I'm in a Real Analysis course at my school right now, and I've only just been introduced to integrals other then the Riemann integral. Some integrals from what I can tell seem to be much more generic, and have very little to do with what my previous notion of an integral was (the inverse of a derivative or the area under a curve).
So my question is what actually makes something an integral? Must it have the Riemann integral as a special case, or are the all in some loose conceptual way related to the area under a curve, or the inverse of a derivative, or am I completely missing the point?
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$\begingroup$This is a very good, astute question. To paraphrase, and pointedly so, the question can be "what do we want an integral to do?" Yes, for example, as in the question, what should an extended notion of integral be compatible with? Certainly all elementary things, e.g., fundamental theorem of calculus, areas under curves, and so on. But/and how can we reasonably capture all this in a rigorous way?
One way, and not the only, and not the only thing we'd want, is a sort of continuity on continuous (scalar-valued, for example) functions of compact support. An easy version of the Riemann integral shows how to integrate such functions very well, compatibly with the fundamental theorem of calculus (and, by design, with area-under-curve computations).
It is not entirely trivial to give the space of continuous, compactly-supported (scalar-valued) functions on a reasonable space (e.g., $\mathbb R$) the correct topology... but if we do so, then the Riesz-Markov-Kakutani theorem says that any continuous linear functional on that space is given by "an integral (against a positive, regular, Borel measure". Good!
And ... uniquely so...
It is certainly true, though, that this characterization does not directly address issues about differentiation in a parameter under the integral, and so on. I would claim that such issues are best addressed thinking in terms of Gelfand's and Pettis' ideas from 1930s, about "weak integrals". Not elementary, but really on-target.
If you have follow-up questions, please do. I've thought about such issues for quite a while now, as they do play a significant role in much of mathematics, even while being dubiously dismissed as "folkloric" or "it's just the definition".
$\endgroup$ 2 $\begingroup$Most integrals seek to "find the area under the curve". The Riemann integral does this, over an interval $[a,b]$, by chopping up the interval $[a,b]$ into consecutive pieces $a = x_0 < x_1 < \dots < x_{n-1} < x_n = b$ and approximating the function below and above on each of the pieces $[x_{i-1},x_i]$. The Riemann integral of a function is the answer you get from approximating below and above (if you don't get the same answer, we say the function is not Riemann integrable).
The Lebesgue integral is a kind of generalization of the Riemann integral in which case we approximate on arbitrary (measurable) subsets of $[a,b]$ - not necessarily consecutive subintervals like we do in the Riemann integral case. And if we can find a family subsets on which we approximate that yield closer and closer upper and lower approximations, we call the common approximation the Lebesgue integral. It turns out that a necessary and sufficient condition to be able to find subsets that yield good lower and upper approximations is measurability of the function in question.
A different type of integral, if you want to call it an integral, is a "path integral". These are actually defined by a "normal" integral (such as a Riemann integral), but path integrals do not seek to find the area under a curve. I think of them as finding a weighted, total displacement along a curve. For example, if you path integrate the function 1 along a circle, you get 0. But if you path integrate a function along a circle, but that function takes higher values only at the beginning of the path, you won't get 0 but rather something closer to the high values of the function.
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