# The Area Between a Curve and the X-axis

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