Riemann sum

A Riemann sum of a bounded function $f:[a,b]\to\mathbb R$ with respect to a tagged partition $(P,\{t_i\})$, where $P=\{x_0,\dots,x_n\}$, is the number

Riemann sums approximate the Riemann integral ; the approximation is controlled by the mesh $\|P\|$ becoming small.

Examples:

• If $f(x)=1$ on $[a,b]$, then every Riemann sum equals $b-a$.
• For $f(x)=x$ on $[0,1]$, take $P=\{0,\tfrac12,1\}$ and midpoint tags $t_1=\tfrac14$, $t_2=\tfrac34$. Then $S(f;P,\{t_i\})=\tfrac14\cdot\tfrac12+\tfrac34\cdot\tfrac12=\tfrac12$.