Vtyjkyuk

Given $r_b$, $h_b$ and $h_t$, what would be the equation for calculating the volume of used space in a bell shaped container. Example sketch

Example sketch

What I have so far:

The shape can be separated into 2 peaces, with the first one being a cylinder for which the volume calculation is easy as
$$V_b=pi *r^2*h_b$$
The second peace is in a shape of a spherical segment for which the volume equation goes as follows
$$V_t=frac16pi h_t(3r_b^2+3r_t^2+h_t^2)$$
where $r_t$ is the radius of the topmost circle. Given that we only have the height $h_t$, we need to calculate $r_t$ using equation
$$r_t=sqrtr_b^2-h_t^2$$
If we input that into the previous equation we get
$$V_t=frac16pi h_t(3r_b^2+3sqrtr_b^2-h_t^2^2+h_t^2)$$
Now we can reduce the equation to get
$$V_t=frac16pi h_t(6r_b^2-2h_t^2)$$
and if we combine both equations we get
$$V_bt=pi r_b^2h_b + frac16pi h_t(6r_b^2-2h_t^2)$$
If we reduce it even further we get the final equation
$$V_bt=pi (r_b^2h_b+h_tr_b^2-frach_t^33)$$

Is this correct? I don't know how to verify it without getting a bucket of water and measuring the volume of a model. Is there a better way of calculating the volume?

You are correct.

Note using a little fact about napkin rings having the same volume at same height can greatly simplify this problem.

https://www.youtube.com/watch?v=J51ncHP_BrY

https://en.wikipedia.org/wiki/Napkin_ring_problem

The volume is therefore the two cylinders of radius $r_b$ and (by pythagoras) $(r_b^2 - h_t^2)^1/2$ plus half the sphere of radius $h_t$