In my previous blog “A Case of Pre-FTC Definite Integration“, we obtained result
without the Fundamental Theorem of Calculus.
Let’s now consider the case of where . Namely, , the area under the curve from 1 to .
Closed form result (1) is not applicable since .
Attempt of finding the limit of a sum quickly bites the dust too due to the fact that .
We see immediately that , i.e.,
Other properties of function can be extracted from (2), as shown below:
Let , we have
Assume when where , we have
Moreover, where ,
With (3-4), (3-5) and (3-6), we conclude that
We will leave this post with the following observation:
This is not difficult to see. , if , then from Fig. 2, we have
area in blue .
The fact that
area in blue
from which (4) is obtained.
When , area in blue is
From Fig. 3 we see that
from which (4) is obtained again.