Riemann sum

Let $I=[a,b]$ be a closed interval, $f:I\rightarrow\mathbb{R}$ be bounded on $I$, $n\in\mathbb{N}$, and $P=\{[x_{0},x_{1}),[x_{1},x_{2}),\dots[x_{n-1},x_{n}]\}$ be a partition of $I$. The Riemann sum of $f$ over $I$ with respect to the partition $P$ is defined as

 $S=\sum_{j=1}^{n}f(c_{j})(x_{j}-x_{j-1})$

where $c_{j}\in[x_{j-1},x_{j}]$ is chosen arbitrary.

If $c_{j}=x_{j-1}$ for all $j$, then $S$ is called a left Riemann sum.

If $c_{j}=x_{j}$ for all $j$, then $S$ is called a Riemann sum.

Equivalently, the Riemann sum can be defined as

 $S=\sum_{j=1}^{n}b_{j}(x_{j}-x_{j-1})$

where $b_{j}\in\{f(x):x\in[x_{j-1},x_{j}]\}$ is chosen arbitrarily.

If $\displaystyle b_{j}=\sup_{x\in[x_{j-1},x_{j}]}f(x)$, then $S$ is called an upper Riemann sum.

If $\displaystyle b_{j}=\inf_{x\in[x_{j-1},x_{j}]}f(x)$, then $S$ is called a lower Riemann sum.

For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.

 Title Riemann sum Canonical name RiemannSum Date of creation 2013-03-22 11:49:17 Last modified on 2013-03-22 11:49:17 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 14 Author Wkbj79 (1863) Entry type Definition Classification msc 26A42 Related topic RiemannIntegral Related topic RiemannStieltjesIntegral Related topic LeftHandRule Related topic RightHandRule Related topic MidpointRule Defines left Riemann sum Defines right Riemann sum Defines upper Riemann sum Defines lower Riemann sum