Integral Triple

[white wadi]

Is the first pic the hyperboloid function?

[inner cipher]

the first pic is the formula of the lateral surface

[white wadi]

How I always think about these integrals is by breaking it down into steps. Starting with the simplest

You want to integrate for mass, so it's literally just $M=\int dm$. Then we need to know the bounds of $m$. The bounds are spatial ($dV$). So how do we related mass and volume? That's where density comes in. We know the density is constant in the hyperboloid cylinder, and 0 outside. Let the density constant be $\rho$. Then $dm=\rho dV$.

[placid cosmos]

SWR

[white wadi]

So the integral becomes $M=\iiint \rho dV$, bounded by the hyperboloid.

[placid cosmos]

SWR

[white wadi]

You are asked to integrate in cylindrical coordinates, which makes sense given the nice symmetry

Do you remember $dV$ in cylindrical coordinates?

[placid cosmos]

SWR

[inner cipher]

yes

we got a progress in a help channel

got this

we got this*

is it right?

in case it is

im having trouble now calculating the moment of inertia respect to the axis y

[white wadi]

the t and z bounds are correct. I need a second to verify the r bounds

yeah the bounds are all correct