Answer
\[0\]
Work Step by Step
\[\begin{align}
& \text{From the graph we can define the region }R\text{ as:} \\
& R=\left\{ \left( x,y \right):-\sqrt{1-{{y}^{2}}}\le x\le \sqrt{1-{{y}^{2}}},\text{ }-1\le y\le 1\text{ } \right\} \\
& \text{Then,} \\
& \iint_{R}{f\left( x,y \right)}=\int_{-1}^{1}{\int_{-\sqrt{1-{{y}^{2}}}}^{\sqrt{1-{{y}^{2}}}}{x{{y}^{2}}}dxdy} \\
& \text{Integrating} \\
& =\int_{-1}^{1}{\left[ \frac{{{x}^{2}}{{y}^{2}}}{2} \right]_{-\sqrt{1-{{y}^{2}}}}^{\sqrt{1-{{y}^{2}}}}dy} \\
& =\int_{-1}^{1}{\left[ \frac{{{\left( \sqrt{1-{{y}^{2}}} \right)}^{2}}{{y}^{2}}}{2}-\frac{{{\left( -\sqrt{1-{{y}^{2}}} \right)}^{2}}{{y}^{2}}}{2} \right]dy} \\
& =\int_{-1}^{1}{\left[ \frac{\left( 1-{{y}^{2}} \right){{y}^{2}}}{2}-\frac{\left( 1-{{y}^{2}} \right){{y}^{2}}}{2} \right]dy} \\
& =\int_{-1}^{1}{0dy} \\
& =0 \\
\end{align}\]
