An exact derivation would be a tough math problem based on a triple integral in cylindrical coordinates. An ASME S.E. head is not a perfect ellipse, so you would need to come up with some complex formulas for your boundary conditions...
If you are just looking for close enough, I think there are good formulas for determining the volume of an ellipse based on the two radii.
When asked, I actually find it a whole lot easier to whip open solidworks and model the head. It takes less than 5mins, and I would feel a lot more comfortable with the results.
I use the 2-radius method.
That's not an exact value for a true ellipse, but then again, the head you buy is not necessarily any closer to that elliptical shape than what your calculation is.
If you model it in 3-d cad, you can't easily allow for thinning, so you still don't have an "exact" number. Just a different approximation.
Aplogies if this post appears twice...experiencing some issues over here...
I calcualte/estimate surface area of a 2:1 head using 1.084D^2, multiply by thickness to get volume of steel, then multiply by density of steel used to get weight. I use 489.6 lbs/cuft for carbon steel. This method does not capture any straight flange that may be present so figure that out using cylinder formulas and add the two together.
It's an estimate but fairly reliable. You can be more precise by using mean diameter of the head based on actual thickness.
The above link has some great info and several equations for doing this calculation (as well as other head shapes).
I usually draw a solid model and get the volume and calculate the weight from there.
Or there is this spread sheet that has many equations for calculating elliptical head info including weight;
