# Volume inside 3 cylinders

Find the volume of the region lying inside all three of the circular cylinders $$x^2+y^2=a^2,$$ $$x^2+z^2=a^2,$$ $$y^2+z^2=a^2$$ Hint: Make a good sketch of the first octant part of the region, and use symmetry whenever possible.

I have trouble in identifying the function to integrate and the boundary of the integral. Can anyone help me?

• The integrand is $1$ and the domain is the intersection of the inside of the cylinders. – Yves Daoust Nov 8 '14 at 9:42
• This comment is to link this post as one of the (abstract) duplicates to the current choice of mother/target post, which merit is not in the content nor being the oldest but merely having an existing link. – Lee David Chung Lin Jan 22 '19 at 12:05

EDIT1: With two cylinder intersection the plane of intersection is $x=y$. Now choose a part of it.