Riemann Sum

analysis

Definition

Riemann Sum

Let $f$ be a bounded function on an interval $[a,b]$, which is partitioned into smaller intervals $[x_0,x_1],\dots ,[x_{n-1},x_n]$ with:

$$a=x_0<x_1<\dots <x_{n-1}<x_n=b$$

The length of the longest sub-interval of a partition $Z$ is called the fineness (or mesh) $\mathcal{F}(Z)$.

Then, we evaluate $f$ in every interval ${\xi }_i\in [x_{i-1},x_i]$ and define the sum:

$$S_n=\sum _{i=1}^{n}f({\xi }_i)(x_i-x_{i-1})$$

The sum corresponding to a partition and a choice of intermediate points (or sample points) is called a Riemann sum.