$C$ is the surface $z=y(e^{-x^2}-y-1/2)$ and conditions $z\geq 0$, $x \in [-1, 1]$ and $\varphi = dxdz+dydz+(e^{-x^2}-2y-1/2)xe^{xz}dxdy$ is a 2-form.
I have to compute $\int_C\varphi$ using Stokes' theorem. I have already computed the integral using common technique and got $0$, but the problem is I must do it with Stokes.
However I can't understand how, because $\varphi$ is not exact, so I can't find a form $\psi$ such that $d\psi=\varphi$. Please help.