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.

"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.
