Definition: Riemann Sum With Respect to a Partition

Let \([a,b]\) be a closed real interval, \(f:[a,b]\mapsto\mathbb R\) be a bounded real function, and \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\) be a partition of the interval \([a,b]\). Further, let \(\xi\in[x_{k-1},x_k]\) (i.e. \(\xi\) is any point in the interval \([x_{k-1},x_k]\), \(1\le k\le n\)). We call

\[\sum_{k=1}^n f(\xi_k)(x_k-x_{k-1})\]

a Riemann sum of \(f\) with respect to the partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\).

Please note that we can have different Riemann sums of \(f\) with respect to the same partition, since the choice of the supporting points \(\xi_k\) is arbitrary in each interval \([x_{k-1},x_k]\).

The mesh \(\mu\) of the partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\) is defined as the maximum length of a single partition interval

\[\mu:=\max_{1\le k\le n}(x_k-x_{k-1}).\]

  1. Proposition: Riemann Sum Converging To the Riemann Integral

Propositions: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983