Answer
$$0$$
Work Step by Step
\begin{align*}
\int_{0}^{2} \int_{0}^{2}\left(x^{2}-y^{2}\right) d y d x&=\int_{0}^{2} \left(x^{2}y-\frac{1}{3}y^{3}\right) \bigg|_{0}^{2} d x\\
&=\int_{0}^{2} \left(2x^{2} -\frac{8}{3} \right) d x\\
&=\frac{2}{3}x^3-\frac{8}{3}x\bigg|_{0}^{2}\\
&=0
\end{align*}
