One way to write it is .
Then
The base is
To find the center of mass:
where is the mass and is the density.
I am to fond the mass of the tetrahedron bound by the xy, yz, and xz planes and x + 2y + 3z = 1 THe density is 1 gm/cm^3 and x,y,z are in cm Then I am to find the center of this.
The bounds of course are my problem!
I think that they all three would be bound from 0 to 1 but I am not sure if it is that easy. Or would x be bound from 0 to 1 and then just solve the equation x + 2y + 3z = 1 for both y and z and use 0 as the lower bounds for all????
There is, by the way, a simple formula that says that the volume of the tetrahedron with vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c) is abc/6.
The plane x+ 2y+ 3z= 1 cuts the axes at (1, 0, 0), (0, 1/2, 0), and (0, 0, 1/3) so the volume of this tetrahedron is (1)(1/2)(1/3)/6= 1/36.
Since the density is just 1, the mass is the volume.
To get the x coordinate of the centroid, integrate x over that tetrahedron and divide by the volume.