(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the mass and center of mass of the solid bounded by the planes x=0, y=0, z=0, x+y+z=1; density[tex]\delta[/tex](x,y,z)=y

2. Relevant equations

[tex]M=\int\int_D\int\delta dV[/tex]

[tex]M_{yz}\int\int_D\int x \delta dV;M_{xz}\int\int_D\int y \delta dV;M_{xy}\int\int_D\int z \delta dV[/tex]

[tex]C.O.M.=(\bar{x},\bar{y},\bar{z})[/tex]

[tex]\bar{x}=\frac{M_{yz}}{M};\bar{y}\frac{M_{xz}}{M};\bar{z}\frac{M_{xy}}{M}[/tex]

3. The attempt at a solution

I'm not sure if it's right, but I took the limits to be from 0 to 1 for x, 0 to 1-x for y, and 0 to 1-x-y for z. This gave me the equation:

[tex]M=\int^1_0\int^{1-x}_0\int^{1-x-y}_0 \delta dzdydx[/tex]

Solving this, I got a mass of -1/3, M_{xy}=-17/180, M_{yz}=41/120, M_{xz}=1/20, and the center of mass at (-41/40, -3/20, 17/60)

Could someone check if I did everything right?

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# Homework Help: Mass and Center of Mass

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