and
Notethat this approximationis very closeto our initialapproximationof However, sincewe took the
maximumvalueof the functionfor a side of eachrectangle,this processtendsto overestimatethe true
value.We couldhaveusedthe minimumvalueof the functionin eachsub-interval.Or we couldhaveused
the valueof the functionat the midpointof eachsub-interval.
Can you see how we are goingto improveour approximationusingsuccessiveiterationslike we did to ap-
proximatethe slopeof the tangentline?(Answer:we will sub-dividethe intervalfrom to
into moreand moresub-intervals,thus creatingsuccessivelysmallerand smallerrectanglesto refine
our estimates.)
Example1:
The followingtableshowsthe areasof the rectanglesand their sum for rectangleshavingwidth
RectangleRi Areaof Ri
R 1
R 2
R 3
R 4
R 5
R 6
R 7
R 8. This valueis approximatelyequalto Hence,the approximationis now quite
a bit less than For sixteenrectangles,the valueis whichis approximatelyequalto Can you
guesswhatthe true area will approach? (Answer:usingour successiveapproximations,the area will
approachthe value )
We call this processof findingthe area undera curveintegrationof overthe interval
Applicationsof IntegralCalculus