A definite integral represents the area under a curve, but as such, it is much more useful than merely a means of calculating irregular areas. To illustrate the importance of this concept to the sciences consider the following example. The work done on a piston, during the power stroke of an internal combustion engine, is equal to the product of the force acting on the piston times the displacement of the piston (the distance the piston travels after ignition). Engineers can easily measure the force on a piston by measuring the pressure in the cylinder (the force is the pressure times the cross sectional area of the piston). At the same time, they measure the displacement of the piston. The work done decreases as the displacement increases, until the piston reaches the bottom of its stroke. Because area is the product of width times height, the area under the curve is equal to the product of force times displacement, or the work done on the piston between the top of the stroke and the bottom.
The area under this curve can be approximated by drawing a number of rectangles, each of them h units wide. The height of each rectangle is equal to the value of the function at the leading edge of each rectangle. Suppose we are interested in finding the work done between two values of the displacement, a and b. Then the area is approximated by Area = hf(a) + hf(a+h) + hf(a+2h) +... + hf(a+(n-1)h) + hf(b-h). In this approximation n corresponds to the number of rectangles. If n is allowed to become very large, then h becomes very small. Applying the theory of limits to this problem shows that in most ordinary cases this results in the sum approaching a limiting value. When this is the case the limiting value is called the value of the integral from a to b and is written:
Where the integral sign (an elongated s) is intended to indicate that it is a sum of areas between x = a and x = b. The notation f(x)dx is intended to convey the fact that these areas have a height given by f(x), and an infinitely small width, denoted by dx.