In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation.Integral calculus is a very well-established mathematical discipline for which there are many sources. See and , for example. 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 variety of scientific fields thereafter.
A definite integral computes 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. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. 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. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations.
Although 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. 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. 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 general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. 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 two points in space. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.
History
Pre-calculus integration The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek mathematicians Democritus and Eudoxus (4th century 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.. 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..
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 later 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 ( AD), extended Archimedes' method of exhaustion, finding the volume of the solid of revolution formed by rotating a parabola around a line perpendicular to its axis. To achieve this, Al-Haytham invented a way of computing the sums of kth powers, which he applied for the sums of squares and the sums of fourth powers...
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, and work by Fermat, began to lay the foundations of modern calculus,. with Cavalieri computing the integrals of up to degree in Cavalieri's quadrature formula.. The case n = −1 required the invention of a function, the hyperbolic logarithm, achieved by quadrature of the hyperbola in 1647.
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 to a general power, including negative powers and fractional powers..
Leibniz and Newton The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton.. The theorem demonstrates a connection between integration and 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 with continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.
Formalization 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 Riemann.. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.
Historical notation The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675.; . He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–1820, reprinted in his book of 1822.; .
Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted..
First use of the term The term was first printed in Latin by Jacob Bernoulli in 1690: "Ergo et horum Integralia aequantur"..
Terminology and notation In general, the integral of a real-valued function with respect to a real variable on an interval is written as
The integral symbol represents integration. The symbol , called the differential of the variable , indicates that the variable of integration is . The function is called the integrand, the points and are called the limits (or bounds) of integration, and the integral is said to be over the interval , called the interval of integration.. A function is said to be if its integral is well-defined and finite.. (The type of integral being considered determines the type of integrability this means, thus Riemann integrable means that the upper and lower Riemann sums converge over the domain, Lebesgue integrable means that the Lebesgue integral exists and is finite, and so on.) If limits are specified, the integral is called a definite integral.
When the limits are omitted, as in
the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand.. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article).
In advanced settings, it is not uncommon to leave out when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof..
Interpretations
Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many infinitesimal pieces, then sum the pieces to achieve an accurate approximation.
As another example, to find the area of the region bounded by the graph of the function between and , one can divide the interval into five pieces (), then construct rectangles using the right end height of each piece (thus ) and sum their areas to get the approximation
which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, ). One writes
which means is the result of a weighted sum of function values, , multiplied by the infinitesimal step widths, denoted by , on the interval .
Formal definitions There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.
Riemann integral
The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval.. A tagged partition of a closed interval on the real line is a finite sequence
This partitions the interval into sub-intervals indexed by , each of which is "tagged" with a specific point . A Riemann sum of a function with respect to such a tagged partition is defined as
thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, . The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, . The Riemann integral of a function over the interval is equal to if:.
For all there exists such that, for any tagged partition with mesh less than ,
When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.
Lebesgue integral