When learning about the Fundamental Theorem of Calculus I relied mainly on deductive reasoning. Deductive reasoning assumes things to be true, and involved breaking things into parts to reason it out. So, I assumed that I have a continuous function that remains constant. Then I trust that if I plug in values for x I would be able to calculate the accumulated area. Everything is dependent on the equation and our understanding of derivatives.
This is such and important theorem because it explains the relationship between an integral and a derivative. It shows how are is dependent on x, so the function is reliant on x. This theorem basically explains why f(x) is f(x).