We can then definethe lowerand uppersums,respectively, over partition , by
where is the minimumvalueof in the intervalof length and is the maximumvalueof
in the intervalof length
The followingexampleshowshow we can use theseto find the area.
Example2:
Showthat the upperand lowersumsfor the function from to approachthe
value
Solution:
Let be a partitionof equalsub intervalsover We will showthe resultfor the uppersums.By
our definitionwe have
We note that eachrectanglewill havewidth and lengths as indi-
cated: