How Integration Works: It’s Just Fancy Addition - dummies

# How Integration Works: It’s Just Fancy Addition

The most fundamental meaning of integration is to add up. And when you depict integration on a graph, you can see the adding up process as a summing up of thin rectangular strips of area to arrive at the total area under that curve, as shown in this figure.

You can calculate the shaded area in the above figure by using this integral:

(Note that everything here involves definite integration as opposed to indefinite integration. Definite integration is where the elongated S integration symbol has limits of integration: the two little constants or numbers at the bottom and the top of the symbol. The elongated S without limits of integration indicates an indefinite integral or antiderivative.)

Look at the thin rectangle in the figure. It has a height of f(x) and a width of dx (a little bit of x), so its area (length times width, of course) is given by f(x) · dx. The above integral tells you to add up the areas of all the narrow rectangular strips between a and b under the curve f(x). As the strips get narrower and narrower, you get a better and better estimate of the area. The power of integration lies in the fact that it gives you the exact area by sort of adding up an infinite number of infinitely thin rectangles.