Surface Integral of Vector Fields

[drecklia]

Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = y i + x j + z^2 k
S is the helicoid (with upward orientation) with vector equation r(u, v) = ucos(v)i + usin(v)j + v k, 0 ≤ u ≤ 3, 0 ≤ v ≤ 3π.

∫∫S f*dS=∫∫D F*(r_u x r_v)dA

i got int[0,3] and int[0,3pi] usin(v)^2-ucos(v)^2+uv^2dvdu
i do not know how to integrate this(yeah, shame on me) and i don't know if this integral is correct

 

[Dick]

It looks ok to me. What's stopping you from doing it? First integrate dv and then the result du. No special tricks needed. Though you could simplify u*(sin(v)^2-cos(v)^2).