how to use the Gauss DivergenceTheorem to calculate the net outward flux of F?

∫∫s F · dS
= ∫∫∫ div F dV, by the Divergence Theorem
= ∫∫∫ (3x^2 + 3y^2 + 3) dV

Now, convert to cylindrical coordinates:
x^2 + y^2 = 4 ==> r = 2, since x^2 + y^2 = r^2
z = x + 1 ==> z = r cos(θ) + 1.

So, the integral transforms to
∫(r = 0 to 2) ∫(θ = 0 to 2π) ∫(z = 0 to r cos(θ) + 1) (3r^2 + 3) * (r dz dθ dr)
= ∫(r = 0 to 2) ∫(θ = 0 to 2π) (3r^3 + r) z {for z = 0 to r cos(θ) + 1} dθ dr
= ∫(r = 0 to 2) ∫(θ = 0 to 2π) (3r^3 + r) (r cos(θ) + 1) dθ dr
= ∫(r = 0 to 2) (3r^3 + r) (r sin(θ) + θ) {for θ = 0 to 2π} dr
= ∫(r = 0 to 2) (3r^3 + r) (2π) dr
= π ((3/2)r^4 + r^2) {for r = 0 to 2}
= 28π.

I hope this helps!

Cheers