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Evaluating an integral using the Jacobian-help!!!

For this question I have managed to sketch the region however I don't understand how to solve this sort of a problem where we have two regions?
Also when we use the change of variable formula with the Jacobian is that for one region only?

Here we have two regions. one where x>0 and one where x<0 but that's all that I have been able to understand from this question and would be grateful if someone could provide some explanation with how to approach this problem.

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Reply 1

Dorkins

You might be over thinking this: try considering it as two separate regions, one for x>0 and the other for x<0. The jacobian will be the same for both, if you use the same functions (u,v) to parametrise the surface.