Riemann Sum

analysis

Definition

Riemann Sum

Let $f:[a,b]\to \mathbb{R}$ be a bounded function. A Riemann sum is an approximation of the area under the curve of $f$ using a finite sum of rectangles.

Let $Z=(x_0,x_1,\dots ,x_n)$ be a partition of $[a,b]$ such that:

$$a=x_0<x_1<\cdots <x_{n-1}<x_n=b$$

For each sub-interval $[x_{i-1},x_i]$, choose a sample point ${\xi }_i\in [x_{i-1},x_i]$. The Riemann sum $S$ associated with partition $Z$ and sample points ${\xi }_i$ is defined as:

$$S=\sum _{i=1}^{n}f({\xi }_i)(x_i-x_{i-1})$$

Mesh and Convergence

The fineness (or mesh) of a partition $Z$, denoted by $\text{mesh}(Z)$, is the length of the longest sub-interval:

$$\text{mesh}(Z)=\underset{1\le i\le n}{\max }(x_i-x_{i-1})$$

As the mesh approaches zero, the Riemann sum of a continuous function converges to the definite integral:

$$\underset{\text{mesh}(Z)\to 0}{\lim }\sum _{i=1}^{n}f({\xi }_i)(x_i-x_{i-1})={\int }_a^{b}f(x)dx$$