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For Students · Riemann Sums

Riemann Sums for a dice distribution

Built for students learning it for a class or exam. See the concept move instead of memorizing formulas — and check your homework intuition. Simulate a dice distribution live below — adjust the inputs and watch it respond, right in your browser.

Riemann SumsLive

approximating the definite integral

Controls

f(x)

Rectangles approximate the area under the curve. Add more and the Riemann sum converges to the true integral.

Presets

Rectangles

▶ Run in Python

Data Inspector

Riemann sum20.7111

Exact integral20.8000

Error0.0889

Governing equation

\int_{- 4}^{4} f (x) d x \approx i = 0 \sum 11 f (x_{i}) Δ x = 20.711, Δ x = 0.667 (midpoint)

Reading this result: With n=12 midpoint rectangles the step is dx = 0.667. The midpoint rule already cancels much of the error at each rectangle. Adding subintervals shrinks the error toward the exact integral; midpoint and trapezoid rules converge fastest (error ~1/n²), so from n=12 the estimate tightens quickly as you push toward n=100.

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Frequently asked questions

Is this good for students?: Yes — this version of "Riemann Sums for a dice distribution" is framed for students learning it for a class or exam. See the concept move instead of memorizing formulas — and check your homework intuition.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.