Answer
$4$
Work Step by Step
When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$
where, $n$ is the unit vector and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
Consider $\iint_S F \cdot n dS=\iint_D 2 (u^2-v^2) dA=2 \times \int_{0}^{1} \int_0^{2} 2 (u^2-v^2) du dv$
or, $\iint_S F \cdot n dS=(2) \int_{0}^{1} [\dfrac{u^3}{3}-uv^2]_0^2 dv= \int_{0}^{1} \dfrac{2^3}{3}-(2)v^2 dv \times $
Hence, we have $\iint_S F \cdot n dS=[\dfrac{8}{3}-\dfrac{2}{3}] \times (2)=4$
