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{{Math-h}}

The '''Riemann integral''' is the mathematical definition of the integral of a function, that is, a measure of the area enclosed by its graph in [[calculus]].  The term, along with '''Riemann sum''', is also a useful method for approximating that area.

The '''Riemann integral''' is the mathematical definition of the integral of a function, that is, a measure of the area enclosed by its graph in [[calculus]].  The term, along with '''Riemann sum''', is also a useful method for approximating that area.

The breaking down of the function graph into rectangular strips, and approximating the area under the graph in terms of rectangles, is a very well-known technique, and is what is commonly called the "Riemann integral", or, more properly, the "Riemann sum".  Most of the rest of this article is about it.  However, Bernhard Riemann's contribution to this, and the reason we name it after him, is his theoretical achievement.  (Reimann was a brilliant theoretical mathematician.  He essentially invented the field of differential geometry, as well as making major contributions to number theory and other areas.)

The breaking down of the function graph into rectangular strips, and approximating the area under the graph in terms of rectangles, is a very well-known technique, and is what is commonly called the "Riemann integral", or, more properly, the "Riemann sum".  Most of the rest of this article is about it.  However, Bernhard Riemann's contribution to this, and the reason we name it after him, is his theoretical achievement.  (Reimann was a brilliant theoretical mathematician.  He essentially invented the field of differential geometry, as well as making major contributions to number theory and other areas.)

What Riemann did was to show that, if one takes the minimum and maximum values (as described above), and makes the strips narrower and narrower, (the mesh finer and finer), the two approximations will [[Convergence|converge]] to each other.  That is, they will approach the same value as a [[Limit_(mathematics)|limit]].  Mathematicians take that limit as the '''definition''' of the (Riemann) integral.  So Riemann's achievement was to take the common computational task of adding up the strip areas, and make a proper (or, as mathematicians say, ''rigorous'') definition out of it.  In so doing, he put the definite integral, and hence integral calculus, on a firm theoretical basis.

What Riemann did was to show that, if one takes the minimum and maximum values (as described above), and makes the strips narrower and narrower, (the mesh finer and finer), the two approximations will [[Convergence|converge]] to each other.  That is, they will approach the same value as a [[Limit (mathematics)|limit]].  Mathematicians take that limit as the '''definition''' of the (Riemann) integral.  So Riemann's achievement was to take the common computational task of adding up the strip areas, and make a proper (or, as mathematicians say, ''rigorous'') definition out of it.  In so doing, he put the definite integral, and hence integral calculus, on a firm theoretical basis.

Riemann's proof is beyond the scope of this article.  It applies to any function which is either continuous or has a finite number of discontinuities.  (At a more advanced level, it applies to any function for which the set of discontinuities has measure zero.)

Riemann's proof is beyond the scope of this article.  It applies to any function which is either continuous or has a finite number of discontinuities.  (At a more advanced level, it applies to any function for which the set of discontinuities has measure zero.)

To define the limit, he had to modify the usual "epsilon/delta" formulation.  The definition is this (see the "[[Limit_(mathematics)|limit]]" article):

To define the limit, he had to modify the usual "epsilon/delta" formulation.  The definition is this (see the "[[Limit (mathematics)|limit]]" article):

:For every epsilon greater than zero, there is a mesh such that, for that mesh and all finer meshes (put in additional subdivision points), the difference between the upper sum (using the maximum function value) and the lower sum (using the minimum value) is less than epsilon.

:For every epsilon greater than zero, there is a mesh such that, for that mesh and all finer meshes (put in additional subdivision points), the difference between the upper sum (using the maximum function value) and the lower sum (using the minimum value) is less than epsilon.

Where <math>a</math> is known as the ''[[constant of integration]]'' which is necessary here since we have performed an [[indefinite integration]] - that is integrated without specifying the limits of integration. This is done because any [[constant]] <math>a</math> will give you a function that has the desired derivative.

Where <math>a</math> is known as the ''[[constant of integration]]'' which is necessary here since we have performed an [[indefinite integration]] - that is integrated without specifying the limits of integration. This is done because any [[constant]] <math>a</math> will give you a function that has the desired derivative.

==See Also==

==See also==

* [[Integral]]

* [[Integral]]

* [[Bernhard Riemann]]

* [[Bernhard Riemann]]

[[Category:Calculus]]

[[Category:Calculus]]

[[Category:integration]]

[[Category:Integration]]