# 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 $A\left( x \right)$ of a shaded region under a curve $y = {x^2}$ as shown in the given diagram.

A small change $\delta x$ in $x$ corresponds to a change in $A$ is $\delta A$, as shown in the given diagram. It is clear from the diagram that

Since $PL = y$, $LM = \delta x$, $SM = y + \delta y$ as shown in the figure, so equation (i) takes the form

Since $\delta y \to 0$ as $\delta x \to 0$, so taking the limit, we have

Since this shows that for the curve $y = {x^2}$, the derivative of the corresponding area function $A$ is

The definition of a definite integral is

Using this definition, we have

Equation (iv) shows that the area under the curve $y = {x^2}$ and between $x = a$ and $x = b$ is as shown in the diagram