Riemann Sum Explorer Mesh ||P|| → 0
Subintervals (n)
10
Subinterval width Δx = (b - a)/n = 0.2000
Interactive Coordinate Canvas
(Click/drag inside slices to reposition sample tags ξᵢ)
Riemann Rectangles
Upper Darboux U(P)
Lower Darboux L(P)
f(x) Curve
Riemann Sum (Sₙ)
2.660000
Rule: Midpoint
Exact Definite Integral (I)
2.666667
∫₀² x² dx = 8/3
Absolute Error |Sₙ - I|
0.006667
Rel: 0.25%
Darboux Gap U(P) - L(P)
0.800000
Integrability Criterion: < ε
The Fundamental Relationship: The definite integral is mathematically defined as the limit of Riemann sums as the mesh norm tends to zero:
∫ₐᵇ f(x) dx ≡ lim_{n → ∞} ∑ᵢ₌₁ⁿ f(ξᵢ) Δx
Gaston Darboux framed integrability via Supremum (Mᵢ = sup f(x)) and Infimum (mᵢ = inf f(x)) over each subinterval [xᵢ₋₁, xᵢ]. A bounded function is Riemann integrable if and only if:
∀ ε > 0, ∃ Partition P such that: U(P) - L(P) = ∑ᵢ₌₁ⁿ (Mᵢ - mᵢ) Δx < ε
Upper Darboux Sum U(P)
3.080000
∑ Mᵢ Δx (circumscribed step envelope)
Lower Darboux Sum L(P)
2.280000
∑ mᵢ Δx (inscribed step envelope)
Sandwich Inequality
L(P) ≤ Sₙ ≤ U(P)
Definite integral I is trapped strictly between L(P) and U(P).
Comparing theoretical asymptotic convergence rates: Endpoint Riemann sums (Left/Right) scale as O(1/n), whereas Midpoint and Trapezoidal rules cancel linear error terms to achieve second-order O(1/n²) convergence.
Current partition coordinates and slice volumes:
| i | [xᵢ₋₁, xᵢ] | Tag ξᵢ | f(ξᵢ) | mᵢ (inf) | Mᵢ (sup) | Slice Area Sᵢ |
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