Today we will do some problems the computation of area the computation of volume position from acceleration cost from marginal cost Here are some more probabilities and distributions averages and expectations 64257nding moments of inertia work from ID: 26656
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Math1A:introductiontofunctionsandcalculusOliverKnill,2012 Lecture24:Applicationsofintegration Hereisalistofapplicationsforintegration.Today,wewilldosomeproblems.thecomputationofareathecomputationofvolumepositionfromaccelerationcostfrommarginalcostHerearesomemore:probabilitiesanddistributionsaveragesandexpectations ndingmomentsofinertiaworkfrompower Probability Inprobabilitytheoryfunctionsareusedasobservablesortode neprobabilities. Assumingourprobabilityspacetobetherealline,aninterval[a;b]iscalledanevent.Givenanonnegativefunctionf(x)whichhasthepropertythatR1 1f(x)dx=1,wecallP[A]=Zb0f(x)dxtheprobabilityoftheevent.Thefunctionf(x)iscalledtheprobabilitydensityfunction Themostimportantprobabilitydensityisthenormaldistribution: Thenormaldistributionhasthedensityf(x)=1 p 2e x2=x Itisthedistributionwhichappearsmostoftenifdatacantakebothpositiveandnegativevalues.Thereasonwhyitappearssooftenisthatifoneobservesdi erentunrelatedquantitieswiththesamestatisticalproperties,thentheirsum,suitablynormalizedbecomesthenormaldistribution.Ifwemeasureerrorsforexample,thentheseerrorsoftenhaveanormaldistribution. a b 1 Theprobabilitydensityfunctionoftheexponentialdistributionisde nedasf(x)=e xforx0andf(x)=0forx0.Itisusedtousedmeasurelengthsofarrivaltimeslikethetimeuntilyougetthenextphonecall.Thedensityiszerofornegativexbecausethereisnowaywecantravelbackintime.Whatistheprobabilitythatyougetaphonecallbetweentimesx=1andtimesx=2fromnow?TheanswerisR21f(x)dx1 2 a b Assumefisaprobabilitydensityfunction(PDF).TheantiderivativeF(x)=Rx 1f(t)dtiscalledthecumulativedistributionfunction(CDF). 2 FortheexponentialfunctionthecumulativedistributionfunctionisZx 1f(x)dx=Zx0f(x)dx= e xjx0=1 e x Theprobabilitydensityfunctionf(x)=1 1 1+x2iscalledtheCauchydistribution. 3 Finditscumulativedistributionfunction.Solution:F(x)=Zx 1f(t)dt=1 arctan(x)jx 1=(1 arctan(x)+1 2) a b Average Hereisanexampleforcomputingtheaverage 4 Assumethelevelinahoneyjarover[02]containingcrystallizedhoneyisgivenbyafunctionf(x)=3+sin(3x)=5+x(2 x)=10.Inordertorestorethehoney,itisplacedintohotwater.Thehoneymeltstoitsnormalstate.Whatheightdoesithave?Solution:TheaverageheightisR20f(x)dx=(2)whichistheareadividedbythebaselength.Inprobabilitytheorywewouldcallf(x)arandomvariableandtheaverageoffwithE[f]theexpectation 3 Momentofinertia IfwespinawireofradiusLofmassdensityf(x)aroundanaxes,themomentofinertiaisde nedasI=RL0x2f(x)dx Thesigni canceisthatifwespinitwithangularvelocityw,thentheenergyisIw=2. 5 Assumeawirehasdensity1+xandlength3.Finditsmomentofinertia.Solution: 6 Flywheelshaveacomebackforpowerplantstoabsorbenergy.Ifthereisnotenoughpower,the\rywheelsarecharged,inpeaktimes,theenergyisrecovered.Theyworkwith80percenteciency.Assumea\rywheelisacylinderofradius1,density1andheight1,thenthemomentofinertiaintegralisR10z2f(z)dx,wheref(z)isthemassindistancez Workfrompower IfP(t)istheamountofpowerproducedattimet,thenRT0P(t)dtisthework=energyproducedinthetimeinterval[0;T Energyistheanti-derivativeofpower. 7 AssumeapowerplantproducespowerP(t)=1000+exp( t)+t2 t.Whatistheenergyproducedfromt=1tot=10?Solution. 4 Wouldn'tbenicetohaveoneofthosebikeswithinteractivetrainingenvironmentsinthegym,allowingtorideinthePeruvianorSwissMountains,theCaliforniacoastorintheItalianTuscany?Additionally,thereshouldbesomecomputergamefeatures,racingotherridersthroughbeaches,desertsorTexanhighways(couldbeongoogleearth).Trainingwouldbesomuchmoreenter-taining.Businessopportunitieseverywhere.The rsto eringsuchtrainingequipmentwillmakeafortune.UntilthenwearestuckwithTVprogramswhichreallysuck. Homework 1 Theprobabilitydistributionwhichdescribesthetimeyouhavetowaitforyournextemailisf(x)=3e 3xwherexistimeinhours.Whatistheprobabilitythatyougetyournextemailinthenext4hours? 2 Assumetheprobabilitydistributionforthewaitingtimetothenextwarmdayisf(x)=(1=4)e x=4,wherexhasdaysasunit.Whatistheprobabilitytogetawarmdaybetweentomorrowandaftertomorrowthatisbetweenx=1andx=2? 3 Aconeofbaseradius1andheight2hastemperaturez2.Whatistheaveragetemperature?RememberthatifA(z)istheareaofasliceatheightzthenV=R20A(z)dzisthevolume.YouhavetocomputeR20z2A(z)dz=V 4 ACDRomhasradius6.Ifwewouldplacethematerialatradiusxontoonepoint,wegetadensityoff(x)=2x.FindthemomentofinertiaIofthedisc.Ifwespinitwithanangularvelocityofw=20roundspersecond.FindtheenergyE=Iw2=2.Withoutcredit:ExplodeaCD:http://www.powerlabs.org/cdexplode.htm.Careful! 5 a)YouareonastationarybikeattheHemenwaygymandpedalwithpowerP(t)=200+100sin(10t) t 300+t2 19440(inWatts=W).Theperiodic\ructuationscomefromahillyroute.Thelineartermisthe"tiringe ect"andthequadratictermisduetoendorphinskickingin.Whatenergy(JoulesJ=Ws)haveyouproducedinthetimet2[01800](s=seconds)?b)Sincewedomathnotphysics,weusuallyignorealltheunitsbutthisoneisjusttoomuchfun.Ifyoudividetheresultby4184,yougetkilocalories=foodcalories.Anapplehasabout80foodcalories.Howmanyapplescanyoueatafteryourhalfhourworkouttocompensatethespentenergy?