370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ( ca. See also: History of calculus Pre-calculus integration In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.Īlthough methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. In this case, they are called indefinite integrals. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function.
Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The process of finding integrals is called integration. In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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