Q:

How do I algebraically find the intersection of the plane $z = 4 + x + y$ with the cylinder $x^2 + y^2 = 4$?

I need to calculate the surface integral $\int\int_{S}(x^2z+y^2z)dS$, where S is the surface defined by the intersection of the plane $z = 4 + x + y$ with the cylinder $x^2 + y^2 = 4$.
I know the procedure for solving the surface integral, and to do so, I need to first find the equation of the intersection, parametrize it, and then carry on with the rest. I'm stuck on finding the equation of the intersection, because it doesn't seem like there's any straightforward way to equate them and solve...

A:

The projection of the surface in the $xy$ plane is the disc 
$$
x^2+y^2 \le 4
$$
So a straightforward parametrization is 
\begin{cases}
x=x\\
y=y\qquad \qquad\text{with}\quad (x,y)\mid x^2+y^2 \le 4\\
z=4+x+y
\end{cases}

