I have the following regions: $S_1=\{(x,y,z):2az=x^2+y^2\}$ and $S_2=\{(x,y,z:x^2+y^2-z^2=a^2\}$ with $a>0$. I'm trying to find the volume of region bounded by $S_1$ and $S_2$ as the book says, but it seem that is not bounded:
The plot for $a=1$
I wonder if the exercise is wrong and I need more restrictions, because I only found one curve of intersection between the two regions. What I did was taking $S_1$ and $S_2$ to have $x^2+y^2-2az=x^2+y^2-z^2-a^2$ which is equivalent to $z^2-2az+a^2=(z-a)^2=0$. Then I have as intersection a circle at height $z=a$ It seems that I'd need to introduce another restriction to have a bounded region, am I missing something?.
Also, are cylindrical coordinates apropiate to calculate a solid that could be bounded by $S_1$ and $S_2$?