Replacing these values into the original position equation, the position vector is
rCO = (24)/(5)i + (18)/(5)j
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
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