Let be the surface of the cap (see figure). Let be the differentiable curve of the cap in the plane. Let be the portion of the plane xy that C contains. And let be the value of the area of . Consider oriented outward. Determine with arrows the sense of the curve and the value of , where .

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My attempt :

The arrow is easy to draw : counter clockwise.

I've calculated the curl of , but now I realize it is likely useless. Anyway here it comes : .

I'm sure I have to calculate (which is worth by Stokes' theorem) but how to proceed? I have almost no information about , I mean I don't know its parametrization. The only way I think I might solve the problem is to realize something obvious that imply that , but I don't see anything.