How were double integrals calculated before Fubini's Theorem?

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?

• 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. – Gerald Edgar Apr 13 '17 at 21:50