S is the part of the cylinder $x^2+y^2=2x$ parametrize $S$

Question

S is the part of the cylinder $x^2+y^2=2x$

with $0 \leq z \leq \sqrt{x^2 + y^2}$

how would I parameterize this ?

$x^2+y^2=2x$ can be made into

$(x-1)^2 + y^2 = 1$

Answer 1

In cylindrical coordinates the surface and its limits are $r=2\cos\theta$ for $x^2+y^2=2x$ and $0\leq z\leq r$ for $0 \leq z \leq \sqrt{x^2 + y^2}$ This suggest to take $\theta$ and $z$ as parameters. But we have the restriction $0 \leq z \leq 2\cos\theta$ limiting the variation for $z$. We can use instead other parameter $t$ in such a way that given its maximum value makes $z$ to reach its own maximum value for that value of $\theta$. So is we can make $z=tr=t2\cos\theta$ with $0\leq t\leq 1$

$$\begin{align*} \theta &= s \\ r &= 2\cos s\\ z &= 2t\cos s \end{align*} \qquad t \in [0,1], \; s \in [-\pi/2,\pi/2)$$

It can easily be expressed in cartesian coordinates:

$$\begin{align*} x &= 2\cos^2 s \\ y &= 2\cos s\sin s\\ z &= 2t\cos s \end{align*} \qquad t \in [0,1], \; s \in [-\pi/2,\pi/2)$$