Fundamental theorem of calculus (idea)

The Fundamental Theorem is "fundamental" because it links the two distinct branches of calculus, derivatives and integrals. As you can see above, it is split into two parts.

• Part I says that you can find the definite integral or (sometimes) area under a curve by evaluating the antiderivative of the function at both edges, and finding the difference.
  Before calculus, areas under curves could be evaluated only through approximation (what we would call Riemann Sums today). Using the FTC on area problems usually results in a huge improvement in speed (and accuracy, because you get an exact answer).
• Part II says that, to take the derivative with respect to x of an integral where the end limit of integration is a function of x, you "take it-stick it-D it". (See below.)

Here's what I mean by "take it-stick it-D it". Look at this example, first:

     /\ u(x)
  d  |
 --  |  f(t) dt 
 dx  |
    \/ a

Taking-sticking-D-ing is a phrase that was "patented" by my high school math teacher, Mr. Noeth. In this example, t is the variable of integration. To T-S-D, you take that end limit ("take it"), replace the variable of integration with it ("stick it"), and take the derivative WRT x of it ("D it"). So you have:

          du
f(u(x)) * --
          dx

Note that you can use any variables, not just x and t.