# 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
$\begin{gathered} {\text{area}}\,{\text{of}}\,PQML < \delta A < {\text{area}}\,{\text{of}}\,RSML \\ \Rightarrow PL \times LM < \delta A < SM \times LM\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Since $PL = y$, $LM = \delta x$, $SM = y + \delta y$ as shown in the figure, so equation (i) takes the form
$\begin{gathered} y\delta x < \delta A < \left( {y + \delta y} \right)\delta x \\ \Rightarrow y < \frac{{\delta A}}{{\delta x}} < y + \delta y \\ \end{gathered}$

Since $\delta y \to 0$ as $\delta x \to 0$, so taking the limit, we have
$\begin{gathered} \mathop {\lim }\limits_{\delta x \to 0} y < \mathop {\lim }\limits_{\delta x \to 0} \frac{{\delta A}}{{\delta x}} < \mathop {\lim }\limits_{\delta x \to 0} \left( {y + \delta y} \right) \\ \Rightarrow y < \frac{{dA}}{{dx}} < y + 0 \\ \Rightarrow y < \frac{{dA}}{{dx}} < y \\ \Rightarrow \frac{{dA}}{{dx}} = y \\ \Rightarrow \frac{{dA}}{{dx}} = {x^2} \\ \end{gathered}$

Since this shows that for the curve $y = {x^2}$, the derivative of the corresponding area function $A$ is
$\frac{{dA}}{{dx}} = {x^2}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)$

The definition of a definite integral is
$\frac{d}{{dx}}\left( {F\left( x \right)} \right) = f\left( x \right) \Rightarrow F\left( b \right) – F\left( a \right) = \int\limits_a^b {f\left( x \right)dx} \,\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right)$

Using this definition, we have
$\frac{d}{{dx}}\left( {A\left( x \right)} \right) = {x^2} \Rightarrow A\left( b \right) – A\left( a \right) = \int\limits_a^b {{x^2}dx} \,\,\,\,{\text{ – – – }}\left( {{\text{iv}}} \right)$ 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
$A = \int\limits_a^b {{x^2}dx}$