What have you tried?
I will give you this much: we can write the cone in parametric coordinates using polar coordinates: , , with from 0 to H and [/tex]\theta[/tex] from 0 to .
With that you can write the "position vector" of a point on the surface of the cone as .
The deriviatives of that with respect to r and :
are in the tangent plane to the cone and their cross product is
The (upward oriented) "vector differential of area" is
Take the dot product of your field with that and integrate.
And what is "Gauss's theorem"?