$$0\le x^2 + z^2 \le 1$$
$$0 \le y^2 + z^2 \le 1$$
I want to compute the volume of the intersection.
Sketching it out on paper is sort of nice: I see cross-sections that are disks, the first cylinder, the y-coordinate is free to vary, and for the second cylinder, the x-coordinate is free to vary.
The intersection, I would guess, seems to be something spherical.
So how can I pin down the actual set of points?
Well, one thing I thought of was to try to manipulate both inequalities to make use of the equation of a sphere, so I try looking at these inequalities instead:
$$y^2\le x^2 + y^2 + z^2 \le 1 +y^2$$
$$x^2 \le x^2 +y^2 + z^2 \le 1+x^2$$
Am I heading in the right direction? Where can I go from here?