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. Imagining a planar region \(R\) as a thin plate of material, a lamina, suppose the function \(\rho(x,y)\) gives the point density at \((x,y)\) in \(R.\) Then the integral \({m = \iint_R \rho(x,y)\,\mathrm{d}A}\) computes the mass of \(R,\) which is the lamina’s zeroth moment of mass since it’s unweighted. The first moments of mass \(M_x\) and \(M_y\) about the \(x\)-axis and \(y\)-axis respectively are computed as the integrals: \[ M_x = \iint_R {\color{maroon} y}\,\rho(x,y)\,\mathrm{d}A \qquad M_y = \iint_R {\color{maroon} x}\,\rho(x,y)\,\mathrm{d}A \] Note the integrating factor of \({\color{maroon} x}\) and \({\color{maroon} y}\) in each, which weigh the integrand by its distance from each respective axis. Then the center of mass, also called the barycenter, of the lamina will be \(\bigl(\overline{x}, \overline{y}\bigr)\) where \(m\overline{x} = M_y\) and \(m\overline{y} = M_x.\) This center of mass is the point at which we may regard the mass as being concentrated for any calculations involving linear forces. When \(\rho(x,y) = 1\) we instead call these moments the moments of area and refer to the center of mass as the centroid. The second moments of mass \(I_x\) and \(I_y\) and \(I_0\), usually called the moments of inertia, about the \(x\)-axis, the \(y\)-axis, and the origin respectively are computed as the integrals: \[ I_x = \iint_R {\color{maroon} y^2}\,\rho(x,y)\,\mathrm{d}A \qquad I_y = \iint_R {\color{maroon} x^2}\,\rho(x,y)\,\mathrm{d}A \qquad I_0 = \iint_R {\color{maroon} \bigl(x^2+y^2\bigr)}\,\rho(x,y)\,\mathrm{d}A \] Note the squared integrating factors in each. The moment \(I_0\) is also called the polar moment of inertia and \(I_0 = I_x+I_y.\) 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. The radius of gyration is the radius at which we may regard the mass as being concentrated for any calculations involving rotational forces. If the mass is concentrated at the point \(\bigl(\overline{\overline{x}}, \overline{\overline{y}}\bigr)\) for \(m\overline{\overline{x}}^2 = I_y\) and \(m\overline{\overline{y}}^2 = I_x\) the inertia about both the \(x\)- and \(y\)-axis will be maintained.