About the presentation: I think I agree: once I'm up to the level of discussing Lagrangians and stationary action I should not re-teach integration; the reader will be familiar with that.
That particular presentation grew over time; I agree it is uneven. I need to scrap a lot of it.
The preceding article http://cleonis.nl/physics/phys256/calculus_variations.php Is more an overarching concept.
Also, I'm active on the stackexchange physics forum. Over the years: Hamilton's stationary action is a recurring question subject. Some weeks ago I went back to the first time a stationary action question was posted, submitting an answer. In that answer: I aimed to work the exposition down to a minimum, presenting a continuous arch. https://physics.stackexchange.com/a/821469/17198
three sections:
1. Work-Energy theorem
2. The central equation of the work 'Mécanique Analytique' by Joseph Louis Lagrange (I discuss _why_ that equation obtains.)
3. Hamilton's stationary action
It's a tricky situation. I'm not assuming the thing I present derivation of, but I can see how it may appear that way.