Definite and indefinite integrals

I find the -indefinite- integral conceptually easier:
the derivative of a function is an operator $D$ on functions;
(careful: it's defined on everywhere differentiable functions;
 if you want to define it on all functions, it only yields a
 -partial- function: $Df(a)$ is not defined if $f$ is not
 differentiable at $a$.
 A more useful generalization is the derivative as a distribution)
The indefinite integral $I$ is a section of this.
  (technically it's a section qua -affine- map, not linear)
  (note that it's well-defined up to constant b/c of the key lemma:
   if $f' \equiv 0$, then $f$ is a constant)
That is, we can define an indefinite integral pointwise,
or we can do it uniformly (linearly, not just affinely)
  via $[I_a f](x) := \int_a^x f$.

Note that if $g$ is a derivative of an everywhere differentiable
function, then it (tautologically) has an indefinite integral, and
this is well-defined up to constant.

Indefinite integrals are pure algebra -- that's why they're formal and
easy (esp. for students).

Note that indefinite integrals of -functions- only make sense in 1
dimension, which is why they become less valued in higher dimensions
(or rather, why people use Fubini so much: to reduce to 1 dimension).
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Definite integrals are trickier b/c you actually need to do analysis.
[and thus are neglected in intro calc classes]
Further, there are many different constructions.

However, they are The Right Way to think about many things,
especially integrals in higher dimensions (where you don't have a
naive Fubini) or on manifolds or other spaces.

Definite integrals are applicable in a rather wider context, b/c you
can integrate functions that aren't derivatives: integrable functions
are very broad.

The Darboux (-> Lebesgue) POV is very elegant and rigorous, and is
closely related to the idea of measure (or content):
measurable set and integrable function (in the sense of Darboux) are
very similar.

The Riemann notion is much vaguer: it doesn't give bounds, just ``an
approximation''.

A definite integral is a -pairing-
$P \times F \to V$:
given a (nice) subset of $X$ and a function from $X$ to $V$ (some
banach space), you get an evaluation.

This is much more sophisticated than the indefinite integral, and we
don't see derivatives anywhere...

Note that if we can take definite integrals over intervals (which we
can), we can get a sort of indefinite integral -- and it'll agree
where-ever the function downstairs is continuous.
(this theory also leads to integrating measures, distributions, etc.)

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What's the connection between indefinite and definite integrals?
1: definite integrals give a way to realize indefinite integrals:
 $F(x) := \int_a^x f$
2: indefinite integrals let you compute definite integrals
 $\int_a^b f = F(b) - F(a)$
 Note: only definite integrals OVER INTERVALS.
 Over other subsets you can't (naively) use indefinite integrals.
