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The term "quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve y = x n. Traditionally important cases are y = x 2 , the quadrature of the parabola , known in antiquity, and y = 1/ x , the quadrature of the hyperbola , whose value is a logarithm .
Integration, the process of computing an integral, is one of the two fundamental operations of calculus, [a] the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide ...
The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb. [2] "Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis.
The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b.This notion of area fits some functions, mainly piecewise continuous functions, including elementary functions, for example polynomials.
Taking an example, the area under the curve y = x 2 over [0, 2] can be procedurally computed using Riemann's method. The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of 2 n {\displaystyle {\tfrac {2}{n}}} ; these are the widths of the Riemann rectangles (hereafter "boxes").
It is assumed that the value of a function f defined on [,] is known at + equally spaced points: < < <.There are two classes of Newton–Cotes quadrature: they are called "closed" when = and =, i.e. they use the function values at the interval endpoints, and "open" when > and <, i.e. they do not use the function values at the endpoints.
The function f(x) (in blue) is approximated by a linear function (in red). In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: ().
The integral as the area of a region under a curve. A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function. The partition does not need to be regular, as shown here.