Saturday, March 21, 2026

Center of mass and moments

 The expressions of mass, center of mass, moments of inertia expressed in double integrals can be modified by replacing the double integrals with triple integrals.
































Example






Solution

The region Q is a tetrahedron meeting the axes at the point (6,0,0), (0,3,0) and (0,0,2) (see figure below). To find the limits of integration, let z = 0 in the slanted plane z = 1/3(6-x-2y). Then for x and y find the projection of Q on the the xy plane which is bounded by the axes and the line x_+ 2y = 6. The mass is calculated as follow:







 

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