Riemann sum

Let $f:[a,b]\to\mathbb{R}$ be bounded and let $(P,T)$ be a tagged partition with $P:a=x_0<\cdots<x_n=b$ and tags $t_i\in[x_{i-1},x_i]$. The Riemann sum of $f$ for $(P,T)$ is

Riemann sums approximate the integral by sampling $f$ at finitely many points and weighting by interval lengths. The Riemann integral (when it exists) is the common limit of these sums as the mesh tends to $0$.

Examples:

• For $f(x)=x$ on $[0,1]$, using the uniform partition $x_i=i/n$ and right-endpoint tags gives $S=\sum_{i=1}^n \frac{i}{n}\cdot\frac{1}{n}=\frac{1}{n^2}\cdot\frac{n(n+1)}{2}\to \frac12.$
• If $f$ is constant $c$, then $S(f;P,T)=c(b-a)$ for every tagged partition.
• For discontinuous $f$, Riemann sums may depend strongly on the tags unless $f$ is integrable.