PX155 - E11 - moment of inertia for continuous rigid bodies

from here $I = \sum_{i} m_{i} r_{i}^{2}$ where $r_{i}$ is the radius of rotation of point $i$ about the axis of rotation

continuous limit

I = \int r^{2} . d m = r^{2} \int ρ . d V $ $ t a k e n o v e r t h e w h o l e b o d y - w i t h $ I $ k n o w n, $ T = \frac{1}{2} I ω^{2} $ a n d $ \vec{L} = I \vec{ω} $ - i f t o r q u e, $ \vec{τ} $ i s p a r a l l e l t o $ ω $ : $ $ \vec{τ} = \frac{d \vec{L}}{d t} = I \frac{d ω}{d t} = I \frac{d^{2} θ}{d t^{2}} . . . [N 2] f o r r o t a t i o n