Quadrature (geometry)

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In mathematics, quadrature is a historic term for the computation of areas and is thus used for computation of integrals.

The word is derived from the Latin quadratus meaning "square". The reason is that, for Ancient Greek mathematicians, the computation of an area consisted of constructing a square of the same area. In this sense, the modern term is squaring. For example, the quadrature of the circle, (or squaring the circle) is a famous old problem that has been shown, in the 19th century, to be impossible with the methods available to the Ancient Greeks,

Integral calculus, introduced in the 17th century, is a general method for computation of areas. Quadrature came to refer to the computation of any integral; such a computation is presently called more often "integral" or "integration". However, the computation of solutions of differential equations and differential systems is also called integration, and quadrature remains useful for distinguish integrals from solutions of differential equations, in contexts where both problems are considered. This is the case in numerical analysis; see numerical quadrature. Also, reduction to quadratures and solving by quadratures means expressing solutions of differential equations in terms of integrals.

The remainder of this article is devoted to the original meaning of quadrature, namely, computation of areas.