Generally a moment is the sum within a system of some physical quantity weighted by (a power of) the distance from some reference point or axis or plane. Specifically if \(f\) measures stuff at a point and \(d\) is the distance from that point to some reference point or axis or plane, the then \(n\)th moment of stuff is computed as \[ \iint\!\!\dotsb\!\!\int {\color{maroon} d^n} f \,\mathrm{d}S \,. \]
Considering a solid expanse \(E\) consisting of some material for which the function \(\rho(x,y,z)\) gives the point density at \((x,y,z)\) in \(E,\) the integral \({m = \iiint_E \rho(x,y,z)\,\mathrm{d}V}\) computes the mass of \(E.\) The first moments of mass about the three coordinate planes are: \[ M_{yz} = \iiint_E {\color{maroon} x}\,\rho(x,y,z)\,\mathrm{d}V \qquad M_{xz} = \iiint_E {\color{maroon} y}\,\rho(x,y,z)\,\mathrm{d}V \qquad M_{xy} = \iiint_E {\color{maroon} z}\,\rho(x,y,z)\,\mathrm{d}V \] Note the integrating factor of \({\color{maroon} x}\) and \({\color{maroon} y}\) and \({\color{maroon} z}\) in each, which weighs the integrand by its distance from each respective plane. The center of mass of the solid expanse \(E\) will be \(\bigl(\overline{x}, \overline{y}, \overline{z}\bigr)\) where \(m\overline{x} = M_{yz}\) and \(m\overline{y} = M_{xz}\) and \(m\overline{z} = M_{xy}.\) Similarly the second moments of mass, the moments of inertia \(I_x\) and \(I_y\) and \(I_z\) about the \(x\)-axis, the \(y\)-axis, and \(z\)-axis respectively are computed as the integrals \[ I_x = \iiint_E {\color{maroon} \bigl(y^2+z^2\bigr)}\,\rho(x,y,z)\,\mathrm{d}V \qquad I_y = \iiint_E {\color{maroon} \bigl(x^2+z^2\bigr)}\,\rho(x,y,z)\,\mathrm{d}V \qquad I_z = \iiint_E {\color{maroon} \bigl(y^2+z^2\bigr)}\,\rho(x,y,z)\,\mathrm{d}V \] The polar moment of inertia is again \(I_0 = I_x+I_y+I_z,\) and the radius of gyration about a reference axis or point is the number \(r\) such that \(m r^2=I_\ast\) for the appropriate moment of inertia \(I_\ast\) for that reference.