How do I calculate this about the z-axis, if the cuboid length is b in the y-direction, a in the x-direction and c in the z -direction?(adsbygoogle = window.adsbygoogle || []).push({});

In my notes, I have I = ∫ r^2 dm = ∫ (x^2 + y^2) dm

dm = ρdV = ρdxdydz

This is what I did:

I = ∫∫∫ (x^2 + y^2)ρ dxdydz

I = ρ∫dz∫dy∫dx (x^2 + y^2)

I = ρ∫∫dy [(1/3)x^3 + xy^2] {0->a}

I = ρ∫∫[(1/3)a^3 + ay^2] dy

I = ρ∫dz [(1/3)ya^3 + (1/3)ay^3] {0->b}

I = ρ∫dz [(1/3)ba^3 + (1/3)ab^3]

I = ρ[(1/3)zba^3 + (1/3)zab^3] {0->c}

I = ρ[(1/3)cba^3 + (1/3)cab^3]

I = (1/3)ρabc(a^2 + b^2)

and ρabc = ρV = M, so I = (1/3)M(a^2 + b^2)

However, the answer has (1/12) instead of (1/3). Where does the other 1/4 come from??

Thanks.

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# Moment of inertia of a cuboid

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