Center Mass Problems

Jan 2010
47
0
a.the region bounded by x + y + z = 2, x = 0, y = 0, z = 0, constant density
b. the region between y=0, y=x^2, where x is from 0 to 1/2, constant density

I know the formulas and everything but what im confused about is the constant density and how that applies.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
The "center of mass" of region A is given by
\(\displaystyle \overline{x}= \frac{\int\int_A\int x \rho(x,y,z) dxdydz}{\int\int_A\int \rho(x,y,z)dxdydz\(\displaystyle
\(\displaystyle \overline{x}= \frac{\int\int_A\int y \rho(x,y,z) dxdydz}{\int\int_A\int \rho(x,y,z)dxdydz\(\displaystyle
\(\displaystyle \overline{x}= \frac{\int\int_A\int z \rho(x,y,z) dxdydz}{\int\int_A\int \rho(x,y,z)dxdydz\(\displaystyle

where \(\displaystyle \rho(x,y,z)\) is the density function. If the density is constant, you can factor it out of the integrals and then cancel. That is, with constant density you can just ignore it (equivalently take it equal to "1").\)\)\)\)\)\)