You might be talking of two different things. Agrarian spoke of "surface with irregularly curved edges". You are talking of integrating "proper expressions". If no formula for the edge is known, calculus is no help, you are back to counting squares in a grid, like the first man.
In either case, I am talking about cardinal necessity of faith in any reasoning about the future, not about issues with precision. Newton (or Ptolemy, or Einstein) depend on faith not because they miss finer points of celestial mechanics, but because, like Aquinas said, in the Christian belief system the whole rig is going to poof, stop, while in most scientific belief systems it keeps turning forever.
You can define any surface with a mathematical formula.
I am talking about cardinal necessity of faith in any reasoning about the future, not about issues with precision...in the Christian belief system the whole rig is going to poof, stop, while in most scientific belief systems it keeps turning forever
What you call "faith" in any reasoning about the future -- it is not blind faith as you have faith in God; it is an expression of calculated confidence based on observed phenomena.
As for the scientific "belief" that the systems will keep on turning forever, that is incorrect. The Big Bang theory certainly predicts that "the whole rig is going to poof, stop," but the reason is different from that given by +Aquinas.
The Church simply surmised a spiritual truth that all that is created ans rendered corrupt with our fall must itself fall (all that has a beginning has an end). The scientists have evidence that a cataclysmic set of events will destroy not only our earth but the whole solar system, and billions of galaxies, each containing billions of stars; that all of this is going to collapse onto itself only to be resurrected from its own ashes.
I do not think that the principle is any different between questions of precision or of reasoning about the future, although the implications for Christianity vis a vis science are obviously much greater for the latter.
My point was that mathematics, while useful, is even in many theoretical situations exact only by being self-referential. And it is always fundamentally rooted in axioms, not to mention unprovable and unmeasurable concepts such as infinity.