The moment of inertia, in this situation, depends only on the total mass and how the mass is distributed around the axis, defined by:
I = ∫r2 dm
where r is the perpendicular distance of each differential unit of mass from the axis of rotation. If you take a "pie slice" of the kind Roman talked about, and rotate it 180o, each differential unit of mass still has the same perpendicular distance from the axis. Hence, nothing about the calculation of I changes, because the mass is still distributed the same way, in terms of perpendicular distances, from the axis.
You can also do the integral (I would recommend polar coordinates). You should find that the dependence of I on k drops out when the integral is fully evaluated.