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
\[A = \int\limits_a^b {{x^2}dx} \]