How Howitzers Howitzer: A Breakdown.

that's some interesting analysis, thanks a lot for sharing!
as hannibal already pointed out we've both been working on very similar scatter and AoE-related projects lately, and i think your approach is a good way to do it.

after reading over your methodology, i'd suggest a few edits to your formulae to get a bit more accurate results though:

i think the approximation you used for the double integral under the AoE curve is quite crude and calculating the precise "volume" is actually not much more complicated than what you've already done.
instead of approximating the area under the curve with rectangles and integrating to get the volumes of the resulting cylinders, you can split the complete area into right trapezoids and calculate the precise volume for the corresponding cone sections via pi*h/3*(r^2+rR+R^2), where r and R are the inner and outer radii and h is the damage at each radius (n, m, f)

with respect to the scatter cone, there is one important varable that you may have missed from what i've read: distance_scatter_max.
this will cap the horizontal scatter you get from distance x dynamic_scatter_ratio (this even caps the FoW multiplier!) and lead to smaller actual scatter areas, especially at max range.

apart from this i think your method works great to get a somewhat reliable impression for raw or "on paper" performance of arty pieces and a handy gauge for comparison. however, relating this to in-game performance is a bit tricky, due to the sheer volume of additional variables factoring in here, as pointed out by others already.
i've been working on a simulation to get actual squad sizes and formations into the equation for quite some time, which i plan to release soon(tm). with a few modifications this would probably work well for artillery and indirect fire, too.

Crunched the numbers. Its a LOT of extra damage. The Priest and Sexton get 22.
4% MORE damage concentration. At Max range (WITHOUT vet 3), the Priest gets a 31.2% scatter region premium. Jesus. Vet makes this all even nastier.

From this, the Sexton AND priest beat out the base-line leIF (AND ANY OTHER ARTILLERY) when fighting at 160 range into FoW. Even the Super Charged rounds are better. And the Priest just bodies it (13% more dps in area). Counter Barrage is still solidly the best artillery in the game, but dang, that max scatter is naaaaaasty.

Super Charged barrages are mildly more effective than leIF and ML/20 into FoW, but worse against spotted. Due to how Super Charged Valentine works, mostly will be in vision (mostly...).

What is the point of this? the Max Distance Scatter makes the mobile artillery precision FoW snipers instead of close support. Mobile howitzers are Back Breakers, not close support (compared to fixed howis).

Jives with my experience. I can snipe enemy rocket artillery reliably with a Sexton at max range (my primary use for it, when not shelling forward bases).

So... Balance Team. What do you want artillery to do?