Ideally you know the divergence thm --an ideal place yo use it

the flux in this case is equal to the volume since div(F) = 1

If not you have some work to do

You'll need 4 flux integrals

1. on the bottom but here z = 0 so the flux is 0

the parabaloid intersects the cylinder in the plane z =4

2. through the sides of the cylinder below z = 4 (split into right and left halves)

3. through the parabaloid above over the circle r = 2