Answer
$$\int_{0}^{2} \int_{y}^{2 y} x y d x d y=6$$
Work Step by Step
Given $$\int_{0}^{2} \int_{y}^{2 y} x y d x d y$$
So, we have
\begin{aligned}I&=\int_{0}^{2} \int_{y}^{2 y} x y d x d y\\&=\int_{0}^{2}\left[\frac{x^{2} y}{2}\right]_{y}^{2 y} d y\\
&=\int_{0}^{2}\left[\frac{(2 y)^{2} y}{2}-\frac{(y)^{2} y}{2}\right] d y\\&=\int_{0}^{2} \frac{3 y^{3}}{2} d y\\
&=\left[\frac{3 y^{4}}{8}\right]_{0}^{2}\\
&=\frac{3(2)^{4}}{8}-0\\
&=6
\end{aligned}