[Tainnor]

As to your first point: In the sense that it's useful to say "every integrable function f has some antiderivative F so that you may compute the integral by computing F at the endpoints", yes, your definition can be useful. On the other hand, it's also an important question to consider "which functions can be derivatives?" and in that sense, the definition is less useful. But definitions are definitions; the most we could objectively argue about is which one is the more standard one.

> Here again I'd argue over semantics, though I'd concede it's much more a matter of personal preference than my argument above, which I think has mathematical weight behind it. Namely, I'd argue that the numerical integration is doing something directly with the non-elementary anti-derivative, namely, evaluating it at a point—just like we call reading off the value of, say, the sine of an angle from our calculator doing something directly with the sine, even though what we're really doing is summing sufficiently many terms in a Taylor-series approximation.

Fundamentally, at a mathematical level, yes. That's what it means for two definitions to be equivalent. But on an algorithmic level, the process of evaluating an integral numerically and the process of finding an antiderivative (especially symbolically) are quite different things.

But in the end, it doesn't seem that we fundamentally disagree.