Riemann sum - Imported Knowlpedia Content

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

S(f;P,\{t_i\})=\sum_{i=1}^n f(t_i)\,(x_i-x_{i-1}).

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 $S(f;P,\{t_i\})=\tfrac14\cdot\tfrac12+\tfrac34\cdot\tfrac12=\tfrac12$ .