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