Answer
$0$
Work Step by Step
$f_{\mathrm{ave}}=\displaystyle \frac{1}{b-a}\int_{a}^{b}f(x)dx$
$=\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin^{2}x\cos^{3}xdx$
$=\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin^{2}x(1-\sin^{2}x)\cos xdx$
$\left[\begin{array}{ll}
u=\sin x & \text{bounds: }0,0 \\
du=\cos x &
\end{array}\right]$
$=\displaystyle \frac{1}{2\pi}\int_{0}^{0}u^{2}(1-u^{2})du$
$=0$
