Riemann sum

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

1. Riemann sums can be computed using left endpoints, right endpoints, or midpoints of subintervals.
2. The accuracy of a Riemann sum improves as the number of subintervals increases.
3. 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.
4. The limit of a Riemann sum as the number of intervals approaches infinity gives the exact value of the definite integral.
5. There are three common types: Left Riemann Sum, Right Riemann Sum, and Midpoint Riemann Sum.

Review Questions

• What are the different types of Riemann sums and how do they differ?
• How does increasing the number of subintervals affect the accuracy of a Riemann sum?
• Explain how a Riemann sum approximates an integral.

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.