Let $f$ be any function defined on the interval [a,b] and let $N$ be a positive integer. The uniform partition of order $N$ of the interval [a, b] is the set of equally spaced points

$x_{i}$ = $a+i.$$\frac{b-a}{N}$ $(0\leq i\leq N)$

that break the interval [a, b] into $N$ equal-length sub intervals

$I_{1}$ = $[x_{0},x_{1}]$, $I_{2}$ = $[x_{1},x_{2}]$,....$I_{N}$ = $[x_{N-1}, x_{N}]$

Here let $\Delta x$ denote the common length $\frac{b-a}{N}$ of these intervals.

A choice of points associated with the uniform partition of the order $N$ is a sequence $S_{N}$ = $(s_{1}, s_{2}...s_{n})$ of points with $s_{i}$ in $I_{i}$ for each $i=1, 2,....N$.

The expression

$R(f,S_{N})$ = $\sum _{i=1}^{N}f(s_{i}).\Delta x$ ....(1)

is called a Riemann sum of $f$. The notation $R(f, S_{N})$ indicates that a Riemann sum depends on the function $f$ and the choice of points $S_{N}$.

Finding areas within shapes gets
complicated fast outside of the standard geometric shapes like
circles, squares triangles and trapezoids. Riemann added up the area
under the complicated curves by using standard geometric shapes.

Different methods of arranging
rectangles under the curve results in more accurate, under-estimate
or over-estimate of area. The smaller and more numerous the
rectangle, the more accurate the estimates.

A Riemann sum can be determined in three different ways as given below