Moment of inertia about arbitrary $E=\{(a,b,c)| a,b,c \in \mathbb R\}$ axis

Question

Calculating the moment of inertia is fairly simple, however, how would I proceed to calculate it about an arbitrary axis? The question asks the moment of inertia of $C=\{(x,y,z)|0\leq x,0\leq y, 0 \leq z,x+y+z\leq 1\}$, so, if I'm not wrong about the boundaries, the moment of inertia about the "usual" Z axis would be: $$I_z=\int_{0}^{1}\int_{0}^{1-x}\int_{0}^{1-x-y}x^2+y^2dzdydx$$

But, what about an arbitrary axis? The question actually asks the moment about the axis $\{(t,t,t)|t\in \mathbb R\}$, but, this is more about the general concept than about the question itself. Any directions would be very welcome.

Answer 1

Just integrate the function $f(x,y,z)$ that equals the square of the distance from $\vec x=(x,y,z)$ to your axis. If $\vec a$ is a unit vector in the direction of the axis, then $$f(x,y,z) = \|\vec x - (\vec x\cdot\vec a)\vec a\|^2 = \|\vec x\|^2 - (\vec x\cdot\vec a)^2.$$

Answer 2

a) Also the variable $z$ shall be limited, presumably $0 \le z$, otherwise the solid is infinite in that direction.

b) Assuming the above, then $C=\{(x,y,z)|0\leq x,0\leq y,0 \le z,x+y+z\leq 1\}$ is a right Tetrahedron, with edges on the axes and on the diagonal plane $x+y+z=1$.

c) The axis $(t,t,t)$ is a symmetry axis of the Tetrahedron, passing through the vertex at the origin and the baricenter of the equilateral triangular face on the diagonal plane.
So it is easy to calculate the moment around that axis as the integral of equilateral triangular slices, parallel to the diagonal plane.

d) The solution for the moment around a generic axis involves the Inertia Matrix, do you know that ?