ΣRiemann Sum Calculator
A Riemann sum is a finite approximation of a definite integral. It partitions the interval [a, b] into n subintervals and approximates the area under the curve using n rectangles. As n → ∞, the Riemann sum converges to the exact integral. Riemann sums are the conceptual foundation for understanding what integration means geometrically.
- 1Divide [a, b] into n equal subintervals of width Δx = (b−a)/n
- 2Left Riemann: use left endpoint of each subinterval — Σ f(xᵢ) · Δx
- 3Right Riemann: use right endpoint — Σ f(xᵢ₊₁) · Δx
- 4Midpoint Riemann: use midpoint — Σ f((xᵢ+xᵢ₊₁)/2) · Δx
- 5Midpoint rule is generally more accurate than left or right for the same n
∫[0,1] x² dx, n=4, midpoint=≈ 0.328125 (exact: 0.3333)Error decreases as 1/n²
∫[0,1] x² dx, n=100, midpoint=≈ 0.333325Approaches exact value as n increases
| Method | Formula | Error Order | Best For |
|---|---|---|---|
| Left | Σ f(xᵢ) · Δx | O(Δx) | Decreasing functions |
| Right | Σ f(xᵢ₊₁) · Δx | O(Δx) | Increasing functions |
| Midpoint | Σ f(mid) · Δx | O(Δx²) | General use |
| Trapezoid | (Δx/2)[f(a) + 2Σf(xᵢ) + f(b)] | O(Δx²) | Smooth functions |
| Simpson | (Δx/3)[f(a)+4f(x₁)+2f(x₂)+...+f(b)] | O(Δx⁴) | Smooth functions, best accuracy |
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