In the screenshot below, I am trying to evaluate two closed line integrals over the regions $C_1=x^2+y^2=1$ and $C_2=4x^2+9y^2=36$. In this specific case, however, the partials of the line integral are equal to each other ($P_y=Q_x$). Thus, since this is a conservative field over a closed path, the integrals should evaluate to 0 (which means they are equal).
The part I do not understand is part B, where we are asked to actually evaluate the two line integrals. Parametrizing the path $x^2+y^2 = 1$ and evaluating it yields $2\pi$ -- which I do not get. If the vector field is conservative, and the path is closed, how does the line integral evaluate to a non-zero value?
Thanks for all the help!