A curve with the area beneath it approximated by n rectangles of equal width; a slider raises n from one to two hundred, and a toggle picks whether each rectangle's height is read at its left or right edge. Three curves are available: x squared from zero to one, whose area is one third; x times three minus x from zero to three, whose area is four and a half; and x minus one from zero to two, which dips below the axis so its rectangles there count negative and the total is zero. Readouts show the sum of the rectangles, the exact area, and the gap between them, which shrinks as n grows.

Add up the slivers

Chop the stretch into n slivers, make each a rectangle, add them. More slivers, less gap — the sum settles onto the area.