Written by the Fiveable Content Team • Last updated August 2025
Written by the Fiveable Content Team • Last updated August 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.
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.
Review Questions
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.