TEX document #7

The density is now

\frac{mass of matter in a square}{area of the square}

in units kg/m^{2} (not kg/m^{3}). (We think of the integration over the thin, uniform third dimension as having already been done.)

Set up the problem so that the body lies in the x–y plane. D is now the two-dimensional region in that plane occupied by the matter. Then the formula for total mass is

M = ∫∫_{D} ρ(x, y) dA,

the coordinates of the center of mass are

\overline{x} = \frac{∫∫_{D} ρ(x, y) x dA}{M} , \overline{y} = \frac{∫∫_{D} ρ(x, y) y dA}{M} ,

and (since z is always 0 inside the body) the three moments of inertia are

I_{z} = ∫∫_{D} ρ (x^{2} + y^{2}) dA,  I_{y} = ∫∫_{D} ρ x^{2} dA,  I_{x} = ∫∫_{D} ρ y^{2} dA.

Notice that if you know any two of the moments of inertia, then you know the third without further calculation! [Insert graphics of lamina spinning about axis perpendicular to lamina and about axis lying in lamina.]