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Calculus 3 Lecture 14.6: How to Solve TRIPLE INTEGRALS (Along with Center of Mass and Volume)
Calculus 3 Lecture 14.6: How to Solve TRIPLE INTEGRALS (Along with Center of Mass and Volume)

Prof Leonard’s notes


McGill University


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Professor Leonard Math Notes II S linesandpanel samedirection sameunitvectororparallel In3Dtodefinealineyouneedapointandavectordirectorliketheslopein 210 Ifyouhave 2 pointsyoucanfindthedirectorvectorbetweenthe points 2 Since the line vectordirectorthenvector Line toJ where tisascalar point Ch GivenapointPocxo yozoandareotherpointPCxy z ontheline so feardirector vector Pott x xo yYo ZZo y leti Lab c be thedirectionvectorwhereabcaredirection s Then ifyouhave 2 pointsPoandP PTI t P c x xoyYo 2201 t cabo xxo yYozzo Latbtat x xo at y yo bt z zo et G x at xo y btIyo j 2 4 20 eaofaline 1 para E xaxo n Y j o z zoea.ofau mericequayMllrice


ThisworksforanypointPocxoyozaandDirvectorf cabc NOTE If a directionnumberClab c 1 0 thenthelinelieson aplane supposethat a othenX Xoandthelinelaysin X Xoplane parallelto42planet 2 linesareparalleliftheirdirectionvectorsareMfscalarmultiplescE mEID ExDFindtheequationofthelinecontainingPC 1 2 yaand QCl 312 Ff 411 312 2 3 423 42,712 stay 2 y y gpeyou canmultiplyherebcweareonlylookingforadirect noneveryoucan’tforadistance youcan’tmultiplythepoint NeedapointPCl 2 42 andavectorRTL4,7 5 X Hut 4 2 7c z z se parametriceeloftheline Tofindapointonthelinepluga valuefort a i 47 s X YET symmetric equation twnatif Eo 4 z thelineisonthe plane 2 112 2 ShowL2 Y 1 Zita is a Ls andbi 1914to VI 2 14,7 55 2 vi isHui b x 7 tot 4 3 114 t z a lot Ex Airplane 1 Li Xi I 24y l sty 4 21 to VT L 2 3 I J notparallelbc NITscalarmultiples 2 3 tta 42 2 2ta 3ta ti e L i 2 i Checkforanintersection X X 2 41 42and21 22 X I 24 3 2 i at z it x 3 i 4 I 34 2 2 ta ta e z ta I 3Cit j a1266J to y a o so theplanes i donotin 2 214 3 ta i t o I t 6 Lines I t z 3 tersect linesDo Intersect Skewlines

Ex 4 XI I Yi 23 2 L xzj2 Yg 3 zzz i 1 Arethey 11 21 Dotheyintersect 4 x l ti y 3 at z t b aist y 3 22 ht 31 Aretheysberu vif i a ly ri 3,2i 4 Not k 2 x l t 23 ta x 112 Y 3 2e 312T Yt t4IYIY u y 22 theselinesintersectat Cla 2 a 2 e t I 3 t4h to I n z 42 andt 42 Findtheanglebetweenplanes ordinesastheycollide Anglebetweenlines Anglesbetween airvectors 1 μ absoluteravegivesmeaaneangee cos 0 go cosO fi I 161 o cos 461 4 a i willHii Texriu ii ii s uu a birthor krill14M Findwhereplane 2 Cline 21 Crossthecoordinateplanes xa 2 13 42 3 12 tz za Itta XYplane set 2 0 outta ta l so f l l o isonthexyplane xz plane set 4 0 0 3 2 ta ite 312 so z o 112 Yzplaneset o 0 21312 to 2g sofog planesin 1123 Normal Avector 1 toaplane Sogivenonepointon aplaneand anormalvectortotheplanein La b c wecan a specificplane define AvectorbetweenPoKoYoZoandPhx4,21ontheplane PTI LxXo Y YoZZo andanormaltotheplaneFi La b c FL Pops n PTI o La b c X Xo Y Yo Z Zo 0 a x Xo 1bCy yo t c ez zod o standardformoftheequationofaplane a n ax axo tby byo t cz ezo o HB with3pointsyoucan ay by or axo tbyo tczo c7 find 2 vectorsand d findthecrossproducts B ax by1oz d Generalformotineegofaplane EX EQofaplanewithPC 3 6 21 and 11 to 2x 134 Z 4 Fia 423 I Ifplanesare 11 theyhavethesamenormalvector Song n’I 2, ly andwehaveapointPC3, 2 Lx 3 t 3 CYb I Cz 121 2x 6 t 3 y H z 2 0 2x 13 y 2 26 equationoftheplane Ex FindEq OftheplanecontainingPC2,3 1 QCl 237 RE1, FI L l 5 u FR L 3 I s

Ex FindEQForlineofIntersectionof2planes Pi 2x3y 142 3 rip L23,41 ninot11nF bcin tri P2 x 144 22 7 ni Li 4 2 ninottnibcni 2 128 0 linesneedaPOINTand a DIRECTIONVECTOR Atonepointthenormalsofeachpoint sonTxni i j k 1 16 Hi C4 4 j 118311 willmeet 2 3 4 Llo8,11 P p paine i 4 Iri Forthepoint setXor yor 2 0 inthesystem P yep If 4 0 2 142 3 7122 3 242 14142 342 82 11 a I 11g ntliesonthexz x 27 7 and x 7 12 7 2,1 141 plane PlayO 84 L x it lot y St z y int g oftheline II 6 Cylindersandsurfaces Cylinders Equations ofcylindershaveonly 2 variablesTheseequations give a traceofthecurve onthe planedenotedbythegivenvariables coordinate 2 Curveisdirected alongtheaxisofthemissingvariables 3 Thecurve trace DOES Notchangealongthedirectionaxis Ex 2 4 Xa 2 variables cylinder directedalongy axis n traceisdrawnontheXzplane o x14 x 12 4 Z x2 14 oparabola t shiftedup 4 onthe 2 axis i fy openingtowards 02 axis 4 I E x Ex 9 2 442 36 2 variables n z tracedrauinonxyplane It 12 1 projected on 2axis 4 9 TH I y ellipseonxyplanewithinterceptsofx 2 and y 13 i x 2 EX 47 1 72 variables cylinder I y Ly t traceon 42 plane


Directed alongxaxis L n X n z Z 0084

Generalsurfaces Haveall 3 variables Tracesoccuroncoordplanesandoronplanesktocoordplanes stilldirectedalonganaxisbutthetracechangesalongtheaxis Step 1 Determine thetypeofsurface 2 Determinethedirectionaxis Findsometracesoncoordinateplane 4 Findatleast A Ellipsoid z s Ex 9 2 44422 36 surfacebiasvariables All Alisa Hasaconstant Ellipsoid Hautolell NOTE xd 1 2 1 1 Intercept _x 12 1 AIL 11 Standardformhelps 4 9 36 y 13 2 power 2 withtraces 2 e 2 3 Hasaconstant Intercepts c d i i Y Is f ly I z c E B OneSheetHyperboloid xa t 2 L Ex 944422 2 36 Allsop ICI const I Homtotell NOTE sheethyperboloid 1 Has 1 0 standardformhelps Taeongyaxis 2 power 2 withtraces iowab are 3Hasaconstant 2 isAlw directedalongthe Traces y O 2 4 1 circleofradius centredatcooo axiswithThe 0 ity 3 x g 24 1 dsetovar oaND t 6222 a 4 2 get 3 Traceswillalwaysbecircles are xjizf i ciraeraraaa orellipses a i i i s i i r y x

Ex 2 2444 wehave 2 sqvar paraboloid It isopeningalong 2 axistowards z 2

2 4 2 2 wsnittpownyor


i If o x 442 0 thereisnomoreanellipse If z 4 7 444 4 1,

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F Hyperbolicparaboloids


ez EI X 422 0 3 variableswithIsa

X ya z asf lethyperboloidparaboloid

Howtotell NOTES along Z

1 3 variableswith2sa2 Degree Isvar Givesdirection setz 1 x y2 HYPERBOLAalongx 2 ONEsadhas 0 Axis set 2 i x y I yaii HyperBoutalongy

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12 I VECTOR FUNCTIONS Parametric Eg X f t y gCtl t isthePARAMETER forsomeintervals I onacommondomain For 1123 x Htt y gut E htt Butvectors arebeaned byi x it y j’t z id so Fct fCHI’t gft thatbi SHHgothhut y z TheTERMINALpoints dthevectorcreatea curvethanSPACE notsurface Forthedomainof E AvectorfunctionisaparametricallydefinedfunctionwheretheTERMINALPoints ofourvectorstracea curvein 3 D Thefactthat t hasacertaindomaingivesiictsanorientation Ee ilto L3,5 21 Terminalpoint 1 3 s 2 bcrectorstartsattheorigin q Exe’s Fct 4ft int x tf y tf andz Int i to t 1 kt 0 SoDomain 10,9101110C FindvectorsratE 2 andt 4 shoulde domain Iet a redla i inD 4 rcul 12 4g in 4 SKETCHING n 1 Identify X yandz Yy 2 Use 1 ormorecomponentstogetAcurveor a surfaceCgetridof t l n FortwocomponentssketchonaMANI Forthreecomponentsthecurveisonasurface i 3 usevaluesof and and on


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limits lifea Title Infant Liemagctblif thisis avector Ex Iif Iti’t Ji’t ftp iYart rioIIafI4IHIma2t II EI F II Iti’t ftp biY l2r 4 Ex lifMgo fcostitTantjttlntbT 1,1 o I 1J lim cost _t otant liem o seinIxf f mosiftYose 1 Iff tht I trot oeh lim tan I slim e’t o Elim YI lim t o t s w d ts a trot Ye t Continuity PCH cost I μtμ te 4 continuouson t o Gtfo at o Gtfo c I ExFct 12 i sin CH j 3ft I continuouson tee1, Geii o btete b tax t 12 DerivativesandIntegrals of Vector Functions a f 11 4 m flxtax ft Q Ax 20 Ax T raise gynisgivesatangentveaortoaspaaae rtct limrcttb Flt Tangentvectorto aspacecurve At 20 At


Instantaneousvelocity bing lt btiigCttbtjthCttDtbi fCtiigCtlJthCtis At Lim fcttbtl flt jt I btro At r’Ct f I t gHj th’CHI Ex Fct titty’t EE SofCt it 2tJt3t2I DirectionvectordtheTANGENTlinetothespacecurve anyt for rut F Ct 2jt6tI

Ex Fct LFit fj’t int13 Ex FCHe etsint it etcostjttan e I FCH La i j if I sinei f’Ct etaint1 e tcost i e t cost sinti g CHLetcosts leftcost e sintlj etlsintiuos F 4 ft Z t Lt j t AI ehits tanH’I a I i Ctl ftp J ta I t soF Lt e t costs i etlcosttsintIt1 12 4TH y Ex 1 SketchFlt L 4 cost 2 sint oete21T r putinitialpointatmeroi 2 FindpointoftangencyCP 0 tl t 31 3 Findthetangentvector P.o p.o,51terminalpointofthevector i r x 4 cost and 4 2sent it T lost mail.siantaII aniiIiIsEeIxyfeane does one to PC4,0 t ILP0, 2 Pluginthevalueoft i F’Czk 14 cos 13, sin1g L 2 B andPOT 2 B 3 F t 4sint it 2cost j thisisthetangentvector r Ig f 23T 1 Thetangentvector givesorientation Ex Fctfete 2T Find thetangentvector to sketch etandyale 24 6 4 I y x d y Pot pot i’lol 41,11 Pci it got i cafetseat tangentrector rus hi 2 ilol u UnitTangentVectorsf1cal Flt F t givesthedirection oftangentline HFttlllrCtJ2sinc2tIii3cosktijt3b t P 42 sin 21 TfP.O 3 cos31,31 4 F z 3 sogo 1 3, Tct t 4 cosCatti 6 sincat i’CtH 4 cosHD’t sincatik F Tyco Li asr HiCMH 4277 3T SoTC F 1 1 2 sirs Li 3535 31M2T38J Ilicmon 3 Tangentlines Need a POTanda tangentvector atthe PO T FCH t T Ex Fct tt2 it th j t f Ta Findtheequationofthetangentline at C L z Flat 42 j 4 so PO 1 12 j f F t z Ct12 at thetasa atit in a ftp aJfftqb sotangentvector t 2 r 2 L at f

12 Arc lenghtandArclenght parameterization Arclenght forparametricequations L fbaffltD lg dt for t E Eab For3D fabfflttftcg 2tCh’CtJ dt for te Fab FUT fits it gltly thatbi F Ct fCHT t gCHj thCH I tangentvectortothiscurve 11 F t 1 fCtD2 Cg’CtIJ2 Ch’ctsJ2 magnitudeofthetangentvector so L fba HFCHH dt 1 Cangivearclength l ift ecab 2 CangivearclengthfunctionChas t lSCH ExFind Arclength Flt 48 intTt StJt 4 costs onOtt221T f Ct 4 costt 3J 4sintbf.lfccogates r II F LtdIl 16 cos2T t at16SinTtt l6t9T 5 L J 5 dt St Y SC21T LOTT EX FCHLetcost etsont et findarclenghtfor t E 0,21 1 F t feltcost sinEl et Sint 1 costJ ety Hr Ctd letcost SintD’tEetCsinttoostDte’T eatfcosts e’t fCcost sint t Sint cost 12 t I et cos 4 2 costsint sinat sinat asintcost cosetti et cosat cosat sinat sinat 2 costsinte2costsinceI et 2cos2Et2sin2t et 2T i et so

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53 et dt Foee I 53 e 53 Ey Foe Tz iz e Tt l For Smoothcurves wecanreparametrizeby arclength Why I wecanwalkthecurve 2 Someformulasarewayeasier 3 F CSIwillalwayshaveamagnitudeof 1 it’s the unittangentvector Ex Plt Itt it Ut2tIj 1 3TI with t 710 1 Findarclengthfunction SCHbcyouhavetwo E Ct f I 2 3 11FCH 11 11497 14T sCt f Ili cutHdu a aiswhere t begins replace t with n SCH f t 14Medu Fir to 14Mt Thisgives C SCI 14M

so FCS tf T t f l 12 μg Jj 3 I s to Ex F Lt etcosti etsintj t et I n t so IIFCtIl 3Tet and SLtd Jta 11 Flushdu SCH ettdu e’tto Get 53 3M et l SCH S 3TetD spy et l Is et l and et JE s ti j t In Ej Stl Pls Fgs 11 cos Inffstill i iff sujsinf bi Ex F t t’t ttcost j t tsintE FindLontocoD fba IIPCHHolt FCtl at T cost tsint j t Sint 1 tcostbi HF tHe 4t2t cost tsint 2 sintettcost 12 412 cosat 2 tcostsinttt2sinattsinat 2tsincostt2cos let2 1 cosat sinat t2Csinattcosset ttT Eth i ge rest t J 051 27 dt fflst TT o r Eano St it Bf tano Jf FET see 20 do dt seatdo Jj sector I DO I Jj see 30 do j f f SeoOLanO t tainf secottanOl Sff f 5ft’T t t In statTts tfto L 8 t in16ft BI O if I Do introrsl unittangentrector TNBFrames Frenet secretFrames Ateverypoint theTMBframegivesusthe unitrectors Iyangent i direction particleisheading Where Nose ofairplaneispointing theseareμn vectors 1Yormal directionparticleisturning 7Where TAILFIN is pointing whichareM 14 orthogo Binormal 1 direction ofparticle’stwist where wingsarepointing hat just likeDjbbutwith amovingparticle HowtofindTMB F unittangent FCss F Cs or TCH F Ct IlF CHH Ja NT unitnormal RICH FCH f r gi taunitrector 1 to T II FLtd 11 M everyvectorwith a constantlenghtis 1 to itstangentvector B T T Futis aUNITvector constantlength Proof let ithave a constantlength v II I 112 T rt Ga c a constant ft ft T dat a O f i s o T I E 2 CT j j

Ex FindF FITI k e eg ofoscplaneandeat ofnormalplane t forFct costT t sint j t th E Fct F’Ct f Ltd Sinti cost Jets IIF Ct 11 HrCEM sin2ttcos sintT 2MsoFLtJ costjtTs 2zIf SintTcostItb 2M BUT if Ct Ez f costT Sinti HFTN I cos ttsins TI N 2 so kilt Fetal costT sin 1 J 03122 N t costT SintJ B Fct xEiCt I I j k Fda Sinti costjetsinattoos’t 2 Sintcost I B Ginty cosey St Sint 0 BE Sinti costj IE I I Fae rEx I f I L K Fct 2mg f Sint 1 costItta N Lt costt Sint J Ct Ea SintT costJtI Point F IL OTt ly tILD P O L Tha T IL LEf lit Oj bi NI 1112 OT Ij PCO 1, B Tbd 2 Ttb Fxrt LE fit b x j Bz i j b rayCitta I o l o l 0 Sotofind BONLY youcanpluginthevalueof tin FandN tomakeiteasier OSCULATINGplanes contains FandIP B istheredirection PC0 I 1112 is apointandin D 2E itb or in Ttb bc itsa scalarmultiple of 157 Eg X O t C 2 TIG 0 Eg X 12 I 2

Normal Plane contains IPandB F isthenormalvector P o l 1112 andin P BegCTtb or in L l O 1 Eq ICx o IC z IL O X 12 Exc F Ltd Letcost etsint et T XT K FCKF Ct et costsentTeet Sint cost Jtetbi cost sint T LsinttcostJtb HrLETH 3 et 3M rLtd Let cost sint et Sintlost et Hr t F et so I Ct 31ydCost Sint Csinttcost 1 N H 7 FCtH FCt Fg Csint cost cost Sint O 117 t Bye osd tostT

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dctldt fletite dt etT etItc J t ett etist c but God L I 2 O so got 1 OJ Eof t c it 2J t I B t c e ti 12J c 2J 1 so if E etT Etbit2J Ibi Let 2 17 ftLtd Ji Ct dt J Let it 2J I etsb dt et it 2tj it ae t Btc FCod IT to f I k t c 3 it Jt 2 c 2T job soF Ct Let121TtfdtHJ lt e t NOTE Wecandefineaccelerationas arectorwithFanaFT let11pctHI v Cspeed a FT KI IT or I F T T Hr xp Il μ tangentialcomponfnt normalcomponent r Hr II ofacceleration ofacceleration 11 i’Ltd 11 1184411 v ell r KTH v HECtHR Projectilemotion r Ct Tocosd t it ht tosin d t Lgta I To initialvelocity heheigth 2 angleofinclination g accelerationofgravity Ex Youshootarifflefroma 200ftcliffwithanangleof30andamuzzlevelocityof 1500 FtlSec Vo 1500 Ftls 2 46 h 200Ft gg 9 ja or32ft PUT 1300 cos If t T t 200 t 1500 sing I 32 T2 J FLH 7503ft Tt 200 1750T Ift2 j x x x range y height 4 Impactvelocity speedwhencit 1 Whenwillthebullethittheground hitstheground weset 4 0 t 47 Isee VCt P Ct 7503T it Go 324J 2 Maximumrange I147 1503 Is 7503TLEI weset 47 D 7505 C 47 D 61 185 Ft Ili 147 1503. 3 Maxheight set 41 0 andreplacevalueof tiny y 7so 32T so c 37 y 3,75g 8989 Ft 4

Meltirraniablends 10. functions 13 I IntrotoMultivariablefunctions DomainSketching levelCurves Tographafunctionyoumusthave 1 Dimensionmoirethanthe ofindependentvariables fCx X11 y x 11 there isoneINDEPENDENT variable sographedin 2D g x y Hey 7 z x2ty 2 3D sothereisatmos 1 DEPENDENTvariable hCXy 2 x2ty2 w x 3 independent variables so 4 D 2 3 z 3 TOgraphthe DOMAIN of afunction youmusthavethesamedirection astheindependentvariable f x D l D 4 41 o 3 D EX f X y z x2t2y2t 3 INDvar x y 2 t lDEPvar W 4 variables sothefunction needs 413 tobegraphed 3DDomainand 40 Graph fl 0,2 l 02 2012 4 177 I flu u ya 11 YCutH u’t 2 u2 2 u 11 13 u’t2utl u’t2u 4 u 2 302 6 642 24 5 includeallINDvariable 9 fall 4 x yF Domain D X y XI y andRange suchthat includeIvarDEPvariable Dange theoutputcalled Z 12 Z oh 2 L to Ex g x y 4427 2 surface 3D Domain 213 be 4 X 122, f E X 2142 ai u bk sq 2 SoDomainD x y x2 4214 andRange 12 z oe z e 2 TOgraphDOMAINYOUmusthave an AXISforeachIMPvariable fad f D lo y EcanstatedomainthiswaywhenweonlyhaveoneIND var KX μ y Domainisanypoint of Y IIya l I I’m LEE X 42 10 y x y 1 1X x

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