Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 3 / 9B / Solution

For the one-to-one transformation $(u,v,w)\mapsto (x,y,z)$, the Jacobian determinant used in the stated formula isLocally, the derivative maps a small rectangular box of volume $dudvdw$ to a parallelepiped of volume $|J|dudvdw$. Partitioning the region into such boxes and passing to the Riemann integral gives the change of variables formula

In the plane $y=0$, the region is inside the unit circle and on the right-hand side of the hyperbolaIt is nonempty exactly when $\alpha \le 1$; it has positive area when $\alpha <1$. In $x>0$, its boundaries are $x=\sqrt{1-z^2}$ and $x=\alpha \sqrt{1+z^2/{\gamma }^2}$. The minimum and maximum $x$ values are $\alpha$ and $1$, both at $z=0$. The largest absolute $z$ occurs where the boundaries meet:

Now set $u=x$, $v=y/k$, and $w=z$. The inverse Jacobian determinant is $k$, and in cylindrical coordinates $r^2=u^2+v^2$ the transformed region isThus $|w|\le z_0$, and each horizontal cross-section is an annulus. Its volume isTherefore, for $\alpha <1$,For $\alpha =1$ the region has zero volume.