Applications: masses, moments, center of mass in the three dimensional space.

Consider the domain $\mathcal{D}$in the space as a solid; if we know at each point what is the density of the material used to build this solid, we can compute the mass, the moments of this object about the coordinate axes, the center of mass of the object, and so on. We denote by $\delta (x,y,z)$the density at the point (x,y,z).

1.

Mass:

$$M= \int \int \int_{\mathcal{D}} \delta (x,y,z) \; dV.$$

2.

First moments about the coordinate planes:
$$\begin{align*}M_{yz} & = \int \int \int_{\mathcal{D}} x \delta (x,y,z) \; dV\\ ... ...V\\ M_{xy} & = \int \int \int_{\mathcal{D}} z \delta (x,y,z) \; dA \end{align*}$$

3.

Coordinates of the center of mass:

$$\overline{x}=\frac {M_{yz}}{M} \qquad ; \qquad \overline{y}=\frac {M_{xz}}{M} \qquad ; \qquad \overline{z}=\frac {M_{xy}}{M}$$

4.

Moments of inertia:
$$\begin{align*}I_x &= \int \int \int_{\mathcal{D}} (y^2+z^2)\delta (x,y,z) \; dV\... ..._L &= \int \int \int_{\mathcal{C}} r^2(x,y,z) \delta (x,y,z) \; dV \end{align*}$$
where L is any line in the plane and r is the distance from the point (x,y,z) to the line L.

5.

Radii of gyration about a line L:

$$R_L= \sqrt{\frac {I_L}{M}}.$$

Example 7.15