# Integral Explorer

Canonical: https://manyspace.aporia.systems/tools/integral
Updated: 2026-10-13

The definite integral is just area — tiled, summed, and refined until the rectangles disappear into it.

## Try this

- Feel convergence. On the parabola with n = 4, note the error. Double to 8, 16, 32 — the error collapses roughly fourfold each time for midpoint. That speedup is the whole reason the method matters.
- Catch the sign. Put the quadratic on [−2, 2]. The middle rectangles hang below the axis in red — negative area eating the positive corners. Reveal exact and confirm the net.
- Cancel a wave. Switch to sine on [0, 6.25] with n = 8 midpoint. Two humps, two signs, one total near zero. Widen or shrink the interval and watch the total breathe.

## How to read an integral

The symbol ∫ hides a humble procedure: chop [a, b] into n slices, build a rectangle on each slice, add their areas. Left, right, and midpoint only disagree about where on each slice to measure the height. As n grows, that choice stops mattering — every method lands on the same number, the exact integral.

Area below the axis counts negative. That is why a full sine period integrates to zero: the top hump and the bottom hump are equal citizens with opposite signs. Whenever your approximation looks wrong, check how much of it lives underground.

## Related tools

- [Derivative Visualizer](https://manyspace.aporia.systems/tools/derivative)
- [Unit Circle Explorer](https://manyspace.aporia.systems/tools/unit-circle)
- [Complex Plane Explorer](https://manyspace.aporia.systems/tools/complex-plane)
- [Probability Lab](https://manyspace.aporia.systems/tools/probability)
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes)
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors)
- [Fourier Series](https://manyspace.aporia.systems/tools/fourier-series)