There are people here who know stuff about math, right? Could I ask someone to help me understand something I'm just not getting? I posted a question (https://math.stackexchange.com/questions/3349594/are-the-definite-and-indefinite-integrals-actually-two-different-things-where-i) on stackexchange but all that's resulted in is conversations in the comments going in circles.
@trwnh Now I have more questions, because isn't a function itself a... "prime object" i guess, i don't know the right word for it, the same class of objects as values
@trwnh but what i'm not getting is that the "sin" of "sin(π)" is still the exact same "sin" as "sin(x)". you don't call it "the definite sine"
@trwnh So how is the situation with the definite and indefinite integrals any different, though?
@trwnh And that's the question I'm trying to get answered. Why do people seem set on considering the definite integral something intrinsically different from the indefinite integral?
@trwnh The whole impetus for me asking that question was in response to this other question (https://matheducators.stackexchange.com/questions/17051/a-different-symbol-for-the-indefinite-integral-antiderivative) that someone seems very intent on making absolutely certain that students understand the two to be distinct things
@trwnh i'm not sure what you're meaning to get at here? They give you two different results but are the same function
@trwnh or meta-function as the case may be. there's probably a proper math term for that. was it functor or something?
@Felthry I'm also an engineer but "Essentially, the definite integral is what you get from running the numbers on the result of an indefinite integral" seems close to me? The trickiness comes when the function you're integrating is discontinuous, because effectively the values of the indefinite integral on either side of the discontinuity get unhooked. I think. I'm not a mathematician.
@CoronaCoreanici @starkatt Ah I should have posted a follow-up, someone ended up posting a good answer on stackexchange! https://math.stackexchange.com/a/3349699/560186
Turns out the confusion was just in terminology--the distinction that was being made was between the integral and the antiderivative, but they were using "indefinite integral" to mean "antiderivative".
@CoronaCoreanici @starkatt if you want to, go right ahead!
thank you for attempting to explain this, everyone who has done so. my heads hurt now and i don't know if i understand things any better or worse than i did initially but maybe we should take a break from this and re-read what people have said later