The Rev Dodgson said:

this post from Quora:

“Why would finding the slope of something be the opposite of finding the area?
Hi Michael,

I see you have a wide range of interests (nearly 600 questions) but, apart from this question, that does not include calculus.

So, rather than some boring technical answer, here is some history that explains maybe the context.

Newton and Leibniz in the 17th century, building on ideas current at the time, realized that their idea of what we now call a derivative could be used to compute areas. The idea was that if you can compute an antiderivative you can use that to compute an area. So, in addition to describing methods of finding derivatives, they also described methods of finding antiderivatives.

Thus derivatives and antiderivatives are related to slopes and areas. So, loosely, the same mathematics that finds slopes will, if you run it backwards, find areas.

By the middle of the 19th century (that’s much later) integration theory abandoned the antiderivative version of integration theory of Newton and Leibnitz in favor of the Riemann integral.

The two theories are compatible but very different in scope. Calculus students learn the Riemann integral, not the Newton-Leibnitz integral. They learn “area” as defined by the Riemann integral, not by the Newton-Leibnitz integral.

That means that since the 1850s students can no longer claim that “area is the opposite of slope” or that “integration is the oppositive of differentiation.”

It is only in special cases that the two theories overlap where this is true. That overlap is taught now in calculus classes as “the fundamental theorem of the calculus.” It expresses the narrow conditions under which the Newton-Leibnitz integration theory connects with the newer Riemann integration theory.

Alas, students memory of the calculus dims rapidly. Many students end up thinking they were actually studying the Newton-Leibnitz integral while we were in vain teaching them the Riemann integral. So you will see (even here on Quora) the sentiment that integration is the oppositive of differentiation (i.e., slopes and areas are opposites). Yes, in special cases but not in general.

To be sure the classical 19th century integration theories and the modern 20th century theories of integration establish a strong nexus between differentiation and integration . But that relation is not at all describable as “opposites.” Well maybe as an answer on Jeopardy that could work.”

Is that the standard mathematical approach to it?

I use integration and differentiation all the time, 1000’s of times a day quite often (or at least my computer does), but I don’t follow that argument at all.

What are the cases where integration is not the opposite of differentiation, and how come I never encounter any in my work?

survivorship bias agrees with us in saying that we too aren’t familiar with the counterexamples because we never use them