SUMMARY / RELATED TOPICS

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, being the other. Given a function f of a real variable x and an interval of the real line, the definite integral ∫ a b f d x can be interpreted informally as the signed area of the region in the xy-plane, bounded by the graph of f, the x-axis and the vertical lines x = a and x = b; the area above the x-axis adds to that below the x-axis subtracts from the total. The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the term integral may refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is written: F = ∫ f d x; the integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval once an antiderivative F of f is known, the definite integral of f over that interval is given by ∫ a b f d x = a b = F − F.

The principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann gave a rigorous mathematical definition of integrals, it is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the 19th century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or more variables, the interval of integration is replaced by a curve connecting the two endpoints. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space; the first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus, 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.

This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle; this method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen derived a formula for the sum of fourth powers, he used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of Indivisibles, work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of xn up to degree n = 9 in Cavalieri's quadrature formula.

Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers; the major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton. Leibniz published his work on calculus before Newton; the theorem demonstrates a connection between differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains.

This framework became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz. While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Rie