Solution
= Solution
The curl is $(2y-y^2,x,0)$, so the field is not conservative. On $C$, $z=1$ and $dz=0$, giving
$$
\boxed{\oint_C(3x\,dx+y^2\,dy)=\oint_Cd\left(\frac32x^2+\frac13y^3\right)=0.}
$$
= Solution
The curl is $(2y-y^2,x,0)$, so the field is not conservative. On $C$, $z=1$ and $dz=0$, giving
$$
\boxed{\oint_C(3x\,dx+y^2\,dy)=\oint_Cd\left(\frac32x^2+\frac13y^3\right)=0.}
$$