Moments and Centers of Mass

We consider a lamina with variable density. Suppose the lamina occupies a region  and has density function . we defined the moment of a particle about an axis as the product of its mass and its directed distance from the axis. we obtain the moment of the entire lamina about the  :

Similarly, the moment about the 

The physical significance is that the lamina behaves as if its entire mass is concentrated at its center of mass. Thus the lamina balances horizontally when supported at its center of mass .The coordinates  of the center of mass of a lamina occupying the region  and having density function  are

where the mass  is given by

EXAMPLE 2: Find the mass and center of mass of a triangular lamina with vertices , and  if the density function is .

The triangle is shown in Figure below. (Note that the equation of the upper boundary is  ) The mass of the lamina is 

Then the formulas in (5) give

The center of mass is at the point  .