Answer
$11\sqrt {14}$
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.
Since, we have $\iint_S f(x,y,z) dS \approx \Sigma_{i=1}^n f[\overline{x}, \overline{y}, \overline{z}] AS_i$
Consider $\iint_S f(x+y+z) dS =\int_{0}^2 \int_{0}^1 (4u+1+v) \times (\sqrt {14}) (dv du)=(\sqrt {14}) \int_{0}^2 [(4uv+v+\dfrac{v^2}{2}) du$
Now, $\iint_S f(x+y+z) dS=(\sqrt {14}) \int_{0}^2 [(4u+1+\dfrac{1}{2}) du=(\sqrt {14}) [2u^2+\dfrac{3u}{2}]_0^2$
Hence, we have $\iint_S f(x+y+z) dS=11\sqrt {14}$
