The mass properties of an object are found using a _volume integral_. The general procedure is as follows 1. **Volume** - The solid is parametrized in a way the volume can be summed by $$V = \int {\rm d}V$$ - There are various common ${\rm d}V$ expressions depending on the geometry, such as ${\rm d}V = {\rm d}x\, {\rm d}y\, {\rm d z}$ or ${\rm d}V = r\,{\rm d}r\,{\rm d}\theta\,{\rm d}z$. - For the most general case if the interior points of the solid can be expressed in terms of a vector with three parameters $\vec{\rm pos}(u,v,w)$ then the volume differential is $${\rm d}V = \left| \frac{\partial \vec{\rm pos}}{\partial u} \cdot (\frac{\partial \vec{\rm pos}}{\partial v} \times \frac{\partial \vec{\rm pos}}{\partial w}) \right| {\rm d}u\,{\rm d}v\,{\rm d}w$$ <sub>Here $\cdot$ is the vector inner product, and $\times$ the vector cross product.</sub> 2. **Mass** - The volume integral is summed over the density function $$ m = \int \rho\,{\rm d}V$$ 3. **Center of Mass** - The volume integral is summed over the position function $$ \vec{\rm com} = \frac{1}{m} \int (\vec{\rm pos})\,\rho {\rm d}V$$ 4. **Mass moment of inertia (tensor) about origin** - The MMOI of the body _about the origin_ is evaluated with the volume integral summing the MMOI of each particle in the solid $$ \mathrm{I}_O = \int ( \vec{\rm pos} \cdot \vec{\rm pos} - \vec{\rm pos} \otimes \vec{\rm pos}) \, \rho\,{\rm d}V$$ <sub>Here $\cdot$ is the vector inner product, and $\otimes$ the vector outer product.</sub> an alternate form of the above is $$ \mathrm{I}_O = \int \begin{vmatrix} z^2+y^2 & -x y & -x z \\ -x y & x^2+z^2 & - y z \\ -x z & -y z & x^2+y^2 \end{vmatrix} \, \rho\,{\rm d}V$$ 5. **Mass moment of inertia (tensor) about the center of mass** - Finally the parallel axis theorem must be applied to transform the MMOI about the origin $\mathrm{I}_O$ to the MMOI about the center of mass $\mathrm{I}_C$ $$\mathrm{I}_C = \mathrm{I}_O - m ( \vec{\rm com} \cdot \vec{\rm com} - \vec{\rm com} \otimes \vec{\rm com})$$