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In my multi-variable calculus class we have been learning how to calculate double integrals over regions. To accomplish this task we usually make use of Fubini's Theorem because it relates the double integral to iterated integrals.

However, upon some research, I found out that Fubini's Theorem is a rather recent result. Wikipedia states that it was formulated in 1907. I was under the impression that calculus was formulated in completion at a much earlier date.

So were double integrals over a region even calculated before? If so, how was it accomplished?

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  • $\begingroup$ Personally, I think it is a bad idea to use the term "Fubini's Theorem" for these facts from calculus. Reserve the name for calculation of multiple Lebesgue integrals or others from measure theory. $\endgroup$ – Gerald Edgar Apr 13 '17 at 21:50
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Double integrals were calculated in the 18th and 19th centuries in the same way they teach you in calculus now, using special cases of what later became Fubini's theorem. Most of these special cases had no special name.

Fubini's theorem is just a name in honor of a person who proved a much more general statement than that which is taught in calculus.

Actually double and triple integrals where calculated even before the notion of integral was formalized, since antiquity, by people like Archimedes and Eudoxus, but they had to invent a new argument for each particular integral. (They essentially approximated integrals by finite sums and then tried to find the limit. The difficulty is in finding the limit explicitly.)

In the 17th century, Cavalieri's Principle (see Wikipedia) was formulated which helps in evaluating some multiple integrals, especially for areas and volumes. Cavalieri principle was still mentioned in high school in 1960s. But modern calculus books prefer to refer to a very general Fubini theorem.

If you wish to see how triple integrals were evaluated before the notion of integral was formalized (I mean before Newton and Leibniz and Cauchy), I recommend to read the remarkable little book by Kepler, Stereometry of wine barrels. I am not sure whether there is an English translation, but there is a detailed exposition in English: http://www.maa.org/press/periodicals/convergence/kepler-the-volume-of-a-wine-barrel-keplers-inova-stereometria-doliorum-vinariorumi

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