Find Moment of Inertia: Z-Axis

[MeMoses]

Homework Statement

Find the moment of inertia of the regoin bounded by z = x^2 + y^2 and z=1 if the density is propertial to the distance from the z axis

Homework Equations

Moment of interia about the z axis = the double integral R of (x^2+y^2)*(density)dV

The Attempt at a Solution

I convert everything to cylindricals, so I get z=r^2 and z=1, but I'm not sure about setting up the integral. What exactly is region R and would the density just be == r? Anyways x^2 + y^2 = r^2, so would the integral be
integral of r^2 * r * r * dθ dr, 0≤θ≤2π, 0≤r≤1
Basically I am just unsure what region R is, right now the limits are just the area bounded by the plane and parabloid but what about the volume underneath? Also why do the moments of inertia about the x and y-axis involve triple integrals and z doesn't or are my notes wrong? Any help is appreciated.

 

[mighty2000]

Hello there!

Firstly, great job converting everything to cylindrical coordinates. That is definitely the way to go for this problem.

To answer your question, the region R in this case would be the volume bounded by the plane z=1 and the paraboloid z=x^2+y^2. This means that the limits of integration for r would be from 0 to 1, as you have correctly identified. However, for θ, the limits would be from 0 to 2π, as this is the full circumference of the circle in the x-y plane.

As for the density, you are correct that it would just be equal to r, as the problem states that the density is proportional to the distance from the z axis.

In terms of the moment of inertia about the x and y axis involving triple integrals, this is because these axes are not the principal axes for this region. The z axis is the only principal axis, so the moment of inertia about this axis can be calculated using a double integral. However, for the x and y axes, the moment of inertia must be calculated using a triple integral because these axes are not aligned with the principal axes of the region.

I hope this helps! Let me know if you have any other questions. Good luck with your calculations!