Final answer:
To find the mass of the solid and the center of mass, we need to integrate the given density function and set up triple integrals over the region. The mass can be calculated by evaluating the integral and the center of mass can be found by calculating the moments about the x, y, and z axes.
Step-by-step explanation:
To find the mass of the solid, we can integrate the density function δ(x,y,z) = 4x over the given region. Since the region is bounded by the coordinate planes and the plane x+y+z=2 in the first octant, we can set up the triple integral as follows:
M = ∫∫∫ δ(x,y,z) dV, where the limits of integration are: 0 ≤ x ≤ 2-y-z, 0 ≤ y ≤ 2-x-z, 0 ≤ z ≤ 2-x-y.
After evaluating this integral, we can find the mass of the solid. To find the center of mass, we need to calculate the moments about the x, y, and z axes by evaluating the triple integrals using the respective coordinates and multiplying by δ(x,y,z). Then, we divide these moments by the total mass to obtain the x, y, and z coordinates of the center of mass.