Answer
$11\sqrt {14}$
Work Step by Step
The flux through a surface can be defined only when the surface is orientable.
We know that $\iint_S F \cdot dS=\iint_S F \cdot n dS$
Here, $n$ denotes the unit vector.
Since, $\iint_S f(x+y+z) dS =\int_{0}^2 \int_{0}^1 (4u+1+v) \sqrt {14} dv du $
$=\sqrt {14} \times \int_{0}^2 [(4uv+v+\dfrac{v^2}{2}) du$
$=\sqrt {14} \times \int_{0}^2 [(4u+1+\dfrac{1}{2}) du$
$=\sqrt {14} [2u^2+\dfrac{3u}{2})_0^2$
$=11\sqrt {14}$
