b) This portion of the problem is resolved with the parallel axis theorem: IOzz = ICzz + md2. Recalling that x2 + y2 = r2, and assuming that the plane of the example figure is the mid plane of the cylinder, the triple integral that describes the moment of inertia can be formed and solved as:
IOzz
=
(T)/(2)⌠⌡ − (T)/(2)2π⌠⌡0R⌠⌡0(x2 + y2)rρ0⎛⎝1 − (r)/(2R)⎞⎠drdθdz + R2∭ρ0dV
=
(T)/(2)⌠⌡ − (T)/(2)2π⌠⌡0R⌠⌡0ρ0r3⎛⎝1 − (r)/(2R)⎞⎠drdθdz + R2(T)/(2)⌠⌡ − (T)/(2)2π⌠⌡0R⌠⌡0ρ0⎛⎝r − (r2)/(2R)⎞⎠drdθdz
=
T2πρ0R⌠⌡0⎛⎝r3 − (r4)/(2R)⎞⎠dr + R2ρ02πTR⌠⌡0⎛⎝r − (r2)/(2R)⎞⎠dr
=
T2πρ0⎡⎣(R4)/(4) − (R4)/(10) + (R4)/(2) − (R4)/(6)⎤⎦
=
R42πρ0T⎡⎣(3)/(4) − (3)/(30) − (5)/(30)⎤⎦ = 2R4πρ0T(29)/(60) = (29)/(30)R4πρ0T