Answer
$18 \pi$
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, $\iint_S f(x,y,z) dS \approx \Sigma_{i=1}^n f(\overline(x), \overline(y), \overline(z)) AS_i$
Area of the upper and bottom part of the disk $=\pi(1)^2=\pi$;
Area of each of the four quarter cylinders $=\dfrac{2 \pi r h}{24}=\dfrac{2 \pi \cdot 1 \cdot 2}{4}=\pi$
Thus, we have $\iint_S F(x,y,z) dS =\pi[2+2+3+3+4+4]=18 \pi$
