calculate the flux of a vector field through a surface

How to calculate the flux of a vector field through a surface in mathematica? I’ve this field:

F = (x, x^2 * y, y^2 * z)  

and this surface:

S = { (x,y,z) ∈ R^3 | 2 * Sqrt[x^2+y^2] <= z <= 1 + x^2 + y^2} 

So, I’m trying:

region = ImplicitRegion[2 * Sqrt[x^2+y^2] <= z <= 1 + x^2 + y^2, {x, y, z}];  Integrate[#, {x,y,z} ∈ region]& /@ ({x, x^2 * y, y^2 * z} . {x, y, z}) 

I expect Pi/30 as a result, but it comes out "Infinity"…