MHB How do you calculate the moments of inertia for a cone?

[Kaspelek]

Hi Guys,

It has been a while since my last post but it's great to be back.

I am having some trouble with part b) of this question. Don't fully understand the concept and what I'm meant to do.

Any guidance or assistance would be greatly appreciated.

Thanks in advance you legends!

 

[I like Serena]

Re: Regions and moments of intertia

Hi Guys,

It has been a while since my last post but it's great to be back.

I am having some trouble with part b) of this question. Don't fully understand the concept and what I'm meant to do.

Any guidance or assistance would be greatly appreciated.

Thanks in advance you legends!

Welcome back Kaspelek! :)

Which intersection points did you find in (a)?Let's start with the cone up to some limit $z_0$.
The cone intersect with any plane with constant $z$ as a circle with radius $z$.
Such a circle can be integrated by running $x$ from $-z$ to $+z$, and by running y from $-\sqrt{z^2-x^2}$ to $+\sqrt{z^2-x^2}$.

\begin{aligned}
\iiint_{\text{Cone}} \mu (x^2+y^2)dV
&= \iiint_{\text{Cone}} (x+z) (x^2+y^2)dV \\
&= \int_0^{z_0}\int_{-z}^{+z}\int_{-\sqrt{z^2-x^2}}^{+\sqrt{z^2-x^2}} (x+z) (x^2+y^2) dydxdz
\end{aligned}
Do you know how to calculate that?