How to desribe a region in cylindrical coordinates

[Pirito]

I have to describe this region in cylindrical coordinates

Ω = {(x,y,z) ∈ R³ ; x²+y²+z²≤1 ; x²+y²≥1−z ; x>0 ; y>0 ; z>0}

With x>0 and y>0 I get θ∈(0,π/2).

My problem comes with the two first restrictions: I know x²+y²=r², so I have:

• r²≤1-z²
• r²≥1-z

And from now on I don't know how to continue to resolve r.
Any help would be much appreciated.

 

[Dr.Peterson]

Why do you need to do more? Those are inequalities in cylindrical coordinates, so haven't you done what was asked? Or, if you wish, you could take square roots.

Just list four (technically, five) inequalities as your answer.

 

[Pirito]

So, the answer would be like Ω = {(r,θ,z) / 1-z≤r²≤1-z², θ∈(0,π/2), z>0}, right?

 

[Dr.Peterson]

Yes, that's one way to say it.