Learning Objectives
- Use triple integrals to locate the center of mass of a three-dimensional object.
All the expressions of double integrals discussed so far can be modified to become triple integrals.
definition
If we have a solid object [latex]Q[/latex] with a density function [latex]{\rho}{({x},{y},{z})}[/latex] at any point[latex](x, y, z)[/latex] in space, then its mass is
[latex]{m} = {\underset{Q}{\displaystyle\iiint}}{\rho}{({x},{y},{z})}{dV}.[/latex]
Its moments about the [latex]xy[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]yz[/latex]-plane are
[latex]\begin{aligned} {{M}_{xy}} = & {\underset{Q}{\displaystyle\iiint}}{z}{\rho}{({x},{y},{z})}{dV}, {{M}_{xz}} = {\underset{Q}{\displaystyle\iiint}}{y}{\rho}{({x},{y},{z})}{dV}, \\ {{M}_{yz}} = & {\underset{Q}{\displaystyle\iiint}}{x}{\rho}{({x},{y},{z})}{dV}. \end{aligned}[/latex]
If the center of mass of the object is the point [latex]{\left ( {\overline{x}},{\overline{y}},{\overline{z}} \right )},[/latex] then
[latex]\large{{\overline{x}} = {\dfrac{{M}_{yz}}{m}}, \ {\overline{y}} = {\dfrac{{M}_{xz}}{m}}, \ {\overline{z}} = {\dfrac{{M}_{xy}}{m}}}.[/latex]
Also, if the solid object is homogeneous (with constant density), then the center of mass becomes the centroid of the solid. Finally, the moments of inertia about the [latex]yz[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]xy[/latex]-plane are
[latex]\begin{aligned} {{I}_{x}} = & {\underset{Q}{\displaystyle\iiint}}{({{y}^{2}} + {{z}^{2}})}{\rho}{({x},{y},{z})}{dV}, \\ {{I}_{y}} = &{\underset{Q}{\displaystyle\iiint}}{({{x}^{2}} + {{z}^{2}})}{\rho}{({x},{y},{z})}{dV}, \\ {{I}_{z}} = & {\underset{Q}{\displaystyle\iiint}}{({{x}^{2}} + {{y}^{2}})}{\rho}{({x},{y},{z})}{dV}. \end{aligned}[/latex]
Example: finding the mass of a solid
Suppose that [latex]Q[/latex] is a solid region bounded by [latex]x+2y+3z=6[/latex] and the coordinate planes and has density [latex]{\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[/latex]. Find the total mass.
try it
Consider the same region [latex]Q[/latex] (Figure 1), and use the density function [latex]{\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[/latex]. Find the mass.
Example: finding the center of Mass of a solid
Suppose [latex]Q[/latex] is a solid region bounded by the plane [latex]x+2y+3z=6[/latex] and the coordinate planes with density [latex]{\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[/latex] (see Figure 1). Find the center of mass using decimal approximation. Use the mass found in Example “Finding the Mass of a Solid”.
Try It
Consider the same region [latex]Q[/latex] (Figure 1) and use the density function [latex]{\rho}{({x},{y},{z})} = xy^{2}z[/latex]. Find the center of mass.
Watch the following video to see the worked solution to the above Try It
We conclude this section with an example of finding moments of inertia [latex]I_x[/latex], [latex]I_y[/latex], and [latex]I_z[/latex].
example: finding the moments of inertia of a solid
Suppose that [latex]Q[/latex] is a solid region and is bounded by [latex]x+2y+3z=6[/latex] and the coordinate planes with density [latex]{\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[/latex] (see Figure 1). Find the moments of inertia of the tetrahedron [latex]Q[/latex] about the [latex]yz[/latex]-plane, the [latex]xz[/latex]-plane, and the [latex]xy[/latex]-plane.
try it
Consider the same region [latex]Q[/latex] (Figure 1), and use the density function [latex]{\rho}{({x},{y},{z})} = {x}{{y}^{2}}{z}[/latex]. Find the moments of inertia about the three coordinate planes.
Watch the following video to see the worked solution to the above Try It
