Small mass element for laminar (moment of inertia)

[jamie.j1989]

Hi, I was just going over the moment of inertia for a 2D lamina, I've been happy with writing the small mass element dM as dM = ρdxdy where ρ is the area density, but for some reason decided on doing it like this,

M(x,y) = ρxy

so

dM = $\frac{∂M}{∂x}$dx + $\frac{∂M}{∂y}$dy
= ρ(ydx + xdy)

This is obviously not the same, and does not give the same answer when substituting in for dM in the moment of inertia tensor. I would like to know if there's a difference between the dM's, and if so why? Thanks.

 

[paisiello2]

M(x,y) = ρxy

This means the mass increases with increasing x and y which I don't think is what you want to model. But if it was then ρ is not constant and should be a function of x and y instead.