In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an integral assigns numbers to
functions in a way that describes
displacement,
area,
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
, and other concepts that arise by combining
infinitesimal data. The process of finding integrals is called integration. Along with
differentiation, integration is a fundamental, essential operation of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
,
[Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example.] 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 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. 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
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
, a function whose derivative is the given function. In this case, they are called indefinite integrals. 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.
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
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
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.
Integrals may be generalized depending on the type of the function as well as the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
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 the two endpoints of the interval. 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 astronomer
Eudoxus (''ca.'' 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. 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
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
and
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, 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
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
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) 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, 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. 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 within continuous domains. This framework eventually became modern
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, 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
George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immater ...
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
The long s , also known as the medial s or initial s, is an archaic form of the lowercase letter . It replaced the single ''s'', or one or both of the letters ''s'' in a 'double ''s sequence (e.g., "ſinfulneſs" for "sinfulness" and "po� ...
), 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 1819–20, 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 sign 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 over its domain is finite. 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
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
) 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.
For example, to find the area of the region bounded by the graph of the function between and , one can cross the interval in five steps (), then fill a rectangle using the right end height of each piece (thus ) and sum their areas to get an approximation of
:
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 increase 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 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 distinguished 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 distinguished 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