This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. 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 differentiation and integration are inverse operations.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. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function in this case, they are also called indefinite integrals. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. The integrals enumerated here are called 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. Today integration is used in a wide variety of scientific fields. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations.
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