Second Moments of Tetrahedron

M¹² = 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

M¹¹ = 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.