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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.

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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.

Generally, banks try to get as much of the interest early on, and move the base to later. This is why it usually makes no sense to pay off a debt after 5 or 10 years when you paid 80% of the interest already (but also why refinancing works for banks: more upfront interest which they put into books as profit right away).

On some things, governments put the cap on how much can early payments be loaded with interest and how much at least should be paying off the base debt — otherwise, you'd be paying only the interest for the first 15 years of a mortgage.

Though, these things work slightly differently in different countries.

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