How to compute the volume of the intersection of two cylinders

Question

I have this problem:

Calculate the volume delimited by $x² + y² = 1$ and $x² + z² = 1$.

I know I have two cylinders, but I can`t find any numeric solution. :(

Any tips on how to get the correct answer?

Thank you!

Answer 1

HINT: What are the cross-sections if you slice perpendicular to the $x$-axis?

Answer 2

You have the Steinmetz Solid.

$\hskip 2.1 in$

There is symmetry across the planes $x = 0, z = y, z = -y, z = 0, y = 0$

Use that to your advantage.

$$16\int_0^1\int_0^\sqrt{1-x^2}\int_0^y \ dz\ dy\ dx$$

Answer 3

I have some working to go with the above answer if that helps make it a bit clearer.

working