pointswithinthe sub-intervalsto find the limit. Hencewe can definethe mostgeneralsituationas follows:
Definition: If is continuouson we dividethe interval into sub-intervalsof equalwidth
with. We let be the endpointsof thesesub-intervalsand let
beanysamplepointsin thesesub-intervals.Thenthedefiniteintegralof from
to is
=
Example1:
Evaluatethe RiemannSum for from to using sub-intervalsand takingthe
samplepointsto be the midpointsof the sub-intervals.
Solution:
If we partitionthe interval into equalsub-intervals,then eachsub-intervalwill havelength
So we have and