Moment of Inertia: Difference Between Equations 3 and 6

[Miike012]

What is the differece between the two formulas numbered 3 and 6 in the paint document?

And what types of questions would I use eq. 3 and eq. 6?

 

[haruspex]

The moment is force * perpendicular distance. Summing gives ∫F(x)x.dx. When F is gravitational and the mass at offset x is y=y(x), that's ∫gyx.dx
The moment of inertia is the factor converting rotation rate, ω, to angular momentum (sometimes called moment of momentum). An element at distance x from the axis is moving at speed xω so has linear momentum xω.dm. The moment of that is x²ω.dm. So this leads to ∫yx².dx.
(Sorry, just realized I've swapped x and y c.w. the link you posted. Can't be bothered to swap back.)