, the function to be integrated may be such that the integral is too complicated to evaluate or may even be impossible to obtain analytically. Secondly, the function may be described only by a data array of values, so that a numerical approximation is the only approach available.

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