Moment of inertia of a cylinder of mass m , length L and radius R about an axis passing through its centre…

Let a cylinder of mass ${\mathrm{m}}_1$, length $\mathrm{L}$ and radius $\mathrm{R}$ then take elementary disc of radius $\mathrm{R}$ and trickiness $\mathrm{dx}$ at distance of $\mathrm{x}$ from axis $OO\text{'}$ then moment of inertia about $OO\text{'}$ as this element

$\mathrm{dl}=\frac{{\mathrm{dmR}}^2}{4}+{\mathrm{dmx}}^2$

$\mathrm{I}=\int \mathrm{dl}=\int \frac{{\mathrm{dmR}}^2}{4}+{\int }_{\mathrm{n}=\mathrm{L}/2}^{\mathrm{n}=-\mathrm{L}/2}\frac{\mathrm{M}}{\mathrm{L}}\mathrm{dx}\times {\mathrm{x}}^2$

$\Rightarrow \mathrm{I}=\frac{{\mathrm{MR}}^2}{4}+\frac{{\mathrm{ML}}^2}{12}$

$\Rightarrow \mathrm{I}=\frac{\mathrm{M}}{4}\times \frac{\mathrm{V}}{\mathrm{\pi L}}+\frac{{\mathrm{ML}}^2}{12}$

$\frac{\mathrm{dI}}{\mathrm{dL}}=-\frac{mV}{4{\mathrm{\pi L}}^2}+\frac{\mathrm{M}\times 2\mathrm{L}}{12}=0$

$$\begin{gathered} \Rightarrow \mathrm{V}=\frac{2}{3}{\mathrm{\pi L}}^3 \\ \Rightarrow {\mathrm{\pi R}}^2\mathrm{L}=\frac{2}{3}{\mathrm{\pi L}}^3 \\ \Rightarrow \frac{\mathrm{L}}{\mathrm{R}}=\sqrt{\frac{3}{2}} \end{gathered}$$$\left(\mathrm{I}=\frac{\mathrm{MV}}{4\mathrm{\pi L}}+\frac{{\mathrm{ML}}^2}{12}\right)$