Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A Riemann sum is a method for approximating the total area under a curve on a graph, otherwise known as an integral. It sums up the areas of multiple rectangles to estimate the value of an integral.
5 Must Know Facts For Your Next Test
Riemann sums can be calculated using left endpoints, right endpoints, or midpoints.
The more rectangles (subintervals) used in the approximation, the more accurate the Riemann sum becomes.
There are three common types of Riemann sums: Left Riemann Sum, Right Riemann Sum, and Midpoint Riemann Sum.
The width of each rectangle in a Riemann sum is called $\Delta x$, which represents the change in x-values over each subinterval.
As the number of subintervals approaches infinity, the Riemann sum approaches the exact value of the definite integral.
The limit of a Riemann sum as the number of subdivisions tends to infinity; it represents the exact area under a curve between two points.
$\Delta x$: $\Delta x$ is the width of each subinterval in a partition of an interval; calculated as $(b - a) / n$, where $a$ and $b$ are bounds and $n$ is number of subintervals.
$f(x_i^*) \cdot \Delta x$: $f(x_i^*) \cdot \Delta x$ represents one term in a Riemann sum, where $x_i^*$ is a sample point within each subinterval.