Multivariable Triple Integral - Calculus Physics/Math Problem

[aligator11]

Homework Statement

Evaluate the Moment of Inertia with respect to Oz axis of the [...] solid A: (x^2+y^2+z^2)^2<=z

Relevant Equations

Formula for moment of inertia: Iz=SSS(x^2+y^2)*constant-density*dxdydz

Cylinder Jacobian: SSS(x,y,z)*|r|dθdrdφ

Cylinder Jacobian substitution parameters:
x=rsin(θ)cos(φ)
y=rsin(θ)sin(φ)
z=rcos(θ)
Jacobian -->|r|=r^2*cos(θ)

Hello everybody.

If anyone could help me solve the calculus problem posted below, I would be greatful.

Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A

Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive something like this: r=(cos(θ))^(1/3) for the definite intergral of my radius. I'm not sure what is the next step I should take...

Thank you all the great souls which are able to help me in that matter.

Cheers!

 

[Gaussian97]

Well, first of all, do you want to integrate using cylindrical coordinates? Or using spherical coordinates? Because you say cylindrical but the variables you define are spherical, so think about this. Also, check the value you get for the Jacobian.

For the other part, I cannot help you if you don't show me what you did to get your answer, sorry.