\( dI = (dm)r^{2}\) \( \Rightarrow (\sigma dA)r^{2}\) \( \Rightarrow (\frac{\sigma_{0}}{r})r^{2}\) \( \Rightarrow (\sigma _{0}2\Pi dr)r^{2}dr\)

I = \( \int dI = \int_{a}^{b}\sigma _{0}2\Pi dr)r^{2}dr\) \( \Rightarrow \sigma _{0}2\Pi(\frac{b^{3} – a^{3}}{3})\)

m = \( \int dm = \int \sigma dA\) \( \Rightarrow \sigma _{0}2\Pi\int_{a}^{b}dr\)

m = \( \sigma _{0}2\Pi(b-a)\)

Radius of gyration

k = \( \sqrt{\frac{I}{m}} = \sqrt{\frac{(b^{3} – a^{3})}{3(b -a)}}\) \( \Rightarrow \sqrt{(\frac{a^{3} + b^{3} + ab}{3})}\)

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