How the definite integral was born — from rectangle approximation to the long S (∫)
The area under a curve has no memorized formula the way a triangle or a circle does. So to measure it, people long ago filled the space under the curve with many very thin rectangles and added up all their areas. A rectangle pokes above or falls short of the curve, so it carries an error, but the finer you slice the pieces, the smaller that overshoot becomes. Slice the pieces infinitely fine and push the error all the way to zero, and the value you reach is the true area. This article confirms, on the easy example of a triangle under a straight line whose answer we already know, that the method really does converge to that answer, and shows with an interactive how the sum of infinitely many thin slices hardened into a long S-shaped symbol, giving birth to the definite integral. The scene where the Greek letter sigma, which once meant addition, transforms into the integral sign meaning the sum of infinitely thin slices is the heart of this article.