I know calculus, and I also know that mortgage payments don't follow that formula - or, maybe they do but they're so buried in noise and extra charges and costs you'd never know it.

Bookkeeping normally computes interest based on some accounting period - days, weeks, months, or quarters. It's a sum of discrete charges, not an integral. It was usually monthly in the paper and pencil era; today it is usually daily. Look up "double declining balance" and "Rule of 73".

Financial calculators do integrate, though. "Present value" is computed by integration.