I'm trying to evaluate the line integral $\displaystyle F(x,y)=\left ({\arctan y,\frac{-xy^{2}}{1+y^{2}}} \right )$ around the curve $\displaystyle 4x^{2}+9y^{2}=36$
Does anybody else get -6Pi?
By Green's theorem
$\displaystyle \oint \limits_C Pdx + Qdy = \iint \limits_R Q_x - P_y dA$
which for your particular vector field gives
$\displaystyle \iint \limits_R -1 dA$ and the area of the ellipse is $\displaystyle 6 \pi$, hence your answer of $\displaystyle - 6 \pi$.