Answer
$$2$$
Work Step by Step
$$\eqalign{
& \int_0^1 {\int_0^2 {\left( {2xy} \right)} dxdy} \cr
& \int_0^1 {\left[ {\int_0^2 {\left( {2xy} \right)} dx} \right]dy} \cr
& \int_0^1 {2y\left[ {\int_0^2 x dx} \right]dy} \cr
& {\text{Evaluate inner integral}} \cr
& \int_0^2 x dx = \left[ {\frac{{{x^2}}}{2}} \right]_{x = 0}^{x = 2} \cr
& = \frac{{{2^2}}}{2} \cr
& = 2 \cr
& {\text{Therefore,}} \cr
& \int_0^1 {2y\left[ {\int_0^2 x dx} \right]dy} = \int_0^1 {2y\left( 2 \right)dy} \cr
& \int_0^1 {4ydy} \cr
& {\text{Integrating}} \cr
& {\text{ = }}\left[ {2{y^2}} \right]_0^1 \cr
& = 2{\left( 1 \right)^2} \cr
& = 2 \cr} $$
