Chapter 5.3, Problem 9P

The following notation is used in the problems: $M\text{ = mass}$, $\bar{x},\bar{y},\bar{z}=$ coordinates of center of mass (or centroid if the density is constant), $I=$ moment of inertia (about axis stated), $I_x,I_y,I_z=$ moments of inertia about x, y, z axes, $I_m=$ moment of inertia (about axis stated) through the center of mass.

Note: It is customary to give answers for $I,I_m,I_x$, etc., as multiples of M (for example, $I=\frac{1}{3}M{l}^2$).

For the pyramid inclosed by the coordinate planes and the plane $x+y+z=1$:

1. Find its volume.
2. Find the coordinates of its centroid.
3. If the density is $z$, find $M$ and $\bar{z}$.