Calculating flux through a surface area

[dan38]

S is a portion of a curve with r(u,v)
where 0 < u < 2 and 0 < v < 2pi

I'm meant to calculate Flux of the vector field F

My Calculations
First found dr/du
then dr/dv

Using the cross product, I found N = (- u cos (v) + 5 sin (v), -5 cos (v) - u sin(v), u)
Then I dot product with the given F (after putting it in terms of "u" and "v")
Then I do the relevant integration

I think my method is correct, however since my cross product yields an extremely long vector it makes all my following calculations really tedious.
So just want to check if there's a easy way to do this.

 

[HallsofIvy]

There is no way to answer that without knowing what r(u,v) is. Some surfaces have "simple" differentials of surface area, some complicated ones. (What you give does NOT seem "extremely long" to me!)

 

[Bipolarity]

Use the divergence theorem or Stokes' theorem.

BiP