The averagevelocityof a fallingobjectfrom to is givenby
HW Problem#10 will give you an opportunityto explorethis relationship.In our discussion,we saw how the
studyof tangentlines to functionsyieldsrich informationaboutfunctions.We now considerthe secondsitu-
ationthat arisesin Calculus,the centralproblemoffindingthe area underthe curveof a function
.
AreaUndera Curve
Firstlet’s describewhatwe meanwhenwe referto the areaundera curve.Let’s reconsiderour basic
quadraticfunction Supposewe are interestedin findingthe area underthe curvefrom
to
We see the cross-hatchedregionthat lies betweenthe graphand the -axis.That is the area we wish to
compute.As with approximatingthe slopeof the tangentline to a function,we will use familiarlinearmethods
to approximatethe area.Thenwe will repeatthe iterativeprocessof findingbetterand betterapproximations.
Can you thinkof any waysthat you wouldbe able to approximatethe area?(Answer:One ideasis that we
couldcomputethe area of the squarethat has a cornerat to be and then take half to find an
area This is one estimateof the area and it is actuallya prettygoodfirst approximation.)
We will use a variationof this coveringof the regionwith quadrilateralsto get betterapproximations.We will
do so by dividingthe -intervalfrom to into equalsub-intervals.Let’s start by usingfour
suchsubintervalsas indicated: