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Open the Numerical Integration Lab →What this project demonstrates
Making the behaviour of a numerical method and its error visible, which is how a calculation gets validated before anyone trusts the result.
The Numerical Integration Lab is an interactive tool that visually shows how to calculate the area under a curve when no simple analytical solution exists. Choose a function, set the integration interval, and watch how three different methods, the midpoint rule, the trapezoidal rule and Simpson's rule, reconstruct the area using rectangles, trapezoids or parabolic arcs.
This is the sixth project published in AI-LAB: the calculation engine, the graphical rendering on canvas and the control interface were generated by conversing with an AI model.
How it was built
The project started from a natural-language description of the three numerical integration methods to compare. The AI implemented safe parsing of mathematical expressions (via the math.js library, never executing arbitrary code), vector drawing of the curve and approximation shapes on canvas, with transition animation between methods, in a single standalone file with no server-side dependencies.
What the tool shows
- Three approximation methods: midpoint (rectangles), trapezoidal, Simpson (parabolic arcs).
- Ready-to-use preset functions (x², sin(x), 1/x, eˣ, √x, polynomials) or a freely written custom function.
- Adjustable number of subdivisions via a slider, to see in real time how the approximation converges to the exact value.
- Estimated area, reference value and numerical error, calculated and updated instantly.
- Error handling for invalid expressions, incorrect intervals or diverging functions.
Privacy note
The tool runs entirely in the browser: no data is ever sent over the internet. Every calculation happens locally via JavaScript, with no external server calls.
AI-LAB publishes exploratory projects: this tool is not part of Valuemate's commercial service portfolio.
Frequently asked questions
When is numerical integration needed?
When the antiderivative of a function cannot be expressed in closed form, or when the function is only known at discrete points, as happens with experimental data. In those cases the integral is approximated by summing elementary areas.
Why is Simpson's rule more accurate than the trapezoidal rule?
Because it approximates the function with parabolic arcs rather than straight segments, so it captures curvature as well. For smooth functions the trapezoidal error falls with the square of the step size and Simpson's with the fourth power: halving the step divides the error by four in the first case and by sixteen in the second.
Do more subdivisions always give a better result?
Only up to a point. Truncation error falls, but rounding error grows as more and more floating-point terms are summed. There is therefore a number of subdivisions beyond which the result stops improving, which is why error analysis matters more than raw computing power.
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