Calculate Volume of $S:= \{(x,y,z) \in \Bbb R^3 \mid z\gt 0, a^2 \lt x^2+y^2+z^2 \lt b^2, z^2 \lt c^2(x^2+y^2) \}$

Question

$0\lt a \lt b$ and $c\gt 0$ and I want to calculate $\lambda ^3(S)$.

I think I have to use some transformation but I already tried transforming into polar coordinates and it didn't work. Any tipps on what transformation to use? Thanks in advance!

Answer 1

I'll assume that $a$, $b$ and $c$ are positive. The set $S$ has a cylindrical symmetry around the $z$ axis, so it can be convenient to change the coordinates as \begin{equation} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} \rho\cos\theta\\ \rho\sin\theta\\ z \end{pmatrix}, \end{equation} with $(\rho,\theta,z)\in(0,+\infty)\times(0,2\pi)\times\mathbb{R}$. Let $g$ be the transformation from cylindrical to rectangular coordinates: it is easy to see that \begin{equation} g^{-1}(S)=\{z>0,a^2<\rho^2+z^2<b^2,z^2<c^2\rho^2\}. \end{equation} It is very useful to visualize this set: remembering that $\rho$ and $z$ are positive in $S$, the intersection of $S$ with a plane of constant $\theta$ (for any value of $\theta$) is

The volume is then \begin{equation} \lambda^3(S)= \int_0^{2\pi}\mathrm{d}\theta\int_U\rho\,\mathrm{d}\lambda' \end{equation} where $\lambda'$ is the Lebesgue measure on the plane and \begin{equation} U=\{(\rho,z)\in\mathbb{R}^2\colon a^2<\rho^2+z^2< b^2, 0<z<c\rho\}. \end{equation} This other integral can be solved resorting again to polar coordinates, this time in $\mathbb{R}^2$. Considering the coordinate transformation $h$ defined as \begin{equation} h \begin{pmatrix} r\\\eta \end{pmatrix} = \begin{pmatrix} r\cos\eta\\ r\sin\eta \end{pmatrix} = \begin{pmatrix} \rho\\ z \end{pmatrix} \end{equation} for $r\in(0,+\infty)$, $\eta\in(0,2\pi)$; the inequality $z<c\rho$ is equivalent to $\sin\eta<c\cos\eta$ which gives (since $\cos\eta>0$) \begin{equation} \eta<\arctan c, \end{equation} so \begin{equation} h^{-1}(U)=\{(r,\eta)\in(0,+\infty)\times(0,2\pi)\colon a<r<b,0<\eta<\arctan c\}, \end{equation} and \begin{equation} \int_U\rho\,\mathrm{d}\lambda'= \int_a^b\int_0^{\arctan c} r^2\cos\eta\,\mathrm{d}\eta\,\mathrm{d} r \end{equation}