I want to ask if my boundaries are correct. My steps and the exercise is one the Photo which I have uploaded.
$$ V= \left\{x\in \mathbb R^{3}|1/2<\sqrt{x^2+y^2+z^2} <1; x>y; x,y>0\right\} $$
$$ \Phi(r,\varphi, \vartheta)=\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} r\cos(\varphi)\cos(\vartheta) \\r\sin(\varphi)\cos(\vartheta) \\ r\sin(\vartheta)\end{pmatrix} $$
$$U=\left\{ (r,\varphi, \vartheta)| r>0; 0<\varphi<2\pi ; -\pi /2<\vartheta<\pi /2\right\} $$
For the Boundaries I have:
$$ -\pi /2<\vartheta<\pi /2 $$ $$1/2<r<1$$ $$\pi/4>\phi>0$$