I have a colleague in the English dept. who is wondering the reason why the word "integral" came to be used to represent the process by which the area under a curve can be found.
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The following etymological notes, from Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, may shed some light on the choice of names.
INTEGRAL CALCULUS. Leibniz originally used the term calculus summatorius (the calculus of summation) in 1684 and 1686.
Johann Bernoulli introduced the term integral calculus.
Cajori (vol. 2, p. 181-182) says:
According to Smith (vol. 2, page 696), Leibniz in 1696 adopted the term calculus integralis, already suggested by Jacques Bernoulli in 1690.
According to Stein and Barcellos (page 311), the term integral calculus is due to Leibniz.
The term "integral calculus" was used by Leo Tolstoy in Anna Karenina, in which a character says, "If they'd told me at college that other people would have understood the integral calculus, and I didn't, then ambition would have come in."
INTEGRABLE is found in English in 1727-41 in Chambers' Cyclopaedia (OED2).
Integrable is also found in 1734 in An Examination of Dr. Burnet’s Theory of the Earth by John Keill and Maupertuis. [Google print search]
The word INTEGRAL first appeared in print by Jacob Bernoulli (1654-1705) in May 1690 in Acta eruditorum, page 218. He wrote, "Ergo et horum Integralia aequantur" (Cajori vol. 2, page 182; Ball). According to the DSB this represents the first use of integral "in its present mathematical sense."
However, Jean I Bernoulli (1667-1748) also claimed to have introduced the term. According to Smith (vol. I, page 430), "the use of the term 'integral' in its technical sense in the calculus" is due to him.
The following terms to classify solutions of nonlinear first order equations are due to Lagrange: complete solution or complete integral, general integral, particular case of the general integral, and singular integral (Kline, page 532).
In mathematics, an integral need not represent the area under the curve of a graph of the function. Also, integral might refer to integers also. For example, integral solutions of an equation means solutions can only be integers. I think mathematicians adopted this word because "to integrate" means to combine things so that they become a whole. This is almost a synonym of "to sum", and integration is practically like a sum, hence the symbol $\int $, which looks like the letter S -- for sum. Overall, the reason why "integral" might mean "area under the curve" and "integration" is the process of finding this area, is merely because it is the most obvious application.