B.3 Moment of inertia tensor

After completing the r integral in eq.(26), the components of the moment of inertia tensor, for a uniform density particle, in terms of the spherical harmonic expansion, are

$$I_{11}=\frac{M}{5V} \int_0^{2\pi} \int_0^{\pi} r^5(\theta,\phi) \sin(\theta)[1 - \sin^2(\theta) \cos^2(\phi)] \: d\theta \: d\phi$$

$$I_{22}=\frac{M}{5V} \int_0^{2\pi} \int_0^{\pi} r^5(\theta,\phi) \sin(\theta) [1 - \sin^2(\theta) \sin^2(\phi)] \: \: d\phi \: d\theta$$

$$I_{33}=\frac{M}{5V} \int_0^{2\pi} \int_0^{\pi} r^5(\theta,\phi) \sin^3(\theta) d\theta \: d\phi$$

$$I_{12}=\frac{-M\:}{5V} \int_0^{2\pi} \int_0^{\pi} r^5(\theta,\phi) \sin^3(\theta) \cos(\phi) \sin(\phi) \: d\theta \: d\phi$$

$$I_{13}=\frac{-M\:}{5V} \int_0^{2\pi} \int_0^{\pi} r^5(\theta,\phi) \cos(\theta) \sin^2(\theta) \cos(\phi) \: d\phi$$

$$I_{23}=\frac{-M\:}{5V} \int_0^{2\pi} \int_0^{\pi} r^5(\theta,\phi) \cos(\theta) \sin^2(\theta) \sin(\phi) \: d\phi$$ (40)