Lecture 16 : Integral as a limit of Riemann sums [Section 16.2]
For a integrable function , the number
can be thought of as the average value of in the interval . Theorem 16.2.6 says that the average value of a function is attained at some point if the function is continuous.
(ii)
Consider given by and .
The average value of equals . But for any . Thus, the continuity hypothesis in theorem 16.2.6 can not be dropped.
PRACTICE EXERCISES
1.
Find an interval a function , and a partition , for which given below is the
, the number 
in the interval 
. Theorem 16.2.6 says that the average value of a function is attained at some point if the function is continuous. 
given by 
and 
. 
equals 
. But 
for any 
. Thus, the continuity hypothesis in theorem 16.2.6 can not be dropped. 
a function 
, and a partition 
, for which 
given below is the 
: 
