Center of Mass Integral

[smatz17]

Find the mass ofa cube with edge length 2 and density equal to the square of the distance from one edge. By definition the coordinates (X,Y,Z) of the center of mass region are the averages of the x, y, and z coordinates of the region. Thus, X= (int, int, int [D] x dV)/volume of D and similarly for Y and Z.

I know the volume of D=8. But I don't know what I'm supposed to solve to get mass. I know I need multiply the density by the volume but the whole (X,Y,Z) coordinates confuse me and I don't know how to set up the density equation.

 

[DrMike]

Find the mass ofa cube with edge length 2 and density equal to the square of the distance from one edge. By definition the coordinates (X,Y,Z) of the center of mass region are the averages of the x, y, and z coordinates of the region. Thus, X= (int, int, int [D] x dV)/volume of D and similarly for Y and Z.

I know the volume of D=8. But I don't know what I'm supposed to solve to get mass. I know I need multiply the density by the volume but the whole (X,Y,Z) coordinates confuse me and I don't know how to set up the density equation.

"Mass = Density x Volume" works fine if the density is the same throughout the whole object. In your case, the density varies throughout the volume, so you need \(\displaystyle Mass = \int\int\int_V Density\ dV\)