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 this area.
5 Must Know Facts For Your Next Test
Riemann sums can be computed using left endpoints, right endpoints, or midpoints of subintervals.
The accuracy of a Riemann sum improves as the number of subintervals increases.
A Riemann sum is expressed as $\sum_{i=1}^{n} f(x_i)\Delta x$ where $f(x_i)$ is the function value at point $x_i$ and $\Delta x$ is the width of each subinterval.
The limit of a Riemann sum as the number of intervals approaches infinity gives the exact value of the definite integral.
There are three common types: Left Riemann Sum, Right Riemann Sum, and Midpoint Riemann Sum.
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Related terms
Definite Integral: The exact area under a curve between two points on a graph, often calculated using limits and represented as $\int_a^b f(x) dx$.
Subinterval: A smaller division within an interval used in numerical methods like Riemann sums to approximate integrals.
Trapezoidal Rule: A numerical method for approximating an integral by dividing the area under a curve into trapezoids rather than rectangles.