I saw a YouTube video that messed up all my intuition. They were calculating the integral of something like -1+x^dx which in the end he solved by taking the limit as dx->0 and he got the solution. But everyone in the comments was saying complex stuff like like “well to be rigorous x^dx can be thought of an element of C*(R,R), the exterior algebra on R”. Huh. When did dx get any different from lim x->0.
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The specific terminology in the comment you mentioned is referencing a concept called “differential forms”.
The exact theoretical explanation is kind of tricky, but the following should give you some idea what this is about: On the number line, a “differential 1-form” like 3x dx basically gives you a “scale” at every point x, kind of like on a measuring tape. For example at x=2, the form 3x dx gives you the scale 6 dx that’s six times as dense as the standard scale; at x=1 the form gives you the scale 3 dx that is only 3 times as dense as the standard scale; at x=-1 the scale -3 dx you get is thrice the usual density, and also increases in the minus direction instead of the plus direction. From this point of view, integrating 3x dx from 1 to 3 is understood as follows: you break up [1,3] into (infinitesimally) tiny segments, check the length of each segment against your scale at the corresponding point, and then add the results up.
For basic integrals over a 1D interval, this sort of conceptual abstraction is kind of overkill. But as one goes further in math, in particular into multivariable calculus, these things turn out to eventually build a beautiful theory that makes some patterns behind the Calculus results we use way more obvious. This approach to differential calculus is less commonly taught at lower levels, since even though it does reveal some patterns, it does have the downside of having to wrap your head around these differential forms that are somehow less intuitive than functions or vector fields. So since it’s not strictly needed, it is left for later, mostly for people that specialize in something that actually really needs this. The typical point where these would be encountered is if one starts working with curved spaces, where the standard approach to calculus starts becoming too cumbersome not to introduce this point of view.
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