From Slices to Totals

If you know how fast water flows into a tank each second, integration tells you the total volume collected over time. If you know the speed of a car at every moment, integration gives the total distance traveled. The pattern is always the same: rate in, accumulated total out.

The integral sign (∫) is actually a stretched-out S, standing for "sum." Leibniz chose it deliberately. Every integral is a sophisticated addition problem — summing contributions so small that you need infinitely many of them to cover the whole picture.