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