178k views
1 vote
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
by
8.2k points

1 Answer

3 votes
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
by
8.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories