Volume of a solid using cylindrical coordinates

Question

Let $G=\{(x,y,z)\in \mathbb R^3:y^2+z^2<4, 0<x<3-y^2-z^2\}$

I want to determine the volume of $G$.

I thought about using some sort of cylindrical coordinates. The transformation for cylindrical coordinates looks like:

$T:(0, \infty)\times(0,2\pi) \times (-\infty, \infty) \to \mathbb R^3$

$T(r, \phi, z)=\begin{pmatrix}r \cos \phi \\r \sin \phi\\z\end{pmatrix}$

In my example, can I use

$T(r, \phi, x)=\begin{pmatrix}x \\r \cos \phi\\r \sin \phi\end{pmatrix}$ ?

The bounds should be

$0\le r\le \sqrt 3$

$0\le \phi\le 2\pi$

$0\le x \le 3-r^2$

Answer 1

In cylindrical coordinates we have

$0\le x \le 3$

$0\le \phi\le 2\pi$

$0\le r\le \sqrt {3-x}$