What is the intersection of these two cylinders?

Question

$$0\le x^2 + z^2 \le 1$$

$$0 \le y^2 + z^2 \le 1$$

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?

Thanks,

Answer 1

The intersection of two cylinders is called a Steinmetz solid. You can give a description of the edges of the solid by

$$x = \pm z, \quad y = \pm \sqrt{1 - z^2}$$

and use these to give corresponding inequalities.

Answer 2

I could not resist to model this in GeoGebra.

Zenith is $(0, 0, 1)$ and nadir $(0, 0, -1)$. A slice at height $z$ is a square with side length $$ a = 2 \sqrt{1-z^2} $$ so $$ dV = (2 \sqrt{1-z^2})^2 dz = 4(1-z^2) dz $$