How can you find the area under a curve by adding up thin slices?

You find the area under a curve by cutting it into thin slices that are almost rectangles, adding up their areas, and then making the slices thinner and thinner until the total settles on one value. That value is the integral. An integral is the continuous analog of a sum, used to calculate areas, volumes and their generalizations.

Start with something you can do already. If a car holds a steady 60 kilometres an hour for two hours, draw its speed on a graph and you get a flat line. The rectangle under that line is 60 tall and 2 wide, and its area, 120, is exactly the distance travelled. Integration was first used for problems just like this, such as finding the area under a curve, or determining displacement from velocity. If the speed changes, the line becomes a curve, and a curve has no simple area formula. So you cheat. Cut the time into short slices, treat the speed as constant within each one, and add up the thin rectangles. A rough job gives a rough answer; thinner slices give a better one.

This is not a modern trick. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek mathematicians Democritus and Eudoxus, which found areas and volumes by breaking them into an infinite number of divisions of known size. Archimedes developed it further and used it to find the area of a circle and the area under a parabola. A similar method was developed independently in China by Liu Hui, who used it to find the area of the circle.

In the late 17th century Newton and Leibniz thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Leibniz also gave us the symbol: his stretched-out S comes from the Latin summa, meaning sum or total.