Let $W$ be a cylinder (the curved 2-dimensional surface only—not the top, bottom or solid inside) defined by $x^2 + y^2 = 16$, with $0 ≤ z ≤ 9.$ Parametrize $W$ using cylindrical coordinates.
Would this just be:
$$x=rcost(\theta),y=rsin(\theta), z=z. 0 \leq \theta \le 2\pi, 0 \leq r \le 4 $$
I don't get the "(the curved 2-dimensional surface only—not the top, bottom or solid inside)" part of the question. Is z just equal to z, since the cylinder's height depends on z? Is my parametrization correct?
Best Answer
Since you want only the surface of the cylinder with height from $0$ to $9$ on the $z$ axis, the parametric equation is: $$ \begin{cases} x=4\cos \theta\\ y=4\sin \theta\\ z=t \end{cases} $$ with $0 ≤\theta < 2\pi$ and $0 ≤ t≤ 9$