Step 1 :The surface integral is given by \(\iint_{S}(z-x) d S\), where \(S\) is the portion of the surface \(z=x+y^{2}\), where \(0 \leq x \leq y, 0 \leq y \leq 1\).
Step 2 :The surface \(S\) is given by \(z = x + y^2\). So, we can write \(z - x = y^2\).
Step 3 :The surface integral can be expressed as \(\iint_{S} y^2 dS\).
Step 4 :The differential area element \(dS\) in Cartesian coordinates is given by \(dS = dx dy\).
Step 5 :So, the surface integral becomes \(\iint_{S} y^2 dx dy\).
Step 6 :The limits of integration for \(x\) are from \(0\) to \(y\) and for \(y\) are from \(0\) to \(1\).
Step 7 :So, the surface integral is \(\int_{0}^{1} \int_{0}^{y} y^2 dx dy\).
Step 8 :Final Answer: The surface integral is set up as \(\boxed{\int_{0}^{1} \int_{0}^{y} y^2 dx dy}\).