= Solution
Take the origin at the <centre of mass>, so $\int_B\rho\mathbf r\,d^3r=0$, and write $r_i=\mathbf r\cdot\mathbf e_i$ in the specified <orthonormal basis>. The <inertia tensor> is
$$
\boxed{I_{ij}=\int_B\rho(\mathbf r)\left(|\mathbf r|^2\delta_{ij}-r_ir_j\right)\,d^3r.}
$$
Here $\delta_{ij}$ is the <Kronecker delta>. For a <rigid body> with <angular velocity> $\boldsymbol\omega$, this <tensor> gives $\mathbf L=\mathbf I\boldsymbol\omega$ and <rotational kinetic energy> $T=\boldsymbol\omega\cdot\mathbf I\boldsymbol\omega/2$.
The <inertia tensor> is a real <symmetric matrix>, so it admits an <orthonormal basis> of <eigenvectors>. The lines in these <eigenvector> directions through the <centre of mass> are its <principal axes>; the corresponding <eigenvalues> $I_1,I_2,I_3$ are its <principal moments of inertia>:
$$
\boxed{\mathbf I\mathbf e_i=I_i\mathbf e_i,\qquad
\mathbf I=\operatorname{diag}(I_1,I_2,I_3)\quad\text{in principal axes}.}
$$
If an <eigenvalue> repeats, the corresponding <principal axes> need not be unique.
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