Fall2016Math1132Q(Section100)-Calculus2

MWF11:15am-12:05pm

Instructor:Dr.AngelynnAlvarezE-mail:angelynn.alvarez@uconn.edu

Office:MONT305OfficeHours:MWF12:15-1:15pm,orbyappt

Section7.7–ApproximateIntegration(PartI)

Ingeneral,itisveryhard(andsometimesimpossible)totaketheantiderivativeofafunction.Hence,itisgenerallyhardandevenimpossibletofindanexactvalueofadefiniteintegral.àThus,weresorttoapproximatingvaluesoftheintegral.Recall:InCalculusI(Section5.1inthetextbook),weusedrectanglestoapproximateadefiniteintegral 𝑓 𝑥 𝑑𝑥!

! by:(1) Dividingtheinterval 𝑎, 𝑏 intosmallerintervalsofequallength

(2) Pluggingeithertheleftendpointorrightendpointofeachinterval

intothefunction𝑓(𝑥)

(3) Multiplyingthelengthofeachintervalby𝑓(𝑥!),where𝑥! isaleftendpointorrightendpoint.

Ifweevaluatedtheleftendpoint,itwascalled“LeftEndpointApproximation”,andifevaluatedtherightend-point,itwascalled“RightEndpointApproximation”.

Moreprecisely:Ifwewanttoapproximate 𝑓 𝑥 𝑑𝑥!! ,wehave:

1. LeftendpointApproximation𝑳𝒏

If𝑛 = thenumberofintervalsyouhave,plugintheleftendpointofeachintervalintothefunctionyouareintegrating.

𝑳𝒏 = 𝒃!𝒂𝒏[𝒇 𝒂 + 𝒇 𝒙𝟏 + 𝒇 𝒙𝟐 + 𝒇 𝒙𝟑 ]

2. RightendpointApproximation𝑹𝒏

If𝑛 = thenumberofintervalsyouhave,plugintherightendpointofeachintervalintothefunctionyouareintegrating.

𝑹𝒏 =𝒃− 𝒂𝒏

[𝒇 𝒙𝟏 + 𝒇 𝒙𝟐 + 𝒇 𝒙𝟑 + 𝒇 𝒃 ]

Example1:Let𝐼 = 𝑓 𝑥!! 𝑑𝑥,where𝑓 𝑥 isthefunctiongraphedbelow.

Compute:L! =𝑅! =

Ifagraphisincreasing….- Theleft-endpointapproximation_____________________________becausetheareaunderthesmallrectanglesislessthantheactualareaunderthegraph.

- Therightendpointapproximation___________________________becausetheareaunderthelargerectanglesismorethantheactualareaunderthegraph.

Ifagraphisdecreasing….- Theleft-endpointapproximation_____________________________becausetheareaunderthelargerectanglesismorethantheactualareaunderthegraph.

- Therightendpointapproximation___________________________becausetheareaunderthesmallrectanglesislessthantheactualareaunderthegraph.

Now,wewilllearn3othermethodsofapproximatingdefiniteintegrals:3. MidpointRule–Usingmidpointsofendpointsofintervals4. TrapezoidalRule–Usingtrapezoidsinsteadofrectangles5. Simpson’sRule–Usingparabolasinsteadoftrapezoids&rectangles3.MidpointRule𝑴𝒏If𝑛 = thenumberofintervalsyouhave,pluginthemidpointofeachintervalintothefunctionyouareintegrating.

𝑀! =𝒃 − 𝒂𝒏

[𝒇 𝑚! + 𝒇 𝑚! + 𝒇 𝑚! + 𝒇 𝑚! ]

Example2:Let𝐼 = 𝑓 𝑥 𝑑𝑥!"! where𝑓(𝑥)isgraphedbelow.

Compute𝑀! =

4.TrapezoidalRule–UsingtrapezoidsinsteadofrectanglesIf𝑛 = thenumberofintervalsyouhave,plug-ineach𝑥-valueintothefunctionweareintegrating,thenmultiplythemiddle𝑥-values(thenon-endpoints)by2.

𝑻𝒏 = 𝒃 − 𝒂𝟐𝒏

[𝒇 𝒂 + 𝟐 𝒇 𝒙𝟏 + 𝒇 𝒙𝟐 + 𝒇 𝒙𝟑 + 𝒇 𝒙𝟒 + 𝒇 𝒃 ]

Example3:Let𝐼 = 𝑓 𝑥 𝑑𝑥!"! where𝑓(𝑥)isgraphedbelow.

Compute𝑇! =

6. Simpson’sRule-Usingparabolasinsteadoftrapezoids&rectangles**Tousethisrule,𝑛mustbeaneveninteger!**If𝑛 = thenumberofintervalsyouhave,plug-ineach𝑥-valueintothefunctionweareintegrating,andnumberthemiddletermsstartingfrom1.Multiplyalltermswithanoddnumberby4,andmultiplyalltermswithanevennumberby2.

𝑺𝒏 =

𝒃− 𝒂𝟑𝒏 [𝒇 𝒂 + 𝟒𝒇 𝒙𝟏 + 𝟐𝒇 𝒙𝟐 + 𝟒𝒇 𝒙𝟑 + 𝟐𝒇 𝒙𝟒 + 𝟒𝒇 𝒙𝟓 + 𝒇(𝒃)]

Example4:Let𝐼 = 𝑓 𝑥 𝑑𝑥!"! where𝑓(𝑥)isgraphedbelow.

Compute𝑆! =

FunFacts:1.Thebiggerthen,themoreaccurateyourapproximationwillbe.2.TheapproximationinSimpson’sRuleistheweightedaveragesofthoseintheTrapezoidalandMidpointRules---thatis:

𝑆! =13𝑇! +

23𝑀!

3.TheTrapezoidalandMidpointRulesaremoreaccuratethantheleftendpointandrightendpointapproximations.4.Simpson’sRuleismoreaccuratethantheTrapezoidalandMidpointRules.

Example5:Thewidths(inmeters)ofakidney-shapedswimmingpoolweremeasuredat4-meterintervalsasindicatedinthefigure.UseSimpson’sruletoestimatetheareaofthepool.(Roundtothenearestsquaremeter.)