Let $M$ be the capped cylindrical surface which is the union of two surfaces, a cylinder given by $x^2 + y^2 = 4,\ 0 \le z \le 1$, and a hemispherical cap defined by $x^2 + y^2 + (z-1)^2 = 4,\ z \ge 1$. For the vector field $\mathbf{F} = (zx + z^2y + 6 y,\ z^3yx + 3 x,\ z^4x^2)$, compute $\iint_M (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ in any way you like.
$\iint_M (\nabla \times \mathbf{F}) \cdot d\mathbf{S} =$