Solution
= Solution
On the surface $y=1+z$, the <integrand> is $x+z+z^2$. Using the <area element of a surface> from part (i),
$$
\begin{aligned}
\iint_S(x+yz)\,dS
&=\sqrt2\int_0^{2\pi}\int_0^1(x+z+z^2)\,dz\,dx\\
&=\sqrt2\int_0^{2\pi}\left(x+\frac56\right)dx\\
&=\boxed{\sqrt2\left(2\pi^2+\frac{5\pi}{3}\right)}.
\end{aligned}
$$