∫ Calculus Lab Riemann Sums & Definite Integrals

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

How slice choices change a Riemann sum

Read the explanation

The canonical source integrates x squared from zero to two. Four equal slices have width zero point five and midpoint tags zero point two five, zero point seven five, one point two five and one point seven five. Their squared heights are zero point zero six two five, zero point five six two five, one point five six two five and three point zero six two five. Adding height times width gives two point six two five. The bars are proportional to these actual sampled heights. With those same four slices, left endpoint heights use zero, zero point five, one and one point five, giving sum one point seven five. Right endpoints give three point seven five. The exact integral is eight thirds, about two point six six seven, between those sums. The bracket follows monotonicity on this specific interval; other functions need their actual extrema. The source estimates extrema numerically inside each slice. For x squared on zero to two the midpoint error is two divided by three n squared. At four slices that is one twenty fourth, about zero point zero four one six seven. At eight it is one ninety sixth, about zero point zero one zero four two. Doubling n quarters the error for this smooth function and rule. The source’s random and custom tags, trapezoids and endpoint rules follow different constructions. Full exported slices preserve the numeric evidence rather than just a displayed total.

Super generates helpful tools and automates fact-checking across the internet proactively. If you enjoyed this tool, build your own with Super and share it with a friend.