Answer
$11\sqrt {14}$
Work Step by Step
Since, $\iint_S F \cdot dS=\iint_S F \cdot n dS$
where, $n$ denotes the unit vector.
and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
The flux through a surface can be defined only when the surface is orientable.
Since, $\iint_S f(x+y+z) dS =\int_{0}^2 \int_{0}^1 (4u+1+v) \times \sqrt {14} dv du =\sqrt {14} [\int_{0}^2 [(4uv+v+\dfrac{v^2}{2}) du]$
or, $=\sqrt {14} \int_{0}^2 [(4u+1+\dfrac{1}{2}) du$
or, $=\sqrt {14} [2u^2+\dfrac{3u}{2}]_0^2$
Thus, we have $\iint_S f(x+y+z) dS=11\sqrt {14}$
