Answer
$11 \pi$
Work Step by Step
Apply Divergence's Theorem: $\iint_S F\cdot dS=\iiint_E div F dV$
and
div $F=3(x^2+y^2+z^2)=3(r^2+z^2)$
For the cylindrical coordinates:
$0 \leq \theta \leq 2 \pi, 0 \leq r \leq 1; 0 \leq z \leq 2$
Now, we have
$\iint_S F\cdot dS=\int_0^{2 \pi} \int_0^{2} \int_0^1 3(r^2+z^2) r dr dz d \theta$
or, $[\theta]_0^{2 \pi} (\int_0^{2} \int_0^1 3r^3+3z^2 r dr dz)$
or, $2 \pi [\int_0^{2}[\dfrac{3r^4}{4}+\dfrac{3z^2 r^2}{2}]_0^1dz]=(2\pi)[(\dfrac{3}{4}) z+(\dfrac{1}{2})z^3]_0^2$
or, $\iint_S F\cdot dS=2\pi[\dfrac{3}{4} (2)+\dfrac{1}{2}(2)^3]$
Thus, $\iint_S F\cdot dS=(3\pi+8\pi)=11 \pi$
