Stokes Theorem for the intersection of $z=x^2+y^2$ and $z=x+2$ and parameterizing the portion of the plane inside of $z=x^2+y^2$

Question

I have been trying to solve the surface integral $$\int \int_S curl(\vec{F}) \cdot \vec{dS}$$ using stokes theorem for the vector field $$F(x,y,z)=\left(xz,yz,xy\right)$$ and where $S$ is the intersection of the surfaces $z=x^2+y^2$ and $z=x+2$. Now, Stokes Theorem states $$\int \int_S curl(\vec{F}) \cdot \vec{dS}=\int_{\partial S} \vec{F} \cdot d \vec{r}.$$ I have since successfully parametrized the curve of intersection as $$\vec{r(\theta)}=\left(\frac12+\frac32\cos(\theta),\frac32\sin(\theta),\frac32\cos(\theta)+2\right)$$ since the equation of the boundary curve is given by $$\left(x-\frac12\right)^2+y^2=\left(\frac32\right)^2.$$ Now, I have tried various times to parameterize of the interior of this disk formed by intersection, but to no avail. I am aware that Stokes Theorem allows for the same results, regardless of if one does the surface integral or the line integral, I am just curious as to how I could parameterize this surface of intersection, that is, the portion of the plane $z=x+2$ inside the paraboloid $z=x^2+y^2$?