Using divergence theorem for F=(xy²,2y²,xy³) over a cylinder

[trelek2]

Use the divergence theorem to compute the surface integral F dot dS , where
F=(xy^2, 2y^2, xy^3) over closed cylindrical surface bounded by x^2+z^2=4 and y is from -1 to 1.

I've tried doing it and got 32pi/3 (i guess its wrong, so how to do it?)
Is it ok to compute Div F in terms of xyz and after that change into cylindrical polar coordinates to calculate volume?

 

[HallsofIvy]

Yes, that's perfectly legitimate.