Consider a function f continuous on a compact interval.


Approximate it by a sequence of polygonal functions (you can).


Then consider a sequence of primitives of the polygonal functions (you can).


At last consider the limit of the latter sequence (you can).


Now you have found a primitive of f (you know) *without integration*.


This is the content of the first part of a not very known note by Lebesgue [*Remarques sur la définition de l'intégrale*](http://gallica.bnf.fr/ark:/12148/bpt6k4862594/f234.image.langEN), Bull.Sci.Math. 29 (1905) 272-275 (see [pdf](http://arxiv.org/pdf/1212.1412v1.pdf) for an exposition in English).


I doubt that such a thing was shown for the first time in 1905.


Lebesgue's good faith is beyond discussion of course.


Do you know something about ?



Newton integration without integration
1. What does this procedure have to do with Newton? Just curious.
2. If you approximate with "polygonal" (=piecewise linear) functions in the most natural way,
that is take points $(x\_k,f(x\_k))$ and connect them with straight line segments, what you obtain
is the "trapezoid rule" for approximate evaluation of integrals.
3. Of course, the trapezoid rule is "a theorem of Adam"; already Gauss knew much more sophisticated rules.
4. Convergence of the procedure with any continuous function and any (reasonable) choice of $x\_k$
is a trivial exercise for modern students, and it was probably in 1905.
I looked at the paper, and it seems to me that in it, Lebesgue only proposes a simple way
to TEACH the integral. There is nothing really new in this paper.


