fall 2016 math 1132q (section 100) - calculus 2 mwf 11 ... · fall 2016 math 1132q (section 100) -...
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Fall2016Math1132Q(Section100)-Calculus2
MWF11:15am-12:05pm
Instructor:Dr.AngelynnAlvarezE-mail:[email protected]
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.)