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
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: