# How can I find the value of these types of integrals?

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?

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Hint: did you try calculating the curl of $\bf F$?
I found curl $\mathbf F=0$, which means that the integral is zero. – zeke Jul 23 '12 at 22:56