Riemann Sum

analysis

Definition

Riemann Sum

Let be a bounded function. A Riemann sum is an approximation of the area under the curve of using a finite sum of rectangles.

Let be a partition of such that:

For each sub-interval , choose a sample point . The Riemann sum associated with partition and sample points is defined as:

Mesh and Convergence

The fineness (or mesh) of a partition , denoted by , is the length of the longest sub-interval:

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