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Moment of inertia tensor

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Find the tensor of inertia about the center of mass of a flat rigid body in the shape of 45 degrees right triangle with uniform mass density. What are the principal axes. What is the moment of inertia with respect to the axis perpendicular to the plane of the triangle through the right angle vertex of the triangle.

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The solution is 4 pages long including illustration and full derivations of the moment of inertia tensor.

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The components of the moment of inertia tensor are:

(1.1)

For a thin lamina, z=0, so this integral becomes a surface integral:
(1.2)
We will construct the system where the triangles hypotenuse is aligned with the x-axis, the ...

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