
In
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an integral assigns numbers to functions in a way that describes displacement,
area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

,
volume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

, and other concepts that arise by combining
infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
data. The process of finding integrals is called integration. Along with
differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in marketing
* Differentiated service, a service that varies with the identity o ...

, 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 generalizations ...

,
[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
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

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
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

of the region in the plane that is bounded by the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

of a given
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
between two points in the
real line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
, a function whose derivative is the given function. In this case, they are called indefinite integrals. The
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
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
Greek mathematics refers to mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
, the principles of integration were formulated independently by
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

and
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath
A polymath ( el, πολυμαθής, ', "having learned much"; Latin
Latin (, or , ...

in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of
infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
width.
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of ...
later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a
curvilinear
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of spac ...
region by breaking the region into thin vertical slabs.
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
*Doma ...
over which the integration is performed. For example, a
line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, the curve is replaced by a piece of a
surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
in
three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
.
History
Pre-calculus integration
The first documented systematic technique capable of determining integrals is the
method of exhaustion
The method of exhaustion (; ) is a method of finding the area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

of the
ancient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...
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
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

in the 3rd century BC and used to calculate the
area of a circle
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

, the
surface area
The surface area of a solid
Solid is one of the four fundamental states of matter
4 (four) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of ...

and
volume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

of a
sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

, area of an
ellipse
In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

, the area under a
parabola
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, the volume of a segment of a
paraboloid
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

of revolution, the volume of a segment of a
hyperboloid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
of revolution, and the area of a
spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are: .
A similar method was independently developed in
China
China (), officially the People's Republic of China (PRC; ), is a country in East Asia
East Asia is the eastern region of Asia
Asia () is Earth's largest and most populous continent, located primarily in the Eastern Hemisphere ...

around the 3rd century AD by
Liu Hui
Liu Hui () was a Chinese mathematician and writer who lived in the state of Cao Wei
Wei (220–266), also known as Cao Wei or Former Wei, was one of the three major states that competed for supremacy over China in the Three Kingdoms perio ...
, 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
Zu or ZU may refer to:
Arts and entertainment Fictional elements
* Zu, a mountain featured in the films '' Zu Warriors from the Magic Mountain'' and '' The Legend of Zu''
* ''ZU'', a " furry" anthology published by MU Press
* Zu, a large birdlike m ...
and
Zu Geng to find the volume of a sphere.
In the Middle East, Hasan Ibn al-Haytham, Latinized as
Alhazen
Ḥasan Ibn al-Haytham (Latinization of names, Latinized as Alhazen ; full name ; ) was a Muslim Arab Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, and Physics in the medieval Islamic world, ...
( AD) derived a formula for the sum of
fourth power
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...
s. 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
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

.
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of
with his
method of Indivisibles
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
, and work by
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607
– 12 January 1665) was a French lawyer at the '' Parlement'' of Toulouse
Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of up to degree in
Cavalieri's quadrature formula
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. T ...
. Further steps were made in the early 17th century by
Barrow
Barrow may refer to:
Places
England
* Barrow-in-Furness, Cumbria
** Borough of Barrow-in-Furness, local authority encompassing the wider area
** Barrow and Furness (UK Parliament constituency)
* Barrow, Cheshire
* Barrow upon Soar, Leicestershire ...

and
, who provided the first hints of a connection between integration and
differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in marketing
* Differentiated service, a service that varies with the identity o ...
. Barrow provided the first proof of the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
.
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
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
by
Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

and
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* Newton (film), ''Newton'' (film), a 2017 Indian fil ...

. 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 generalizations ...

, 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
Rigour (British English
British English (BrE) is the standard dialect of the English language
English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, wh ...
.
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 people, Anglo-Irish philosopher whose primary achievement was the advancement of a theory ...

memorably attacked the vanishing increments used by Newton, calling them "
". Calculus acquired a firmer footing with the development of
limits
Limit or Limits may refer to:
Arts and media
* Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
. Integration was first rigorously formalized, using limits, by
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of ...
. 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
—to which Riemann's definition does not apply, and
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a France, French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area betwe ...
formulated a
different definition of integral, founded in
measure theory
Measure is a fundamental concept of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
(a subfield of
real analysis
200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

). 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
system.
Historical notation
The notation for the indefinite integral was introduced by
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath
A polymath ( el, πολυμαθής, ', "having learned much"; Latin
Latin (, or , ...

in 1675. He adapted the
integral symbol
The integral symbol:
:∫ (Unicode
Unicode is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's ...
, ∫, from the letter ''ſ'' (
long s
The long s, , is an archaism, archaic form of the letter case, lower case 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 "poſſeſs" or "poſseſs" fo ...

), 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
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French Republic (french: link=no, R ...

in ''Mémoires'' of the French Academy around 1819–20, reprinted in his book of 1822.
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

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
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) inclu ...
in 1690: "Ergo et horum Integralia aequantur".
Terminology and notation
In general, the integral of a
real-valued function
Mass measured in grams is a function from this collection of weight to positive number">positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, i ...
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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
) whose derivative is the integrand. The
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
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
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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 sum
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of functions with respect to ''tagged partitions'' of an interval. A tagged partition of a
closed interval
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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