The fundamental theorem of calculus
Derivatives measure rates and integrals add up amounts. Why should these two separate ideas turn out to undo each other?
Where you are 2 inside
What this area is about
The definite integral is a limit of sums, which can take a page to compute. The fundamental theorem of calculus links it to the derivative, so that integrals can be found from antiderivatives in one line. It has two halves, and together they make differential and integral calculus one subject.
How these fit together
Both children are results, each a theorem with the procedures that follow from it.
The first theorem. The first fundamental theorem of calculus says that if f is continuous on an interval I and a is in I, then A(x) = \int_a^x f(t)\,dt has A'(x) = f(x) at every interior point x of I. It also attaches the chain rule when the upper limit is itself a function of x.
The second theorem. The second fundamental theorem of calculus says that if f is continuous on [a, b] and F is an antiderivative of f there, then \int_a^b f(x)\,dx = F(b) - F(a). This is how nearly every exact integral is found.
A path through it
The first theorem comes first: it shows that every continuous function has an antiderivative, and the second theorem is proved from it.
Next come accumulation and net change, substitution, areas between curves and improper integrals.