Moment of Inertia using Triple Integral

[jj2443]

Homework Statement

Compute the moment of inertia around the z-axis of the solid unit box [0,1]x[0,1]x[0,1] with density given by \delta=x^{2}+y^{2}+z^{2}.

Homework Equations

I=\int\int\intr^{2} \delta dV

The Attempt at a Solution

I know that the distance r^{2} from the z-axis would be x^{2}+y^{2}. I don't know how to determine the bounds and the order of the three integrals. Could someone please explain to me how to determine which order I should integrate, and then how I go about finding the bounds of integration for each of the three integrals.

Thanks!

 

[Clever-Name]

Since none of the boundaries on your volume is dependent on each other you can just write it as a simple 'volume' integral if you want to think of it that way, as if you're finding the volume of the cube but with the other terms involved: r^{2} and {\delta}

So in my mind it should be set up as follows:

\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}(x^{2}+y^{2})(x^{2}+y^{2}+z^{2})dzdydx

The order of integration won't matter.

 

[jj2443]

Oh, that's much easier than I was trying to make it. Thank you!