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).

> 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)

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

If you think of it as being an accumulator function it can feel a bit more natural - the _definition_ of this accumulator is that, F(x) is the area from 0 to x

The fact that the derivative of this accumulator function is equal to the original function, this is the fundamental theorem of calculus, and I violently agree with you that this part is shockingly, unexpectedly beautiful

If I tell you I have function f with f(a) = 10 and on it's path from a to b, the graph first increaes by 5 units then by another 10, and then later on drops by 25 units, you can immediately deduce that f(b) = f(a) + (+5 +10 -25) = 0. The fundamental theorem of calculus uses the same concept:

To see why \int_a^b f(x) dx = F(b) - F(a) with F'(x) = f(x),

we replace f with f' (and hence F with f) and get

\int_a^b f'(x) dx = f(b) - f(a).

Re-arranging terms, we get

f(b) = f(a) + \int_a^b f'(x) dx.

The last line just says: The value of function f at point b is is the value at point a plus the sum of all the infinitely many changes the function goes through on its path from a to b.

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.

I have internalised that in mathematics nice things come in bouquets. If there is a thing defined with properties A, B, C, and there is an other thing defined with properties D, E, F, then chances are that those 2 things are the same thing, because there are only so few nice concepts.

There are many types of examples, and many different reasons why I don't find a particular connection or connection type surprising. So I can concentrate on memorising them, and building intuition.

For the Fundamental Theorem of Calculus:

  - int f' = f: the sum of the change is the thing itself. E.g. pour water in the bathtub, if you sum the rate you pour, that's the total water in the bathtub
  - int f' = f(t2) - f(t1) : same but water differences between 2 times.
  - (int f)' = f: the rate of the sum is the function itself. If you go and integrate your function f, the integrate function's change rate at x is f(x) 
  - and so on. 

> the area under an entire curve being related to the derivative at only two points

This is a very wrong sentence. The area under f on [a,b] is not related to the derivative of f at a and b. The area under f on [0,x] is a real function F(x) by definition, and there is nothing surprising that the area of f on [a,b] is F(b)-F(a). Simple interval arithmetic.

Now F, the sum, is related to f: F' = f.

tl;dr : in the "fundamental theorem of calculus" there are 2 main observations:

  - the operators 'summing' and 'change rate' are each other's inverse and commute: 
  
       (int f)' = int (f') = f
                              
       F' = f  <==>  int f = F
  
  - from interval arithmetics:
  
       S(a,b) = S(0,b) - S(0,a)

You meant antiderivative?

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