Let $R=\{(x,y,z): -2\leqslant z\leqslant xy\;\mathrm{ and }\;x^2+y^2=1\}\subset\mathbf R^3$ and consider the vector field $\mathbf F(x,y,z)=-x\mathbf i+y\mathbf j+\exp(z^2)\mathbf k.$ I want to find $\iint_R\mathrm{curl}\;\mathbf F\;\mathrm d S,$ but I do not know how to choose a suitable parametrisation.
How do you solve these type of integrals is general?