To find the mass, moments, and center of mass of a solid E with the given density function p, we need to evaluate the triple integrals of the respective functions over the region E.
The region E is bounded by the parabolic cylinder z = 1 - y^2 and the planes x + 6y = 6, x = 0, and z = 0.
To find the mass, we need to integrate the density function p(x, y, z) = 5 over the region E:
m = ∭E p(x, y, z) dV = ∭E 5 dV
We can evaluate this integral using cylindrical coordinates:
m = ∫0^1 ∫0^π/2 ∫0^(6-6r cos θ)/5 5 r dz dθ dr
m = 15π/2
To find the moments, we need to integrate the respective functions over the region E:
Myz = ∭E x p(x, y, z) dV = ∭E 5x dV
Mxz = ∭E y p(x, y, z) dV = ∭E 5y dV
Mxy = ∭E z p(x, y, z) dV = ∭E 5z dV
We can evaluate these integrals using cylindrical coordinates:
Myz = ∫0^1 ∫0^π/2 ∫0^(6-6r cos θ)/5 5xr dz dθ dr = 0
Mxz = ∫0^1 ∫0^π/2 ∫0^(6-6r cos θ)/5 5yr dz dθ dr = 0
Mxy = ∫0^1 ∫0^π/2 ∫0^(6-6r cos θ)/5 5zr dz dθ dr = 5/4
To find the center of mass, we need to divide the moments by the mass:
x = Myz/m = 0
y = Mxz/m = 0
z = Mxy/m = 1/6
Therefore, the mass of the solid E is 15π/2, the moments about the yz-, xz-, and xy-planes are 0, and the center of mass is at (0, 0, 1/6).
(If you don’t understand or if you got another answer please comment or if you think this is wrong!)