Riemann Sum

Let a Closed Interval $[a,b]$ be partitioned by points $a<x_1<x_2<\ldots<x_{n-1}<b$, the lengths of the resulting intervals between the points are denoted $\Delta x_1$, $\Delta x_2$, ..., $\Delta x_n$. Then the quantity

$$\sum_{k=1}^n f(x_k^*)\Delta x_k$$

is called a Riemann sum for a given function $f(x)$ and partition. The value $\max\Delta x_k$ is called the Mesh Size of the partition. If the Limit $\max\Delta x_k\to 0$ exists, this limit is known as the Riemann Integral of $f(x)$ over the interval $[a,b]$. The shaded areas in the above plots show the Lower and Upper Sums for a constant Mesh Size.

See also Lower Sum, Riemann Integral, Upper Sum