Let us consider an example to illustrate the application of the definite integral to find the area function of a shaded region under a curve as shown in the given diagram.
A small change in corresponds to 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 the limit, we have
Since this shows that for the curve , the derivative of the corresponding area function is
The definition of a definite integral is
Using this definition, we have
Equation (iv) shows that the area under the curve and between and is as shown in the diagram