Riemann sumRiemannSum2001-10-19 01:20:272007-05-18 01:58:00Definitionleft Riemann sumright Riemann sumupper Riemann sumlower Riemann sum\usepackage{amssymb}
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\usepackage{xypic}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 \emph{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 \emph{left Riemann sum}.
If $c_j=x_j$ for all $j$, then $S$ is called a \emph{\PMlinkescapetext{right} 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 \emph{upper Riemann sum}.
If $\displaystyle b_j=\inf_{x\in[x_{j-1},x_j]} f(x)$, then $S$ is called a \emph{lower Riemann sum}.
For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.