1. The problem statement, all variables and given/known data
Hi there. Well, I have the next exercise, which I've solved, but I don't know if the solution I got is the right one.

It says Given the next region on spheric coordinates find the expression for it in rectangular coordinates, and plot.


Well, In think it is a part of a cone, without a sphere of radius 1, with a sphere of radius 2 on the top.

2. Relevant equations
\begin{Bmatrix}{ x=\rho\cos\theta\sin\phi}\\y=\rho\sin\theta\sin\ph  i\\z=\rho\cos\phi \end{matrix}

3. The attempt at a solution
x^2+y^2=\rho^2\sin^2\phi(\cos^2\theta+\sin^2\theta  )\Rightarrow{x^2+y^2=\rho\sin^2\phi}

z^2=\rho^2\cos^2\phi\Rightarrow{\rho^2=\displaysty  le\frac{z^2}{\cos^\phi}}

x^2+y^2=z^2\displaystyle\frac{\sin^2\phi}{\cos^2\p  hi}\theta}\Rightarrow{x^2+y^2=z^2\tan^2\phi}\Right  arrow{x^2+y^2\leq{\displaystyle\frac{z^2}{3}}}

f(x,y)=\begin{Bmatrix}{ x^2+y^2\leq{\frac{z^2}{3}}\forall{}z\in{[\frac{\sqrt[ ]{3}}{2}},\sqrt[ ]{3}]}\\x^2+y^2+z^2\leq{4}\forall{z\in{(\sqrt[ ]{3},2]}}\\x^2+y^2+z^2\geq{1} \end{matrix},x>0,y>0

Is this right?