Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle 0,y,-z \rangle\, dS$.
Okay so I know I have to break the disk and paraboloid apart. I have to find their surface integral and then add them together. Looking at the paraboloid I found $\dfrac{dy}{dx} = 2x$ and $\dfrac{dy}{dz} = 2z$. I'm going to plug this into the equation: the double integral of $$\sqrt{\left(\dfrac{dy}{dx}\right)^2 + \left(\dfrac{dy}{dz}\right)^2 +1}\,dA$$ I'm just having trouble figuring out the domain of integration.