Answer
$$\int_{0}^{2} \int_{y}^{2 y}\left(10+2 x^{2}+2 y^{2}\right) d x d y=\frac{140}{3}$$
Work Step by Step
Given $$\int_{0}^{2} \int_{y}^{2 y}\left(10+2 x^{2}+2 y^{2}\right) d x d y$$
So, we get
\begin{aligned}
&\int_{0}^{2} \int_{y}^{2 y}\left(10+2 x^{2}+2 y^{2}\right) d x d y \\&=\int_{0}^{2}\left[10 x+\frac{2 x^{3}}{3}+2 y^{2} x\right]_{y}^{2 y} d y\\
&=\int_{0}^{2}\left[\left(20 y+\frac{16}{3} y^{3}+4 y^{3}\right)-\left(10 y+\frac{2}{3} y^{3}+2 y^{3}\right)\right] d y \\ &=\int_{0}^{2}\left[10 y+\frac{20}{3} y^{3}\right] d y\\
&=\left[5 y^{2}+\frac{5 y^{4}}{3}\right]_{0}^{2}\\
&=20+\frac{80}{3}=\frac{140}{3}
\end{aligned}
