# Integral

**integrationintegral calculusdefinite integralintegratingintegratedintegrateintegrable functionintegralsintegrandarea under the curve**

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.wikipedia

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### Calculus

**infinitesimal calculusdifferential and integral calculusclassical calculus**

Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other.

It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves).

### Infinitesimal

**infinitesimalsinfinitely closeinfinitesimally**

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Infinitely many infinitesimals are summed to produce an integral.

### Interval (mathematics)

**intervalopen intervalclosed interval**

Given a function of a real variable and an interval It is the fundamental theorem of calculus that connects differentiation with the definite integral: if is a continuous real-valued function defined on a closed interval

Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define.

### Derivative

**differentiationdifferentiablefirst derivative**

Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other.

The fundamental theorem of calculus states that antidifferentiation is the same as integration.

### Line integral

**path integralcontour integralcontour integration**

A line integral is defined for functions of two or more variables, and the interval of integration

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.

### Fundamental theorem of calculus

**fundamental theorem of real calculusfundamental theorem of the calculusFundamental- of Calculus**

It is the fundamental theorem of calculus that connects differentiation with the definite integral: if is a continuous real-valued function defined on a closed interval

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

### Antiderivative

**indefinite integralindefinite integrationantidifferentiation**

For this reason, the term integral may also refer to the related notion of the antiderivative, a function whose derivative is the given function.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

### Differential calculus

**differentiationdifferentialcalculus**

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.

It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve.

### Cavalieri's quadrature formula

in Cavalieri's quadrature formula.

In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral

### Cavalieri's principle

**method of indivisiblesindivisiblesa similar method**

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

Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration.

### Area of a circle

**area enclosed by a circlearea of a diskarea inside a circle**

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.

Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis.

### Integral symbol

**integral sign∫⌠**

He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for "sum" or "total").

is used to denote integrals and antiderivatives in mathematics.

### Bonaventura Cavalieri

**CavalieriCavalieri, Bonaventura**

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

Cavalieri's principle in geometry partially anticipated integral calculus.

### Limit (mathematics)

**limitlimitsconverge**

Calculus acquired a firmer footing with the development of limits.

Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

### Probability density function

**probability densitydensity functiondensity**

Moreover, the integral under an entire probability density function must equal 1, which provides a test of whether a function with no negative values could be a density function or not.

This probability is given by the integral of this variable’s PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range.

### Differential form

**differential forms2-formtwo-form**

Some common interpretations of dx include: an integrator function in Riemann-Stieltjes integration (indicated by dα(x) in general), a measure in Lebesgue theory (indicated by dμ in general), or a differential form in exterior calculus (indicated by in general).

Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds.

### Mathematics

**mathematicalmathmathematician**

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics.

### Measure (mathematics)

**measuremeasure theorymeasurable**

Some common interpretations of dx include: an integrator function in Riemann-Stieltjes integration (indicated by dα(x) in general), a measure in Lebesgue theory (indicated by dμ in general), or a differential form in exterior calculus (indicated by in general). 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).

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

### Haar measure

**unimodularHaar measuresaverage over all possible rotations**

The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

### Riemann integral

**RiemannRiemann integrableRiemann-integrable**

The Darboux integral, which is constructed using Darboux sums and is equivalent to a Riemann integral, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being simpler to define than Riemann integrals.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

### Volume integral

**integral over spacevolumecomputing the volume**

Integrals can be used for computing the area of a two-dimensional region that has a curved boundary, as well as computing the volume of a three-dimensional object that has a curved boundary.

In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals.

### Real analysis

**realtheory of functions of a real variablefunction theory**

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).

(This value can include the symbols \pm\infty when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a limit is fundamental to calculus (and mathematical analysis in general) and its formal definition is used in turn to define notions like continuity, derivatives, and integrals.

### Henstock–Kurzweil integral

**Denjoy integralgauge integralHenstock–Kurzweil**

The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.

In mathematics, the Henstock–Kurzweil integral or gauge integral (also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral) is one of a number of definitions of the integral of a function.

### Ralph Henstock

**Henstock, Ralph**

The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.

As an Integration theorist, he is notable for Henstock–Kurzweil integral.

### Archimedes

**Archimedes of SyracuseArchimedeanArchimedes Heat Ray**

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.

Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus.