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
Equation (iv) shows that the area under the curve and between and is given by the diagram