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Area under the curve

Example 1

Find the area between \(y=7-x^2\) and the \(x\)-axis between the values \(x=-1\) and \(x=2\).

Firstly, draw the graph:

The area will be:

\begin{eqnarray*}  
&&\int_{-1}^2 7-x^2 \; \mathrm{d} x \\  
&=& \left[ 7x - \frac{x^3}{3} \right]_{-1}^2 \\  
&=& \left[ 7\times 2 - \frac{2^3}{3} \right] - \left[7\times -1 - \frac{(-1)^3}{3} \right] \\  
&=& 18
\end{eqnarray*}