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