Consider the integral: $\int_S curl(F)n dS$ of the vector field $F(xy^2, yz^2, zx^2)$ over a triangle defined by the vertices $(1,0,0) $, $(0,1,0)$ and $(0,0,1)$.
From Stokes' Theorem, this is the same as the line-integral defined by that triangle. Can I assume that is the same as the sum of the 3 line-integrals of each of the sides of the triangle?
Since each line is on one of the coordinate planes, this would really simplify the calculations.
You can integrate the boundary of the triangle piecewise where each piece is in one coordinate plane. Just make sure you integrate each piece in the correct direction.