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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?

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    Since you only want the surface of the cylinder, $r$ should just be equal to $4$. Anything smaller than $4$ parametrises the inside of the cylinder instead of its surface. Otherwise, you're good. – 2017-01-13
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    roll up a sheet of paper and tape the two opposite edges together; that's "(the curved 2-dimensional surface only—not the top, bottom or solid inside)" – 2018-06-09

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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 $0 ≤ t≤ 9$

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    How would we parametrize this if we did include the entire boundary: sides, top, and bottom? – 2018-08-22