### Second Moments of Tetrahedron

M^{12} = Int[x = (0:1)]Int[y = (0:1-x)]xy
= Int[x = (0:1)]Dif[y = (0:1-x)] xy^2/2

= Int[x = (0:1)] x (1-x)^2/2
= Int[x = (0:1)] (x - 2x^2 + x^3)/2

= Dif[x = (0:1)] (x^2/2 - 2x^3/3 + x^4/4)/2
= 1/24
M^{11} = Int[x = (0:1)]Int[y = (0:1-x)]x^2
= Int[x = (0:1)]Dif[y = (0:1-x)]yx^2

= Int[x = (0:1)] (1-x)x^2
= Int[x = (0:1)] x^2 - x^3

= Dif[x = (0:1)]x^3/3 - x^4/4 = 1/12

Indeed!

Late night thought:
The math for the 2nd moment of a body should be a 2nd order contravarient tensor and thus transform as such.
This provides the motive of the stuff above.