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The parallel axis theorem relates moment of inertia of an object about an arbitrary axis to its moment of inertia about axis through center of mass

$$\boxed{I_0 = I_\text{cm} + M d^2} \tag 6$$

where $M$ is total mass of the object and $d$ is distance between the two axis. Note that the two axes must be parallel for Eq. (6) to work!

The parallel axis theorem, also known as Huygens-Steiner theorem, relates moment of inertia of an object about axis through center of mass to moment of inertia about arbitrary parallel axis

$$\boxed{I = I_\text{cm} + M d^2} \tag 6$$

where $M$ is total mass of the object and $d$ is distance between the two parallel axis.

$$\boxed{I = \iiint_S r^2 dm} \tag 1$$

where $S$ is space in which the object is defined. Now the only thing you have to do is to represent the infinitesimally small mass $dm$ as a function of distance $r$. For objects whose density $\rho$ is constant (homogeneous objects), the mass can be expressed as

$$\boxed{I = \rho \iiint_S r^2 dV} \tag 2$$

$$\boxed{I = \sigma \iint_S r^2 dA} \tag 3$$

The integration is usually done either in Cartesian coordinates or polar coordinates, whichever is more suitable for a specific problem.

$$\boxed{I = \sigma \iint_S (x^2 + y^2) dx dy} \tag 4$$

$$\boxed{I = \sigma \iint_S r^2 \cdot r dr d\theta} \tag 5$$

$$\boxed{I = \iiint_\mathcal{V} r^2 dm} \tag 1$$

where $\mathcal{V}$ is the volume that object occupies in 3D. Now the only thing you have to do is to represent the infinitesimally small mass $dm$ as a function of distance $r$. For objects whose density $\rho$ is constant (homogeneous objects), the mass can be expressed as

$$\boxed{I = \rho \iiint_\mathcal{V} r^2 dV} \tag 2$$

$$\boxed{I = \sigma \iint_\mathcal{A} r^2 dA} \tag 3$$

where $\mathcal{A}$ is the area that object occupies in 2D. The double integrals are solved either in Cartesian coordinates or polar coordinates, whichever is more suitable for a specific problem. Except for these two, the triple integrals can also be solved in spherical coordinates, but I will focus here only on double integrals.

$$\boxed{I = \sigma \iint_\mathcal{A} (x^2 + y^2) dx dy} \tag 4$$

$$\boxed{I = \sigma \iint_\mathcal{A} r^2 \cdot r dr d\theta} \tag 5$$

The parallel axis theorem relates moment of inertia of an object about an arbitrary axis $I_0$ to its moment of inertia about axis through center of mass $I_\text{cm}$

where $M$ is total mass of the object and $d$ is distance between the two axis. Note that the two axes are parallel in this definition!

The parallel axis theorem relates moment of inertia of an object about an arbitrary axis to its moment of inertia about axis through center of mass

where $M$ is total mass of the object and $d$ is distance between the two axis. Note that the two axes must be parallel for Eq. (6) to work!