Through a homogeneous hemisphere with radius $R$ a hole is drilled with radius $R_0=R/2$ centrally so that the hole axis coincides with the hemisphere's axis of symmetry.
(a) How much is the remaining volume of the hemisphere after the hole has been drilled out?
(b) How much has the hemisphere mass center moved and towards which direction due to drilling?
I don't know how to set up the integral or generally how to solve this problem.
HINT:
Due to symmetry it can be integrated as a single variable. Calculate and remove volumes of cap and cylinder from that of hemisphere.
It so happens remaining volume can be expressed in terms of the height of the remaining cylinder only.