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The word integral in mathematics is a function that expresses the area under the graph of its derivative.

In English integral can mean necessary or essential.

Do either of these terms derive from the other? They do not seem to be related but perhaps there is a historical similarity?

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    A mathematical integral "integrates" the area contained within a curve into a single "complete" number.
    – Hot Licks
    Oct 4 '16 at 2:57
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    That'd be a definite integral which is of a portion of a function.
    – Aequitas
    Oct 4 '16 at 3:01
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    Your question has a definitive answer here: math.stackexchange.com/questions/808285/…
    – deadrat
    Oct 4 '16 at 3:07
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    Incorrect statement. The word integral in mathematics is a function that expresses the area under the graph of it's derivative. Under the graph (not derivative). Derivative is opposite of integral.
    – paparazzo
    Oct 4 '16 at 6:44
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    @EricLippert - That's easy -- they're both painful to deal with.
    – Hot Licks
    Oct 4 '16 at 22:48
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The sense of the English word "integral" that relates to the mathematical usage is the adjectival meaning "whole". From Wiktionary:

Constituting a whole together with other parts or factors; not omittable or removable

Something not omittable or removable can easily be seen to be necessary or essential.

The mathematical notions of "integers" and "integrals" relate to this meaning. Another term for integers is "whole numbers", in the sense that they have no fractional part. They don't have pieces--they are whole.

Integration in math is the process of summing up (in the limit) small pieces or slices of a shape (represented as the area under a function curve) into a whole area. In other words, from the pieces a whole is made.

The mathematical usage derives from the original meaning of "whole".

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