Upper sum (Riemann)

A weighted sum of suprema of f over subintervals of a partition.

Upper sum (Riemann)

Let $f:[a,b]\to\mathbb{R}$ be bounded and let $P:a=x_0<\cdots<x_n=b$ be a partition. For each subinterval, define

M_i := \sup\{f(x): x\in[x_{i-1},x_i]\}.

The upper sum of $f$ with respect to $P$ is

U(f,P) := \sum_{i=1}^n M_i\, (x_i-x_{i-1}).

Upper sums approximate the integral from above. As the partition is refined, upper sums decrease (or stay the same).

Examples:

If $f(x)=x$ on $[0,1]$ and $P:0<1/2<1$ , then $M_1=1/2$ , $M_2=1$ , so $U(f,P)=\frac12\cdot\frac12+1\cdot\frac12=\frac34$ .
If $f$ is constant, $f(x)=c$ , then $U(f,P)=c(b-a)$ for every $P$ .
For a bounded but highly oscillatory $f$ , $U(f,P)$ may remain far above any candidate limit unless the oscillations are controlled.