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$$\int$$\int$r$^{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!