Divergence theroem I have a problem applying the divergence theorem in two problems.
The first asks me to calculate $\int \int F · N dS$ where $F(x,y,z)=(x^2 + \sin z, x y + \cos z, e^y)$ in  in the cylinder $x^2 + y^2=4$  limited by the planes $XY$ and $x+z=6$.
I compute divergence and $\mathop{div}{F} = 3 x$.
To compute $\int \int \int \mathop{div}{F} \ dV $, I use cylindrical coordinates:
$x= r \cos( \alpha)$
$y=r \sin (\alpha)$
$z=t$
for $r \in [0,2]$, $\alpha \in [0, 2 \pi]$ and $t \in [0, 6- r \cos(\alpha)]$.
As the Jacobian is $r$ I have the following:
$$ I=\int \int F · N dS$=  $\int \int \int \mathop{div}{F}  \space dV =  \int_{0}^{2 \pi} \int_{0}^{2} \int_{0}^{6- r \cos(\alpha)} 3 r \cos( \alpha) r \ dt \ dr \ d\alpha $$
I compute it using Mathematica and I have I=0 however I know that the result is not that. Where have I been wrong?
The other problem is similar.
I have to compute I have to calculate the flow F that passes through S where $F(x,y,z)=(x^2 + \sin(yz), y- x e^{-z},z^2)$ and $S$ is the boundery of cylinder $x^2+y^2=4$ limited by planes $z=0$ and $x+z=2$. I have applied the divergence theorem as in the previous case and I have to
$$ I=\int \int F · N dS= \int_{0}^{2 \pi} \int_{0}^ {2} \int _{0}^{2-r \cos(\alpha)} (2 r \cos (\alpha) + 1 + 2z )r\ dz\ dr\ d\alpha=\frac{64 \pi}{3} $$
And as in the previous case, it is not the solution either. Where this error?
Thank you very much to all.

I am not following your conversion to polar coordinates.
$\iiint 3x \ dz\ dy\ dx$
$x = r\cos \theta\\
y = r\sin \theta\\
z = z$
Jacobian:
$dx\ dy\ dz = r\ dz\ dr\ d\theta$
$\int_0^{2\pi}\int_0^2\int_0^{6-r\cos\theta} 3r^2\cos\theta \ dz\ dr\ d\theta$
