Two hollow cylinders have the same inner and outer diameters and the same mass, but they have different lengths because one is made of low density wood and the other of high density brass. Which cylinder has the greater moment of inertia about its cylindrical center axis?

Both cylinders will have the same moment of inertia.

Explanation:

The moment of inertia of a rigid body depends on the distribution of mass around the axis of rotation, i.e at what radius from the axis how much mass is located. What the moment of of inertia DOES NOT depend in is the distribution of mass parallel to the axis of rotation. This means that two hollow cylinders of the same mass but with different lengths will have the same moment of inertia!

For completeness, the momentum of inertia of a hollow cylinder is

from which we clearly see that   only depends on the mass and the radius of the hollow cylinder and not on its height; Hence, both the wooden and the brass cylinders will have the same moment of inertia.

to be honest as much as i would love to answer this. i feel as if something's missing. just saying.

where are the vectors ?