Perhaps linking integrals/derivatives into stats could be done from a more discrete approach as opposed to a continuous one first.
For example: Starting with a representation of an integral as sum of areas of rectangles (something most students can do) and then saying what happens if the width of all the rectangles gets smaller and smaller. A la Riemann
This could then tie into the pdf quite reasonably?
When we got to continuous distributions, we kind of hit a wall. Here's the puzzle. Suppose the data are recorded to two decimal places, and are reasonably bounded, for instance they are between 0 and 100. That's a discrete distribution. All of the data in the examples and problem sets were fixed-point.
I literally couldn't justify to someone who had never taken calculus, why a wholly different kind of distributions with its own collection of formulas, was necessary or useful. The teacher had never explained it either.
Instead, to deal with this 'a lot of really small things' we need a slightly different approach. Then draw a pdf, and talk about area under the curve. Perhaps relate this to the discrete case where the curve is just really blocky.
Vs
Trigonometric substitute to solve analytically.
For example.
These days you can solve many practical statistical problems with Monte Carlo, but understanding why those curves look the way they do is still very useful.