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Let ![$ I=[a,b]$](http://images.planetmath.org:8080/cache/objects/368/l2h/img1.png "$ I=[a,b]$") be a closed interval,
 be bounded on  ,
 , and
![$ P = \{[x_0, x_1), [x_1, x_2), \dots [x_{n-1}, x_n]\}$](http://images.planetmath.org:8080/cache/objects/368/l2h/img5.png "$ P = \{[x_0, x_1), [x_1, x_2), \dots [x_{n-1}, x_n]\}$") be a partition of  . The Riemann sum of  over  with respect to the partition  is defined as
where
![$ c_j \in [x_{j-1},x_j]$](http://images.planetmath.org:8080/cache/objects/368/l2h/img11.png "$ c_j \in [x_{j-1},x_j]$") is chosen arbitrary.
If
 for all  , then  is called a left Riemann sum.
If  for all  , then  is called a right Riemann sum.
Equivalently, the Riemann sum can be defined as
where
![$ b_j \in \{ f(x):x\in[x_{j-1},x_j]\}$](http://images.planetmath.org:8080/cache/objects/368/l2h/img19.png "$ b_j \in \{ f(x):x\in[x_{j-1},x_j]\}$") is chosen arbitrarily.
If
![$ \displaystyle b_j=\sup_{x\in[x_{j-1},x_j]} f(x)$](http://images.planetmath.org:8080/cache/objects/368/l2h/img20.png "$ \displaystyle b_j=\sup_{x\in[x_{j-1},x_j]} f(x)$") , then  is called an upper Riemann sum.
If
![$ \displaystyle b_j=\inf_{x\in[x_{j-1},x_j]} f(x)$](http://images.planetmath.org:8080/cache/objects/368/l2h/img22.png "$ \displaystyle b_j=\inf_{x\in[x_{j-1},x_j]} f(x)$") , then  is called a lower Riemann sum.
For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.
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