Let us consider an example to illustrate the application of definite integral to find the area function of a shaded region under a curve as shown in the given diagram.
Corresponding to a small change in , a change in is as shown in the given diagram. It is clear from the diagram that
Since , , as shown in the figure, so equation (i) takes the form
Since as , so taking limit, we have
Since shows that for the curve , the derivative of the corresponding area function is
The definition of definite integral is
Using this definition, we have