Why are integrals called integrals? What is the historical background for this term? 
I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve.  It seems to me a strange choice of word.

Can be useful to trace back the usage of the word integral.
From Florian Cajori, A history of mathematical notations (1928), page 181 of II vol of Dover reprint :

At one time Leibniz and Johann Bernoulli discussed in their letters both the name and the principal symbol of the integral calculus. Leibniz favored the name calculus summatorius
  and the long letter $\int$ as the symbol. Bernoulli favored the name calculus integralis and the capital letter $I$ as the sign of integration.
The word "integral" had been used in print first by Jakob Bernoulli [footnote : Jakob Bernoulli in Acta eruditorum (1690), p. 218], although Johann claimed for himself the introduction of the term. Leibniz and Johann Bernoulli finally reached a happy compromise,
  adopting Bernoulli's name "integral calculus," and Leibniz' symbol of integration.

