Question

Find a parametric representation for the surface. the part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 9. (enter your answer as a comma-separated list of equations. let x, y, and z be in terms of s and/or θ.)

Answer 1

The cylinder is a clue to use cylindrical coordinates. Taking $x=3\cos\theta$ and $y=3\sin\theta$, we then are forced to use $z=3\cos\theta+3$. So the parameterization of the intersection of the plane and cylinder is

$\mathbf s(\theta)=(3\cos\theta,3\sin\theta,3\cos\theta+3)$

To get the surface, we can introduce a second parameter $r$ that "contracts" the elliptical intersection to a point. The simplest way to do this is to use

$\mathbf s(r,\theta)=(3r\cos\theta,3r\sin\theta,3r\cos\theta+3)$

with $0\le r\le1$ and $0\le\theta\le2\pi$.