# Prof Leonard’s notes

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

## University

McGill University

## Course

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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

### in3pJ

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

###### μ

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

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### Howtotell NOTES along Z

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##### f

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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

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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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