Riemann sums

Math

Definition

An approximation of the area under a curve by dividing the region into rectangles (or trapezoids) and summing their areas. As the number of rectangles increases, the approximation approaches the exact integral.

How It Works

  1. Divide the interval [a, b] into n equal subintervals of width Δx = (b − a)/n.
  2. Choose a sample point in each subinterval (left endpoint, right endpoint, or midpoint).
  3. Evaluate f at each sample point to get the height of each rectangle.
  4. Multiply each height by Δx to get each rectangle's area.
  5. Sum all the rectangle areas to get the Riemann sum approximation.

Examples

  • Approximating ∫ from 0 to 2 of x² dx using 4 right-endpoint rectangles
  • Estimating total distance traveled from a velocity-time table using left Riemann sums
Key Fact

Σ f(xᵢ)·Δx approximates ∫ from a to b of f(x) dx

Related Terms

Fundamental theorem Sigma notation Antiderivatives Area between curves

Study This Concept

Practice Riemann sums with free review games in these units:

AP Calculus AB Unit 6: Integration and Accumulation