Never thought about it before, but it's mildly amusing.

http://www.etymonline.com/index.php?term=integral

Integral comes from the Latin meaning "forming a whole."

Integer is from the Latin "integer" which means "whole," BUT the "whole numbers" are not the same as the Integers. Rather, they are a subset of the Integers.

N = Set of Natural numbers = {1, 2, 3, ... }
W = Whole numbers = Natural #s plus 0 = {0, 1, 2, 3, ... }
I = Integers = { ..., -3, -2, -1, 0, 1, 2, 3, ... }

Some books use "J" instead of "I" to represent the Integers. Usually, books use a typeface called "blackboard bold" (just looked it up) with letters that represent sets. I note without evidence or explanation that the meanings of the terms "whole number" and "natural number" may have changed in the last 40 or 50 years.