## Spheric to cartesian coordinates

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.

$\phi\in{(0,\displaystyle\frac{\pi}{6}}),\rho\in{[1,2],\theta\in{(0,\displaystyle\frac{\pi}{2}})}$

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?