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Find the moment of inertia of a circular...

Find the moment of inertia of a circular disk or solid cylinder of radius `R` about the axis through the centre and perpendicular to the flat surface.

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The figure shows that the appropriate mass element is a circular ring of radius r and width dr.Its area is dA = `2pirdr` and its mass is `dm = sigmadA` , where `sigma- = M/A is the areal mass density. The moment of inertia of this element is `dI=dmr^2=2pisigmar^3dr`For the whole body, `I=2pisigma int_(0)^(R)r^3dr=1/2pisigmaR^4` The mass of the whole disk or cylinder is M = `sigmaA=sigmapiR^2`, and so `I=1/2MR^2`.
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