# Example of an Area Under a Curve

In this tutorial we shall find the area bounded by the curve. In the given diagram we have two regions: region $A$ and region $B$

First we find the area under the curve of region A:

Area of the region $A = \int\limits_3^4 {ydx}$
$\begin{gathered} A = \int\limits_3^4 {\frac{{{x^2}}}{2}dx = \frac{1}{2}} \int\limits_3^4 {{x^2}dx} \\ \Rightarrow A = \int\limits_3^4 {\frac{{{x^2}}}{2}dx = \frac{1}{2}} \left| {\frac{{{x^3}}}{3}} \right|_3^4 = \frac{1}{6}\left( {{4^3} – {3^3}} \right) \\ Area = \frac{1}{6}\left( {64 – 27} \right) = \frac{{37}}{6} \\ \end{gathered}$

Next we find the area under the curve of region B:

Area of the region $A = \int\limits_4^5 {ydx}$
$\begin{gathered} A = \int\limits_4^5 {\frac{{{x^2}}}{2}dx = \frac{1}{2}} \int\limits_4^5 {{x^2}dx} \\ \Rightarrow A = \int\limits_4^5 {\frac{{{x^2}}}{2}dx = \frac{1}{2}} \left| {\frac{{{x^3}}}{3}} \right|_4^5 = \frac{1}{6}\left( {{5^3} – {4^3}} \right) \\ \Rightarrow Area = \frac{1}{6}\left( {125 – 64} \right) = \frac{{61}}{6} \\ \end{gathered}$