So...I have found a problem where I have to solve the integral of the vector field:
$F(x,y)=(\sin(x)ln(x)+y^2) a_x + (\cos(y)e^y-x^2) a_y$
Along the borders of the region bounded by the circumferences:
C1 with R=1 centered on $(1,0)$
C2 with R=2.centered on $(2,0)$
Which look like this (a half moon):
The vector field is not conservative, but it shouldn't be such a big deal since I could either plan the integral or just apply Green's theorem. The issue comes from having two circumferences there since so far I have only made line integrals over a single curve, so I suspect I should apply Green's theorem, though I am not sure how to show (if) that the region is enclosed by a smooth, closed, positively oriented curve. Or maybe there are other means to solve it?
Any help is welcome