Answer
\[ = \frac{1}{3}\]
Work Step by Step
\[\begin{gathered}
\int_0^{\frac{\pi }{2}} {{{\sin }^2}\theta \cos \theta d\theta } \hfill \\
\hfill \\
u = \sin \theta \,\,\,\,\,\,\,then\,\,\,\,du = \cos \theta d\theta \hfill \\
\hfill \\
{\text{Changing limits of integration}} \hfill \\
\hfill \\
\theta = 0\,\,\,\,\,\,implies\,\,u = 0 \hfill \\
\theta = \frac{\pi }{2}\,\,\,\,implies\,\,u = 1 \hfill \\
\hfill \\
apply\,\,the\,\,\,substitution \hfill \\
\hfill \\
= \int_0^1 {{u^2}\,du} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \,\,\left[ {\frac{{{u^3}}}{3}} \right]_0^1 \hfill \\
\hfill \\
Fundamental\,\,theorem \hfill \\
\hfill \\
= \frac{1}{3} - 0\, = \frac{1}{3} \hfill \\
\hfill \\
\end{gathered} \]
