Center Mass Problems

larryboi7

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
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").\)\)\)\)\)\)