Hencewe musttake to achievethe desiredaccuracy.
Fromthe last example,we see one of the weaknessesof the TrapezoidalRule—itis not very accuratefor
functionswherestraightline segments(andtrapezoidtiles)do not lead to a goodestimateof area.It is
reasonableto think that othermethodsof approximatingcurvesmightbe moreapplicablefor somefunctions.
Simpson’s Ruleis a methodthat usesparabolasto approximatethe curve.
Simpson’s Rule:
As was true with the TrapezoidalRule, we divide the interval into sub-intervalsof length
We then constructparabolasthrougheachgroupof threeconsecutivepointson the graph.
The graphbelowshowsthis processfor the first threesuchparabolasfor the caseof sub-intervals.
You can see that everyintervalexceptthe first and last containstwo estimates,one too high and one too
low, so the resultingestimatewill be moreaccurate.
Usingparabolasin this way producesthe followingestimateof the area from Simpson’s Rule:
We note that it has a similarappearanceto the TrapezoidalRule.However, thereis one distinctionwe need
to note.The processof usingthreeconsecutive to approximateparabolaswill requirethat we assume
that mustalwaysbe an evennumber.
ErrorEstimatesfor the TrapezoidalRule
As with the TrapezoidalRule,we havea formulathat suggestshow we can choose to ensurethat the
errorsare withinacceptableboundaries.The followingmethodillustrateshow we can choosea sufficiently
large