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==Riemann integral==

==Riemann integral==

{{main|Riemann Integral}}

{{main|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]] under a curve, the Riemann integral consists of dividing the area under the curve of the function into slices. 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 Integral==

==Lebesgue Integral==

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.

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 technique.

==See Also==

==See Also==

*[[Indefinite integral]]

*[[Indefinite integral]]

===External Links===

===External Links===

[http://mathworld.wolfram.com/Integral.html Integrals - Wolfram MathWorld]

*[http://mathworld.wolfram.com/Integral.html Integrals - Wolfram MathWorld]

*[http://www.relativitycalculator.com/mathematical_references.shtml Some Quick and Dirty Mathematical References]

*[http://www.relativitycalculator.com/mathematical_references.shtml Some Quick and Dirty Mathematical References]