Doubt about the domain of a parameterization of a plane inside a cylinder

Question

I have to parameterize the portion of the plane $x + z = 4$ inside the cylinder $x^2 + y^2 = 4x$.

The parameterization I chose was -

$g(x,y) = (x,y,4-x)$ and the domain $D = \{(x,y): 0 \le (x-\frac{3}{2})^2+y^2≤25/4\}$

Is that correct?

Answer 1

Your domain does not seem right to me. This is a circular cylinder along $z$ axis with center at $(2,0,z)$ and radius $2$. The plane $ x + z = 4$ intersects it between points $(4,0,0)$ and $(0,0,4)$ in an ellipse. The minor axis of the ellipse is $4$ ($-2 \le y \le 2$) along $y$ axis. The major axis is $4\sqrt2$ along $X$ and $Z$ axis with $45^0$ angle to the axis of the cylinder, between points $(4,0,0)$ and $(0,0,4)$.