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For First Responders · Riemann Sums

Riemann Sums for a bell curve

Built for first responders planning or training for real incidents. Run fast what-if scenarios for response planning and training — no software to install in the field. Simulate a bell curve 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 first responders?: Yes — this version of "Riemann Sums for a bell curve" is framed for first responders planning or training for real incidents. Run fast what-if scenarios for response planning and training — no software to install in the field.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.