Hi. I have a problem with this exercise. I wanted to verify the greens theorem for the vector field over the path

The Green theorem says:

and I have:

So then I've made the line integral:

Creo que esto esta bien, la integral la resolví con la computadora para no tener problemas :p

And then the double integral:

The problem is clearly with the signi. The mistake I think I've committed was putting on the reverse the integral limits for the double integral. I think that x should go from 1 to -1, but the thing is I don't know why. So I'm not pretty sure on how to determine the integral limits on this cases. I've tried to think about the parametrization, but I don't know what to do. Its clear to me that the parametrization plays an important role in the sign of the integral. But I don't know how to reason this, so I wanted some help and suggestions ;D

Bye there.