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Home Multiple Integrals Triple Integrals Examples Example 4
 
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Example 4

An object has constant density and the shape of a tetrahedron with vertices at the four points

(0,0,0), (1,0,0), (0,1,0), (0,0,1). Find the center of mass.

Step 1

The region is sketched in Figure 12.6.10.

12_multiple_integrals-357.gif

Figure 12.6.10

Step 2

The region E is the solid bounded by the coordinate planes and the plane a- + y + z = 1 which passes through (1,0,0), (0, 1,0), (0,0,1). Solving for z, the plane is

z = 1 - x - y.

This plane meets the plane z = 0 at the line 1 - x - y = 0, or y = 1 - x. Therefore E is the region

0 ≤ x ≤ 1, 0 ≤ y ≤ 1 - x, 0 ≤ z ≤ 1 - x - y.

Step 3

Let the density be ρ = 1.

12_multiple_integrals-358.gif

Similarly 12_multiple_integrals-359.gif The center of mass is

12_multiple_integrals-360.gif

Home Multiple Integrals Triple Integrals Examples Example 4

Last Update: 2006-11-15