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.
 A: "I cannot quite see what is *integral* about an integral"

From your statement above, it appears you are thinking of an alternate meaning of the word "integral." Specifically, A is integral to B if it is a necessary component of B (e.g., "this scene is integral to the plot").
But that is not how it is used in mathematics. Think instead of integration in society (as contrasted with segregation) - to bring pieces together into a whole. An integral is a mathematical instance of integration in this sense.
A: It seems to be taken from the Latin word integratus taken from here (etymonline).

1630s, "to render (something) whole," from Latin integratus or integrare, past participle of integrare "make whole," from integer "whole" (see integer). Meaning "to put together parts or elements and combine them into a whole" is from 1802.

A: 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.

A: The derivative represents the momentary variation of a function, whereas the integral yields the sum-total of all these $($small$)$ momentary variations put together, giving the value of the entire variation, over the whole amount of time.
A: The usual meaning of integral in non-mathematical contexts is that an integral is a whole.  It is a sum of infinitely many infinitely small parts that make up a whole.  And that is exactly what something like $\displaystyle\int_a^b f(x)\,dx$ is.
A: I too have long noticed that many terms in mathematics are not suggestive of what they refer to. Consider for example the term "experiment" in classical probability theory. We all know what an "experiment" is and it is most certainly not an instance of the action of a random mechanism. Or the word "probability" itself which originally meant "relative frequency" and then was extended to mean any belief, supposition, guess, or arbitrary value in so-called "subjective probability" which is a very peculiar metaphor. Consider the suggestiveness of such terms as "fraction," "number," "real," "complex," "imaginary," "algebra," "derivative," and - as the comedian said - "Away we go!"
