Riemann sum

A Riemann sum approximates an integral by a finite sum.

Formally,
let $f$ be a real function on a closed interval $[a,b]$ with partition $P=(x_0,x_1,\dots ,x_n)$.
Then the Riemann sum $S$ is

where $\mathrm{\Delta }{x}_i=x_i-x_{i-1}$,
and $x_i^{\ast }\in [x_{i-1},x_i]$ is well-defined.

Choices of $x_i^{\ast }$ give different Riemann sums:

• $x_i^{\ast }=x_{i-1}$ is the left Riemann sum
• $x_i^{\ast }=x_i$ is the right Riemann sum
• $x_i^{\ast }=\frac{\mathrm{\Delta }x}{2}$ is the middle Riemann sum
• $f(x_i^{\ast })=inf\{f(x):x\in [x_{i-1},x_i]\}$ is the lower Riemann sum
• $f(x_i^{\ast })=sup\{f(x):x\in [x_{i-1},x_i]\}$ is the upper Riemann sum