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}