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----------------|+F$-F^in6$Q+Time:~%a~|+F$-F`^em6#%#%cG>Fgq-%*piecewiseG622FfqF 6Q5(very-late-)~eveningF$31F6Ffq2FfqF8Q6(very-early-)~morningF$31F8Ffq2FfqF:Q1( early-)~morningF$31F:Ffq2Ffq"#6Q(morningF$31F^`emFfq2FfqFb][lQ$dayF$31Fb][lFfq2 Ffq"#>Q*afternoonF$31Fg`emFfq2Ffq"#AQ(eveningF$31F\aemFfq1Ffq"#BQ0(late-)~eveni ngF$>F`bl-Fa`dl6#-Ff`dl6$/Fj`dlQ*GreetingsF$7$-Faadl6$-Feadl6%Q&Good~F$FgqQF!~I ~am~Integrator6~-~at~your~service!F$Fhadl-Ficdl6$Q2OK~-~take~it~easyF$F`^el-Fea l6#Fd\el>F^el7&Q&curveF$Q.planar~domainF$Q(surfaceF$Q*3D~domainF$>Fgel7&Q&Curve F$Q%PlanF$Q%SurfF$F[cem>F_em7&Q3A~curve~in~3-spaceF$Q0A~planar~domainF$Q5A~surf ace~in~3-spaceF$Q7A~3D~domain~in~3-spaceF$>Fbr-Fa`dl6#-Ff`dl6$/Fj`dlQ>Main~geom etric~object~requestF$7%-Faadl6$Q]pPlease~specify~|+~the~main~geometric~object~ |+~under~consideration~in~this~exercise:F$Fhadl-&_Fb`dl%,DropDownBoxG6#.%'DDB11 1G6$/F\a[mFh`oF_em-Ficdl6$F\ddl-F^ddl6#7#Fgdem>%%ans1G-Fgb[m6$&-Fdddl6#FbrF[tF_ em-F^in6$Q2Main~object:~%a~|+F$&F_em6#Faeem@%/FaeemF4C+>F^_dlF4>F__dlF6>%#L2G7) QEIntegrate~a~function~along~the~curveF$QXFind~approximate~integral~of~a~functi on~along~the~curveF$QUFind~the~tangential~curve~integral~of~a~vector~fieldF$QJF ind~the~center~of~mass~of~weighted~curveF$QboFind~torque~of~a~vector~field~on~w eighted~curve~with~respect~to~a~pointF$QNFind~moment~of~inertia~with~respect~to ~a~lineF$QLFind~flow~of~the~curve~along~a~vector~fieldF$>%$L2aG7)Q#C1F$Q#C2F$Q# C3F$Q#C4F$Q#C5F$Q#C6F$Q#C7F$>%$L2bG7)Q)kurveIntF$Q/kurveIntApproxF$Q-tangKurveI ntF$Q(kurveCmF$Q(kurveKmF$Q(kurveImF$Q*kurveFlowF$>%$L2cG7)7#Q"fF$Fbhem7#Q"VF$F bhem7%FchemFehemQ#ptF$7%FchemFghemQ"eF$7$FehemQ"TF$>%$L2dG7)7%Q,Fig1:~CurveF$Q8 Fig2:~Curve~with~weightF$Q6Calculation:~IntegralF$7&F`iemFaiemQ4Fig3:~Approxima tionF$QMValue:~Approximated~integral~and~exact~valueF$7&F`iemQ3Fig2:~Vector~fie ldF$Q%$L 2eG7)F4F5F4F4F4F4Fhz>Fa[n7)7$F4F5Febx7$F4F6F^[fmF^h]lF^[fm7&F4F6F7F8F$@%/FaeemF 5C+>F^_dlF5>F__dlF5>Fbfem7'QMIntegrate~a~function~along~the~planar~domainF$QjnF ind~approximate~integral~of~a~function~along~the~planar~domainF$QNFind~center~o f~mass~of~weighted~planar~domainF$QGShow~wire~frame~model~of~planar~domainF$Q;S how~gray~wire~frame~modelF$>F\gem7'Q#P1F$Q#P2F$Q#P3F$Q#P4F$Q#P5F$>Ffgem7'Q(plan IntF$Q.planIntApproxF$Q'planCmF$Q/traadModelPlanF$Q3traadModelPlanGrayF$>F`hem7 'FbhemFbhemFbhem7#Q$nixF$F]]fm>F]iem7'7&Q7Fig1:~Parameter~domainF$Q9Fig2:~Param etrized~imageF$FdiemFbiem7&Fb]fmFc]fmQFa[n7'F^h]lFebxF^h]l7#F4F_^fm>Fjjem7 'F4F5F4FhzFhzF$@%/FaeemF6C+>F^_dlF5>F__dlF6>Fbfem7/QFIntegrate~a~function~over~ the~surfaceF$QYFind~approximate~integral~of~a~function~over~the~surfaceF$QHFind ~center~of~mass~of~weighted~surfaceF$Q\oFind~moment~of~inertia~of~weighted~surf ace~with~respect~to~a~lineF$QfoFind~torque~from~a~vector~field~on~weighted~surf ace~with~respect~to~a~pointF$QTFind~the~flux~of~a~vector~field~through~the~surf aceF$QjnFind~the~flux~of~the~curl~of~a~vector~field~through~the~surfaceF$Q_oFin d~the~circulation~of~a~vector~field~along~boundary~of~the~surfaceF$QPFind~the~f low~of~a~surface~along~a~vector~fieldF$QVFind~the~total~wring~of~a~vector~field ~on~the~surfaceF$QNShow~wire~frame~model~of~parametrized~surfaceF$F\\fmQHShow~w ire~frame~model~and~full~boundaryF$>F\gem7/Q#S1F$Q#S2F$Q#S3F$Q#S4F$Q#S5F$Q#S6F$ Q#S7F$Q#S8F$Q#S9F$Q$S10F$Q$S11F$Q$S12F$Q$S13F$>Ffgem7/Q)fladeIntF$Q/fladeIntApp roxF$Q(fladeCmF$Q(fladeImF$Q(fladeKmF$Q(fluxIntF$Q+StokesFluxF$Q.StokesRandIntF $Q*fladeFlowF$Q/fladeTotalVridF$Q0traadModelFladeF$Q4traadModelFladeGrayF$Q7tra adModelFladeMedRandF$>F`hem7/FbhemFbhemFbhemFhhemFfhemFdhemFdhemFdhemFjhemFdhem F]]fmF]]fmF]]fm>F]iem7/7'Fb]fmQLFig2:~Parametrized~image~surface~wire~frameF$QF Fig3:~Parametrized~full~image~surfaceF$Q4Fig4:~ApproximationF$Q>Calculation:~Su rface~integralF$7&Fb]fmFc]fmFdiemFeiemFf]fm7&Fb]fmQCFig2:~Parametrized~image~wi th~axisF$Q>Fig3:~Approximation~with~axisF$Fbjem7'Q;Fig1:~Parametrized~surfaceF$ FgiemQWFig3:~Vector~field~at~wire~surface~and~torque~at~pointF$QWFig4:~Vector~f ield~at~full~surface~and~torque~at~pointF$Q7Value:~Torque~at~pointF$7'FabfmFgie mQCFig3:~Vector~field~at~wire~surfaceF$QCFig4:~Vector~field~at~full~surfaceF$QR Calculation:~Flux~of~vector~field~through~surfaceF$7(FabfmFgiemQ;Fig3:~Curl~of~ vector~fieldF$QQFig4:~Curl~of~vector~field~at~wire~frame~surfaceF$QKFig5:~Curl~ of~vector~field~at~full~surfaceF$QZCalculation:~Flux~of~curl~of~vector~field~th rough~surfaceF$7'FabfmFgiemQCFig3:~Wire~frame~and~full~boundaryF$QDFig4:~Vector ~field~at~full~boundaryF$QXValue:~Total~circulation~of~vector~field~along~bound aryF$7)FabfmQMFig2:~Vector~field~around~wire~frame~surfaceF$QGFig3:~Vector~fiel d~around~full~surfaceF$QFFig4:~Integral~curves~through~surfaceF$Q>Fig5:~Forward ~flow~of~surfaceF$QTAnimation:~Total~flow~of~surface~along~vector~fieldF$QVValu e:~Expression~for~integral~curves~of~vector~fieldF$7)FabfmFgiemQIFig3:~Vector~f ield~at~wire~frame~surfaceF$FgbfmQOFig5:~Wring~vector~field~at~wire~frame~surfa ceF$QIFig6:~Wring~vector~field~at~full~surfaceF$QQValue:~Wring~vector~field~and ~total~wring~vectorF$7#QXFig:~Colored~wire~frame~version~of~parametrized~surfac eF$7#Q=Fig:~Gray~wire~frame~versionF$7#QJFig:~Wire~frame~surface~and~full~bound aryF$>Fa[n7/FebxFebxF^h]lF^h]l7%F4F6F7Ffdfm7%F4F7F8FfdfmFi[_m7'F4F6F7F8F9F_^fmF _^fmF_^fm>Fjjem7/F4F5F4F4F4F4F4F4FhzF4FhzFhzFhzF$@%/FaeemF7C+>F^_dlF6>F__dlF6>F bfem71QBIntegrate~a~function~in~3D~domainF$QUFind~approximate~integral~of~a~fun ction~in~3D~domainF$QJFind~center~of~mass~of~weighted~3D~domainF$Q^oFind~moment ~of~inertia~of~weighted~3D~domain~with~respect~to~a~lineF$QjoFind~torque~from~a ~vector~field~on~a~weighted~3D~domain~with~respect~to~a~pointF$QYFind~the~total ~divergence~of~a~vector~field~in~3D~domainF$QgoFind~the~total~flux~of~a~vector~ field~out~through~the~surface~of~a~3D~domainF$QNFind~flow~of~a~3D~domain~along~ a~vector~fieldF$QSFind~the~total~curl~of~a~vector~field~in~3D~domainF$Q^oFind~t he~total~wring~of~a~vector~field~on~the~boundary~of~3D~domainF$QIShow~wire~fram e~version~of~the~3D~domainF$QNShow~gray~wire~frame~version~of~the~3D~domainF$Q9 Show~3D~boundary~surfaceF$QOShow~wire~frame~version~of~3D~boundary~surfaceF$QTS how~gray~wire~frame~version~of~3D~boundary~surfaceF$>F\gem71Q#B1F$Q#B2F$Q#B3F$Q #B4F$Q#B5F$Q#B6F$Q#B7F$Q#B8F$Q#B9F$Q$B10F$Q$B11F$Q$B12F$Q$B13F$Q$B14F$Q$B15F$>F fgem71Q'rumIntF$Q-rumIntApproxF$Q&rumCmF$Q&rumImF$Q&rumKmF$Q'divIntF$Q*GaussFlu xF$Q(rumFlowF$Q,rumTotalRotF$Q1rumRandTotalVridF$Q.traadModelRumF$Q2traadModelR umGrayF$Q+sideFladerF$Q5traadModelSideFladerF$Q9traadModelSideFladerGrayF$>F`he m71FbhemFbhemFbhemFhhemFfhemFdhemFdhemFjhemFdhemFdhemF]]fmF]]fmF]]fmF]]fmF]]fm> F]iem717'Fb]fmQNFig2:~Parametrized~image~by~boundary~surfacesF$QGFig3:~Parametr ized~image~by~wire~frameF$FjafmFbiem7'Fb]fmFhhfmFihfmQ:Fig4:~(1/8)-approximatio nF$Feiem7'Fb]fmFhhfmQZFig3:~Parametrized~image~by~wire~frame~and~center~of~mass F$QMFig4:~(1/8)-approximation~and~center~of~massF$QTValue:~Expression~for~total ~mass~and~center~of~massF$7'Fb]fmFhhfmQMFig3:~Parametrized~wire~frame~image~and ~axisF$QCFig4:~(1/8)-approximation~and~axisF$Fbjem7&FabfmQZFig2:~Vector~field,~ wire~frame~3D~domain,~torque~at~pointF$QOFig3:~Wire~frame~3D~domain~and~torque~ at~pointF$QUValue:~Torque~expression~for~full~3D~domain~at~pointF$7&QNFig1:~Par ametrized~image~by~boundary~surfacesF$QQFig2:~Parametrized~wire~frame~image~of~ 3D~domainF$Q3Fig3:~Vector~fieldF$QKCalculation:~Total~divergence~in~3D~domainF$ 7&FhifmQDFig2:~Parametrized~wire~frame~imageF$FjifmQQValue:~Total~out-flux~thro ugh~3D~domain~boundaryF$7*QQFig1:~Parametrized~(3D~domain)-boundary~surfacesF$Q ;Fig2:~Wire~frame~3D~domainF$QLFig3:~Wire~frame~3D~domain~and~vector~fieldF$QFF ig4:~Full~3D~domain~and~vector~fieldF$QSFig5:~Integral~curves~through~wire~fram e~3D~domainF$Q@Fig6:~Forward~flow~of~3D~domainF$QOAnimation:~Total~flow~of~3D~d omain~along~fieldF$Fhjem7&F`jfmQLFig2:~Wire~frame~3D~domain~and~vector~fieldF$Q QFig3:~Curl~vector~field~and~wire~frame~3D~domainF$QNValue:~Expression~for~tota l~curl~of~3D~domainF$7&FhifmQGFig2:~Parametrized~image~by~wire~frameF$FjifmQSVa lue:~Total~wring~vector~for~(3D~domain)-boundaryF$7#QBFig:~Colored~wire~frame~3 D~domainF$7#Q?Fig:~Gray~wire~frame~3D~domainF$7#QIFig:~Colored~3D~domain~bounda ry~surfacesF$7#QNFig:~Wire~frame~(3D~domain)-boundary~surfacesF$7#QSFig:~Gray~w ire~frame~(3D~domain)-boundary~surfacesF$>Fa[n71FebxFa_oFebxFebxF^h]lF^[fmF^[fm F3F^h]lF^[fmF_^fmF_^fmF_^fmF_^fmF_^fm>Fjjem71F4F5F4F4F4F4F4FhzF4F4FhzFhzFhzFhzF hzF$>Fcr-Fa`dl6#-Ff`dl6$/Fj`dlQ%%ans2G-Fgb[m6$&-Fdddl6#FcrF[tFbfem-F^in6 $Q0Main~task:~%a~|+F$&Fbfem6#Fj]gm>F1-Fd^[m6$&F]iemFd^gm&FjjemFd^gm>Fdr-Fa`dl6# -Ff`dl6$/Fj`dlQ4Input~request~alertF$7$-Faadl6$QeoI~need~some~input~for~that~op eration~|+~.......~|+~Are~you~ready~to~deliver?F$Fhadl-Ficdl6$Q,OK~-~ready:F$F` ^el-Feal6#-Fdddl6#Fdr-F^in6#Q:Input~delivery:~-------~|+F$@%2Fhz-Fhfl6#-%*inter sectG6$-Faq6$&Fa[nFd^gm%$setG-Faq6$F1Fi`gm>FgbnF4>FgbnFhz@%3/FgbnF4/F^_dlF4C$>F hdl-Fa`dl6#-Ff`dl6$/Fj`dlQ:Warning:~Please~note~--->F$7$-Faadl6$Q^ySince~your~r equest~involves~figure(s)~|+~your~input~must~be~prepared~accordingly.|+~....... ~|+|+For~example,~|+~the~parametrization~must~be~expressed~in~the~variable~u~on ly,~|+~the~u-interval~must~have~numeric~bounds,~etc.|+~.......|+~F$Fhadl-Ficdl6 $F\ddlF`^el-Feal6#-Fdddl6#FhdlF$@%3F`agm/F^_dlF5C$>Fhdl-Fa`dl6#-Ff`dl6$Fhagm7$- Faadl6$QjzSince~your~request~involves~figure(s)~|+~your~input~must~be~prepared~ accordingly~for~that~task.|+~.......|+|+For~example,~|+~the~parametrization~mus t~be~expressed~in~the~variables~u~and~v~only,~|+~the~u-~and~v-intervals~must~ha ve~numeric~bounds,~etc.|+~.......~|+F$FhadlF^bgmF`bgmF$@%3F`agm/F^_dlF6C$>Fhdl- Fa`dl6#-Ff`dl6$Fhagm7$-Faadl6$Qb[lSince~your~request~involves~figure(s)~|+~your ~input~must~be~prepared~accordingly~for~that~task.|+~.......|+|+For~example,~|+ ~the~parametrization~must~be~expressed~in~the~variables~u,~v,~and~w~only,~|+~th e~u-,~v-~and~w-intervals~must~have~numeric~bounds,~etc.|+~.......~|+F$FhadlF^bg mF`bgmF$>Fep-F[_dlF]_dl@%3Faagm/F__dlF6-F^in6$QKParametrization~of~the~curve:~r :=~u->~%a~|+F$-FepFefoF$@%3Ffbgm/F__dlF5-F^in6$QWParametrization~of~the~planar~ domain:~r:=~(u,v)->~%a~|+F$-FepF`[mF$@%3FfbgmFbdgm-F^in6$QQParametrization~of~t he~surface:~r:=~(u,v)->~%a~|+F$F]egmF$@%3FccgmFbdgm-F^in6$QSParametrization~of~ the~surface:~r:=~(u,v,w)->~%a~|+F$-FepF\]lF$>Ficn-%*Call_BnetG6$F^_dlFgbn>F+F[` o-F^in6$Q;Parameter~domain:~B:=~%a~|+F$F+@%-F\dn6$Fchem&F`hemFd^gmC$>Fh^z-%'Cal l_fG6#F__dl@%Fbdgm-F^in6$QNWeight~function~integrand:~f:=~(x,y,z)->~%a~|+F$-Fh^ 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