# Mass and Center of Mass

## Homework Statement

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$$\delta$$(x,y,z)=y

## Homework Equations

$$M=\int\int_D\int\delta dV$$
$$M_{yz}\int\int_D\int x \delta dV;M_{xz}\int\int_D\int y \delta dV;M_{xy}\int\int_D\int z \delta dV$$
$$C.O.M.=(\bar{x},\bar{y},\bar{z})$$
$$\bar{x}=\frac{M_{yz}}{M};\bar{y}\frac{M_{xz}}{M};\bar{z}\frac{M_{xy}}{M}$$

## 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:
$$M=\int^1_0\int^{1-x}_0\int^{1-x-y}_0 \delta dzdydx$$
Solving this, I got a mass of -1/3, Mxy=-17/180, Myz=41/120, Mxz=1/20, and the center of mass at (-41/40, -3/20, 17/60)
Could someone check if I did everything right?

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
The limits of integration are correct. But I do NOT get "-1/3" as the mass! In fact, it should be obvious that the integral of the function y over a region in the first octant cannot be negative.