What is the difference between an antiderivative and an integral?
It depends on a couple of things. Which antiderivative, the general or a particular? which integral definite or indefinite? And, who are we asking?
General Antiderivative and Indefinite Integral:
Many mathematicians do not distinguish the indefinite integral and the general antiderivative. In either case for function
Some (for instance, textbook author James Stewart) make a distinction. What Stewart refers to as "the most general" antiderivative of
The indefinite integral of
A particular antiderivative of
is a particular antidervative of
is a different particular antidervative of
The definite integral of
(To further complicate matters, this definite integral can be found, using the Fundamental Theorem of Calculus, Part 2, by finding the/an indefinite integral / general antiderivative first, then doing somearithmetic.)
Your question is related to what was truly the "key insight" in the development of calculus by Isaac Newton and Gottfried Leibniz.
Focusing on functions that are never negative, this insight can be phrased as: "Antiderivatives can be used to find areas (integrals) and areas (integrals) can be used to define antiderivatives". This is the essence of the Fundamental Theorem of Calculus.
Without worrying about Riemann sums (after all, Bernhard Riemann lived almost 200 years after Newton and Leibniz anyway) and taking the notion of area as an intuitive (undefined) concept, for a continuous non-negative function
Conversely, if we make the upper limit of the integral symbol a variable, call it
In the case where