Parametrization of boundary of set

Question

I know how to parametrize and calculate the area of boundaries of simple subset of $\mathbb{R}^3$ like the sphere and I would like to know how to do the same for more complex set like this one: $\Omega :=\{(x,y,z)\in\mathbb{R}^3:x^2-y^2-z^2\leq 1, 0\leq x\leq 2-y^2-z^2\}$ (which I think is the intersection between a two-sheeted hyperboloid and a paraboloid); I want to parametrize $\partial\Omega$ so that I can later calculate its area. In general I'm interested in how one can parametrize the boundary of sets such as $\Omega$ which are defined by more than one "condition" (in the case of $\Omega$ there are two: $x^2-y^2-z^2\leq 1$ and $0\leq x\leq 2-y^2-z^2$).

(Initially I thought about $\varphi\colon [0,2-y^2-z^2]\times\mathbb{R}\to\mathbb{R}^3, \varphi(x,y):=(x,y,\sqrt{x^2-y^2-1})$ but it doesn't look very promising)

Can someone explain to me how should I deal with these kinds of sets?

Best regards,

lorenzo.

Answer 1

In this special case, it is not that difficult to visualize. Basically, this is two pieces of surface of revolution and one disk in the back (which I didn't illustrate), This good enough for you?

Also note that it is possible to parametrize the two pieces illustrated by projecting them onto the yz plane. You should use cylindrical coordinates in this case: $$(r,\theta)\mapsto(x(r),r,\theta)\mapsto (x,r\cos\theta,r\sin\theta)$$.