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the cylinder (x^2)+(y^2)=(R^2)

the parabola x = y-((y^2)/R)

the planes z = H, y = 0, and z = 0

and the velocity field is:

F = yz(i)+xz(j)+xy(k)

and we need to calculate the outward flux of the field of the region at z = H (the top of the region).

Ive tried doing this 2 ways:

Doing the double integral of div(F)

Doing the double integral of F (dot) k dA

the solution to this problem uses the 2nd of the two; my question is this:

why are the 2 methods in disagreement with each other? (when i take div(F) i get zero, so flux is zero)