I want to calculate the surface integral
$$\iint_{\partial S}\left(\nabla\times\mathbf{F}\right)\cdot\mathbf{n}\,\mathrm{d}A,$$
when
$$\mathbf{F}(x,y,z) = {y\,z\,{\bf i}-x\,z\,{\bf j}+\left(x^2-y^2\right)\,z\,{\bf k}}$$
and
$$S = \left\{ (x,y,z)\in\mathbb{R}^3 \mid x^2+y^2+(z-{2})^2\le{25},\,0\le z\le{2} \right\}.$$
So the z-component of the surface normal n is positive everywhere.
$$\nabla\times\mathbf{F}=(x-2zy)\mathbf i +(y-2xz)\mathbf j + (-2z)\mathbf k$$
We can see that $r^2=25\to r=5$
But how do I proceed from here?