Riemann sums approximate ∫ₐᵇ f(x)dx by dividing [a,b] into n subintervals of width Δx=(b−a)/n. Left sum uses f(xᵢ), right uses f(xᵢ₊₁), midpoint uses f((xᵢ+xᵢ₊₁)/2). All converge to the integral as n→∞.
How to Use
Set up the integral:
- Function: Choose f(x)
- Interval: [a, b]
- Subdivisions: n
Methods Compared
- Left: underestimates increasing f
- Right: overestimates increasing f
- Midpoint: O(1/n²) error
- Trapezoidal: O(1/n²) error
Convergence
As n→∞, all methods converge to the exact integral. Midpoint and trapezoidal converge faster (quadratically) than left/right (linearly).
Step-by-Step Instructions
- 1Select a function.
- 2Set interval [a,b].
- 3Choose n subdivisions.
- 4Select method.
- 5View approximation.