Moment of inertia of a cylinder of mass m, length L and radius R...

Let a cylinder of mass ${m}_{1}$, length $L$ and radius $R$ then take elementary disc of radius $R$ and trickiness $dx$ at distance of $x$ from axis $OO '$ then moment of inertia about $OO '$ as this element

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

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

$⇒ I = \dfrac{{MR}^{2}}{4} + \dfrac{{ML}^{2}}{12}$

$⇒ I = \dfrac{M}{4} \times \dfrac{V}{\pi L} + \dfrac{{ML}^{2}}{12}$

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

$⇒ V = \dfrac{2}{3}{\pi L}^{3} ⇒ {\pi R}^{2}L = \dfrac{2}{3}{\pi L}^{3} ⇒ \dfrac{L}{R} = \sqrt{\dfrac{3}{2}}$$\left(I = \dfrac{MV}{4\pi L} + \dfrac{{ML}^{2}}{12}\right)$