I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions.

I feel similar about the trace of a matrix being equal to the sum of eigenvalues.

Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

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.

You meant antiderivative?

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. Granted F, the integral function, is related to f: F' = f.

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

  - the operators 'sum of' 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)

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