This notation for the area is only slightly different from that used already
for the antiderivative. It is used because the two concepts are very closely
related. In fact the "Fundamental Theorem of Calculus" states that
the area in the region described above as afunction of b is an antiderivative
of the integrand f. We will discuss this theorem in the next lecture.

We will now discuss something that at first glance is silly: the formal definition
of "the area under a curve". We all have good _{ }intuitive
ideas of what area is; and this discussion seems pointless. Our purpose in introducing
it is to give us a way to determine _{ }which functions have well defined
integrals and which do not. You can get into trouble if you try to integrate
a function whose _{ }integral does not exist. Even more importantly,
this definition gives us a way to generalize the concept, for example to the
case

Example