> there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative

the discrete version is much clearer to me. Suppose you have a function f(n) defined at integer positions n. Its "derivative" is just the difference of consecutive values

     f'(n) = f(n+1) - f(n)

Then the fundamental theorem is just a telescopic sum:

     f(b) - f(a) = \sum_a^b f'(n)

It is not determined by the derivative, it's the antiderivative, as someone else mentioned. The derivative is the rate of change of a function. The "area under a curve" of the graph of a function measures how much the function is "accumulating", which is intuitively a sum of rates of change (taken to an infinitesimal limit).

The antiderivative at x is defined as the area under the curve from 0 to x, which the Riemann sum gives a nice intuition for how you can get from the derivative.

So to get the area under the curve between a and b, you calculate the area under the curve from 0 to b (antiderivative at b) and subtract the area under the curve from 0 to a (antiderivative at a).

At least that's my sleep deprived take.

There is some geometric intuition in wikipedia page for this theorem you may like :)

You meant antiderivative?