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The Fundamental Theorem of Calculus links the concepts of differentiation and integration together.
How did mathematicians of the past see the link between these two concepts? Integration is used to compute areas while differentiation is used to compute speeds. How did they link them together?
They didn't really "link them together". As late as the first half of the 17th century the topics that we classify under "calculus" today were treated by using separate techniques. This applies not merely to area-finding as opposed to what we call "differential calculus", but also within the latter the techniques of finding tangent lines were often different from the techniques of finding maxima and minima. When Fermat was working on problems of maxima and minima he wasn't necessarily thinking of "doing the same thing" as when he was looking for tangents to curves, though in the end he solved both by various adaptations of a basic technique called adequality. It was the generation after Fermat, including Barrow, and as some modern scholars argued also James Gregory, that began to realize the importance of the relation between what we consider to be the two "inverse" procedures.
