14. Let S be the closed region in the first octant of R³ bounded by the coordinate planes and the plane x + y + z = 1. The mass density at any point in S is equal to the square of the distance from the point to the xy-plane. What is the mass of S ?

Since the solid is in the first octant it is bounded above by
0 <= z <= 1-x-y. It is bounded in the z = 0 plane by the line
0 <= x <= 1 and the line 0 <= y <= 1-x.

If the density function rho(x,y,z) of a solid object that occupies the region S in R³ is given at any point (x,y,z) in S, then the mass m of S is given by the following triple integral:

Consequently, the mass is given by