Finding a parametric representation for a boundary of $S$

Question

the question is: Let $\Omega$ denote the conical region $\sqrt{x^2 + y^2} \leq z \leq 2$.
Find a parametric representation $x(u,v)$ for $S = \partial \Omega$, the boundary of $\Omega$. (You'll need to split it into two parts).
I have no clue how to do this. What does it mean by boundary? Does it mean $z=2$ and $z = \sqrt{x^2 + y^2}$?How would I approach finding a parametric for this?

Answer 1

I plotted the region for you. The region $\Omega$ is this solid cone. The boundary $\partial \Omega$ has two parts:

• the curved outer surface, given by the equation $z = \sqrt {x^2 + y^2}$

• the flat upper surface, given by $z = 2$.

To parametrise each of these boundary surfaces, you could first parametrise the $x$ and $y$ coordinates using plane polar coordinates. (If you decide to take this approach, you should think carefully about the range of the radial coordinate.) And once you have specified your parametrisation of $x$ and $y$, the $z$ coordinate is then determined by the equation of the surface, which is different depending on whether you're doing the curved surface or the flat surface. I don't want to say more at this stage, but feel free to leave a comment if you would like more clarification.