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Find the mass, the moments, and the center of mass of the solid E with the given density function p.

E is bounded by the parabolic cylinder z = 1 - y2 and the planes x + 62 = 6, x = 0, and z = 0; p(x, y, z) = 5

m =

Myz =

Mxz =

Mху =

(x, y, z) =

User Duygu
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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!)
User Abhinandan Sahgal
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