In numerical analysis, **numerical integration** constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term **numerical quadrature** (often abbreviated to *quadrature*) is more or less a synonym for *numerical integration*, especially as applied to one-dimensional integrals. Numerical integration over more than one dimension is sometimes described as **cubature**, although the meaning of *quadrature* is understood for higher dimensional integration as well.

The basic problem considered by numerical integration is to compute an approximate solution to a definite integral:

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

Read more about Numerical Integration: Reasons For Numerical Integration, Methods For One-dimensional Integrals, Multidimensional Integrals, Connection With Differential Equations

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“The more specific idea of evolution now reached is—a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and *integration* of matter.”

—Herbert Spencer (1820–1903)

“The terrible tabulation of the French statists brings every piece of whim and humor to be reducible also to exact *numerical* ratios. If one man in twenty thousand, or in thirty thousand, eats shoes, or marries his grandmother, then, in every twenty thousand, or thirty thousand, is found one man who eats shoes, or marries his grandmother.”

—Ralph Waldo Emerson (1803–1882)