Example 3

Find the area between \(y = \sin(x)\) and the \(x\)-axis between the values \(x=0\) and \(x=2\pi\).

Firstly, draw the graph:

Notice from the graph (above) that the curve crosses the \(x\)-axis at \(x=\pi\), and that the first area will be positive (above the \(x\)-axis) and the second area will be negative (below the \(x\)-axis). Set up the integration for each area:

\begin{eqnarray*}  
\mbox{Area} &=& \int_{0}^{\pi} \sin(x) \; \mathrm{d} x + \left| \int_{\pi}^{2\pi} \sin(x) \; \mathrm{d} x \right| \\  
&=& \bigg[ -\cos(x) \bigg]_{0}^{\pi} + \left| \bigg[ -\cos(x) \bigg]_{\pi}^{2\pi} \right| \\  
&=& \left( -\cos(\pi) + \cos (0) \right) + \left|\left( -\cos(2\pi) + \cos(\pi) \right) \right| \\  
&=& (-(-1) + 1) + | -1 + -1 | \\  
&=& 2 + | -2| \\  
&=& 2+2 = 4
\end{eqnarray*}

Therefore, the area between \(y = \sin(x)\) and the \(x\)-axis between the values \(x=0\) and \(x=2\pi\) is equal to 4 units\(^{2}\).