= Solution
The <parallel axis theorem> relates the <moment of inertia> $I$ about an axis to $I_{\rm CM}$ about a parallel axis through the <centre of mass>, separated by perpendicular distance $d$:
$$
\boxed{I=I_{\rm CM}+M_{\rm tot}d^2.}
$$
In the defining mass <integral>, expanding the squared distance produces a cross term whose first mass moment about the <centre of mass> is zero.
For a thin planar <rigid body>, the <perpendicular axis theorem> applies to two perpendicular axes $x,y$ in its plane and the perpendicular axis $z$, all meeting at the same point:
$$
\boxed{I_z=I_x+I_y.}
$$
Indeed, $I_x=\int y^2\,dm$, $I_y=\int x^2\,dm$, and $I_z=\int(x^2+y^2)\,dm$. The planar condition is essential; this is not an identity for an arbitrary three-dimensional <rigid body>.
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