Integral Explorer
The definite integral is just area — tiled, summed, and refined until the rectangles disappear into it.
Tip: put the sine on a full period — the humps cancel and the signed total reads zero.
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.
Calculus makes sense once you have tiled an area yourself
This explorer is one small piece of how ManySpace teaches mathematics — every idea as something you can run, refine, and push to the extreme.