Calculating probabilities is a very important application of integration in the physical, social, and life sciences. To understand the basics, consider the game of darts in which a player throws a dart at a board and tries to hit a particular target. Let us suppose that a dart board is in the form of a disk
with radius 10 inches. If we assume that a player throws a dart at random, and is not aiming at any particular point, then it is equally probable that the dart will strike any single point on the board. For instance, the probability that the dart will strike a particular 1 square inch region is
or the ratio of the area of the desired target to the total area of
(assuming that the dart thrower always hits the board itself at some point). Similarly, the probability that the dart strikes a point in the disk
of radius 3 inches is given by the area of
divided by the area of
In other words, the probability that the dart strikes the disk
is