I always thought integrals and anti derivatives were the same thing but I recently read that they are separate things that can be related by the Fundamental Theorem of Calculus. How does that work? Tried reading a couple explanations but they didn’t dumb it down enough for me I think.
In: Mathematics
Integrals measure areas and volumes, almost by definition. Often the one under a graph, but it can actually be something more general.
Antiderivatives try to find a function F that has a given function f as its rate of change. Accumulating the value of f as the change of F.
Those two concepts turn out to be almost the same: accumulating the change proposed by f is essentially the same as finding the are under the graph of f. After all, both methods ultimately sum up values of f, scaled to an infinitely small step size. This argument when put into formality is actually a proof of the Fundamental Theorem of Calculus.
However, it should be noted that integration _is_ really more general. There are functions and shapes where this notion of volume still makes sense, but “rate of change” does not.
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