Riemann sum

Let $I=[a,b]$ be a closed interval, $f:I\to ℝ$ be bounded on $I$, $n\in \mathbb{N}$, and $P=\{[x_0,x_1),[x_1,x_2),\mathrm{\dots }[x_{n-1},x_n]\}$ 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 $b_j=\underset{x\in [x_{j-1},x_j]}{sup}f(x)$, then $S$ is called an upper Riemann sum.

If $b_j=\underset{x\in [x_{j-1},x_j]}{inf}f(x)$, 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