Let be a continuous function on the interval .
The following theorem
is incredibly useful in mathematics, physics, biology, etc.
If is any differentiable function on such that
One reason this is amazing, is because it says that the area under the
entire curve is completely determined by the values of a (``magic'')
auxiliary function at only points. It's hard to believe. It
reduces computing (2.1.2) to finding a single function ,
which one can often do algebraically, in practice.Whether or not
one should use this theorem to evaluate an integral depends a lot on
the application at hand, of course. One can also use a partial limit via a
computer for certain applications (numerical integration).
I've always wondered exactly what the area is
under a ``hump'' of the graph of
. Let's figure it out,
But does such an always exist? The surprising answer is ``yes''.
Note that a ``nice formula'' for can be hard to find or even
The proof of Theorem 2.1.5 is somewhat complicated but is
given in complete detail in Stewart's book, and you should definitely
read and understand it.
[Sketch of Proof]
We use the definition of derivative.
is essentially constant,
(this can be made precise using
the extreme value theorem). Thus
which proves the theorem.