Example of Area Under a Curve

In this tutorial we shall find the area bounded by the curve in the given diagram we have two regions that is region A and region B


example-area-under-curve

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}


First 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}

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