Answer
\[ = \frac{1}{2}\]
Work Step by Step
\[\begin{gathered}
f\,\left( \theta \right) = \cos \theta \sin \theta \hfill \\
f\,\left( \theta \right) = 0\,\,\,,\,\,\,\,\left( {axis - \theta } \right) \hfill \\
\hfill \\
Let\,\,\theta = 0\,\,\,\,\,\,and\,\,\,\,\theta = \frac{\pi }{2} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
A = \int_0^{\frac{\pi }{2}} {\cos \theta \sin \theta d\theta } \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \,\,\left[ {\frac{{{{\sin }^2}\theta }}{2}} \right]_0^{\frac{\pi }{2}} \hfill \\
\hfill \\
Fundamental\,\,theorem \hfill \\
\hfill \\
= \frac{{{{\sin }^2}\,\left( {\frac{\pi }{2}} \right)}}{2} - \frac{{{{\sin }^2}\,\left( 0 \right)}}{2} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{{\,{{\left( 1 \right)}^2}}}{2} - \frac{{{0^2}}}{2} \hfill \\
\hfill \\
= \frac{1}{2} \hfill \\
\hfill \\
\end{gathered} \]
