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An '''integral''' is a mathematical construction used in [[Calculus]] to represent the area of a region in a plane. Integrals use the following notation:  

An '''integral''' is a mathematical construction used in [[calculus]] to represent the area of a region in a plane bounded by the graph of a [[function]] in one [[real]] variable. Integrals use the following notation:  

<math>\int_a^b f(x)dx</math>

<math>\int_a^b f(x)dx</math>

where ''a'' and ''b'' represent the lower and upper bounds of the interval being integrated over, ''f(x)'' represents the function being integrated (the '''integrand'''), and ''dx'' represents a dummy variable given various definitions, depending on the context of the integral.

where ''a'' and ''b'' represent the lower and upper bounds of the [[interval]] being integrated over, ''f(x)'' represents the function being integrated (the '''integrand'''), and ''dx'' represents a dummy variable given various definitions, depending on the context of the integral.

There are two types of integrals.  Definite integrals are integrals that are evaluated over limits of integration.  Indefinite integrals are not evaluated over limits of integration.  Evaluating an indefinite integral yields the antiderivative of the integrand plus a constant of integration.

There are two types of integrals.  Definite integrals are integrals that are evaluated over limits of integration.  Indefinite integrals are not evaluated over limits of integration.  Evaluating an indefinite integral yields the antiderivative of the integrand plus a constant of integration.

Integration has many physical applications.  The indefinite integral of a time function of acceleration with respect to time gives the velocity function defined to within a constant, while the definite integral of a time function with respect to time gives the change in velocity between the upper and lower limits of integration.  Likewise, the indefinite integral of a time function of velocity with respect to time gives the position function defined to within a constant, and the definite integral of this velocity function will give the change in position between the two limits of integration.

Integration has many physical applications.  The indefinite integral of an [[acceleration]] function with respect to time gives the [[velocity]] function defined to within a constant, while the definite integral of an acceleration function with respect to time gives the change in velocity between the upper and lower limits of integration.  Likewise, the indefinite integral of a time function of velocity with respect to time gives the position function defined to within a constant, and the definite integral of this velocity function will give the change in position between the two limits of integration.

Integration is the inverse function of the [[derivative]], and is related to it by the [[Fundamental Theorem of Calculus]].

Integration is the inverse function of the [[derivative]] and the two notions are related by the [[Fundamental Theorem of Calculus]].

==Properties of intergrals==

 

==Properties of integrals==

Intergation has the following properties<ref>[http://www.sosmath.com/calculus/integ/integ02/integ02.html Properties of Intergrals]</ref>

Intergation has the following properties<ref>[http://www.sosmath.com/calculus/integ/integ02/integ02.html Properties of Intergrals]</ref>

<math>\int_a^b f(x)dx=g(b)-g(a)</math>

<math>\int_a^b f(x)dx=g(b)-g(a)</math>

This works for the kind of functions encountered in late high school and early university mathematics. It is, however, an incomplete method. For example one cannot write the anti-derivative of <math>e^{x^{2}}</math> in terms of familiar functions (such as trigonometric functions, exponentials, and logarithms) and function operations.

This works for the kind of functions encountered in late high school and early university mathematics. It is, however, an incomplete method. For example one cannot write the anti-derivative of <math>e^{x^{2}}</math> in terms of familiar functions (such as [[trigonometric function]]s, [[exponential]]s, and [[logarithm]]s) and function operations.

==Riemann intergral==

==Riemann integral==

As a geometric interpretation of the integral of the [[area]] of a curve, the Riemann integral consists of dividing the area under the curve of the function into rectangles. The [[domain]] of the function is partioned into N segments of width <math>\frac{b-a}{N}</math>. The height of the segment is dependent on which side of the rectangle is taken. The lower sum takes the lower side of the rectangle, the upper sum the higher side of the rectangle. In the [[limit]] of <math>N\rightarrow\infty</math> these two [[series (mathematics)|series]] become the integral. If they approach the same value then the integral exists, otherwise it is undefined.

As a geometric interpretation of the integral of the [[area]] of a curve, the Riemann integral consists of dividing the area under the curve of the function into rectangles. The [[domain]] of the function is partioned into N segments of width <math>\frac{b-a}{N}</math>. The height of the segment is dependent on which side of the rectangle is taken. The lower sum takes the lower side of the rectangle, the upper sum the higher side of the rectangle. In the [[limit]] of <math>N\rightarrow\infty</math> these two [[series (mathematics)|series]] become the integral. If they approach the same value then the integral exists, otherwise it is undefined.

==Lebesgue Intergral==

==Lebesgue Integral==

The Lebesgue intergral is usually introduced in late university or early postgraduate mathematics. It is naively described as rotating the Reimann intergral, in that it is the range instead of the domain that is partioned. An understanding of [[measure theory]] is required to understand this techniques.

The Lebesgue integral is usually introduced in late university or early postgraduate mathematics. It is naively described as rotating the Reimann integral, in that it is the range instead of the domain that is partitioned. An understanding of [[measure theory]] is required to understand this techniques.

==See Also==

==See Also==

==References==

==References==

{{reflist}}

{{reflist}}

[[Category:Calculus]]

[[Category:Calculus]]