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"" 0 "" {TEXT -1 34 "Centres of gravity and centroids \+ " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Moments and centres of gravity" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 115 "A ruler balan ces at its mid point and a billiard cue balances at a point closer to \+ the thick end than the thin end." }}{PARA 0 "" 0 "" {TEXT -1 33 "This \+ balance point is called the " }{TEXT 259 17 "centre of gravity" } {TEXT -1 15 " of the object." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 268 "Imagine that the shape below has been cut from a uniform sheet of a material such as metal, plastic, wood or cardboa rd, and supported horizontally. The first view is from directly above \+ the plane object. (If the object is thin, we can think of it as lying \+ in a plane.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "We can regard the total mass M of this object as being made up of a large number of elementary masses " }{XPPEDIT 18 0 "m[1],m[2],m [3],` . . . `,m[k];" "6'&%\"mG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G&F $6#%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M = m[1]+m[2]+m[3]+` . . . `+m[k];" "6#/%\"MG,,&%\"mG6#\"\"\"F)&F'6#\"\"#F)&F'6#\"\"$F)%(~ .~.~.~GF)&F'6#%\"kGF)" }{TEXT -1 1 "." }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "By Newton's second law of motion, the force of gravity on each element is " }{XPPEDIT 18 0 "m[1]*`.`*g,m[2] *`.`*g,m[3]*`.`*g,` . . . `,m[k]*`.`*g;" "6'*(&%\"mG6#\"\"\"F'%\".GF'% \"gGF'*(&F%6#\"\"#F'F(F'F)F'*(&F%6#\"\"$F'F(F'F)F'%(~.~.~.~G*(&F%6#%\" kGF'F(F'F)F'" }{TEXT -1 8 ", where " }{TEXT 260 1 "g" }{TEXT -1 85 " i s the acceleration due to gravity. These are the weights of the elemen tary masses. 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F[xF]x-%%TEXTG6&7$$\"#cF*$\"#YF*Q\"x6\"Fg\\q-%%FONTG6$%*HELVETICAGF[[n -Fc`q6&7$$\"#pF*$\"#CF*Fj`qFg\\qF\\aq-Fc`q6&7$$F`\\mFdv$\"#7F*Fj`qFg\\ qF\\aq-Fc`q6&7$$\"$v&Fit$\"#XF*Q\"1F[aqFg\\q-F]aq6$F_aqFjw-Fc`q6&7$$\" $0(Fit$\"#BF*Q\"2F[aqFg\\qFebq-Fc`q6&7$$\"$:%Fit$\"#6F*Q\"3F[aqFg\\qFe bq-Fc`q6&7$$\"#*)F*$!\"$F*Q&m~~.gF[aqFbvF\\aq-Fc`q6&7$$\"$1\"F*$!#bFit F^dqFbvF\\aq-Fc`q6&7$$\"$N'Fit$F^tF*F^dqFbvF\\aq-Fc`q6&7$$Fg^oF*$F\\^o F*Q\"AF[aqFbvF\\aq-Fc`q6&7$FhboFeclQ\"BF[aqFbvF\\aq-Fc`q6&7$$\"$#*)Fit $!\"%F*FdbqFbvFebq-Fc`q6&7$$\"%i5Fit$!#lFitF^cqFbvFebq-Fc`q6&7$$\"$P'F it$F0F*FfcqFbvFebq-Fc`q6&7$Fecl$\"#NF*Fj`qFe^qF\\aq-Fc`q6&7$$\"#&)F*$F _wF*Q$M.gF[aqFe^qF\\aq-Fc`q6&7$$\"$D'Fit$\"#MF*Q\"GF[aqFe^qFebq-%*AXES STYLEG6#%%NONEG-%+AXESLABELSG6%Q!F[aqFehq-F]aq6#%(DEFAULTG-%%VIEWG6$Fh hqFhhq" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" " Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curv e 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27 " "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "C urve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 259 31 "moment of a force about an axis" }{TEXT -1 97 " is the p roduct of the force and the perpendicular distance of its point of act ion from the axis." }}{PARA 0 "" 0 "" {TEXT -1 28 "It can be thought o f as the " }{TEXT 259 14 "turning effect" }{TEXT 256 1 " " }{TEXT -1 28 "of the force about the axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 245 "A see-saw will remain in balance when a \+ child is sitting on one side, and an adult is on the other side, provi ded that the heavier adult moves closer to the central pivot, so that \+ the moments of the child and the adult about the pivot are equal." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Suppose t hat the individual masses " }{XPPEDIT 18 0 "m[1],m[2],m[3],` . . . `,m [k];" "6'&%\"mG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G&F$6#%\"kG" } {TEXT -1 32 " are at perpendicular distances " }{XPPEDIT 18 0 "x[1],x[ 2],x[3],` . . . `,x[k];" "6'&%\"xG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~. ~G&F$6#%\"kG" }{TEXT -1 94 " from the horizontal axis AB in the plane \+ of the object. The moments of the individual forces " }{XPPEDIT 18 0 " m[1]*`.`*g,m[2]*`.`*g,m[3]*`.`*g,` . . . `,m[k]*`.`*g;" "6'*(&%\"mG6# \"\"\"F'%\".GF'%\"gGF'*(&F%6#\"\"#F'F(F'F)F'*(&F%6#\"\"$F'F(F'F)F'%(~. ~.~.~G*(&F%6#%\"kGF'F(F'F)F'" }{TEXT -1 22 " about this axis are " } {XPPEDIT 18 0 "m[1]*`.`*g*`.`*x[1],m[2]*`.`*g*`.`*x[2],m[3]*`.`*g*`.`* x[3],` . . . `,m[k]*`.`*g*`.`*x[k];" "6'*,&%\"mG6#\"\"\"F'%\".GF'%\"gG F'F(F'&%\"xG6#F'F'*,&F%6#\"\"#F'F(F'F)F'F(F'&F+6#F0F'*,&F%6#\"\"$F'F(F 'F)F'F(F'&F+6#F6F'%(~.~.~.~G*,&F%6#%\"kGF'F(F'F)F'F(F'&F+6#F=F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 17 "centre of gravity" }{TEXT -1 188 " of the object is a point G in the plane of the object which has the char acteristic property that, for any choice of axis such as AB in the pla ne of the object, the perpendicular distance " }{XPPEDIT 18 0 "x[G];" "6#&%\"xG6#%\"GG" }{TEXT -1 155 " of the centre of gravity G from the axis provides the distance to be used in calculating the total moment of the object about AB, using the total mass M." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "m[1]*`.`*g*`.`*x[1]+m[2]*`.`*g*`.`*x [2]+m[3]*`.`*g*`.`*x[3]+` . . . `+m[k]*`.`*g*`.`*x[k] = M*`.`*g*`.`*x[ G];" "6#/,,*,&%\"mG6#\"\"\"F)%\".GF)%\"gGF)F*F)&%\"xG6#F)F)F)*,&F'6#\" \"#F)F*F)F+F)F*F)&F-6#F2F)F)*,&F'6#\"\"$F)F*F)F+F)F*F)&F-6#F8F)F)%(~.~ .~.~GF)*,&F'6#%\"kGF)F*F)F+F)F*F)&F-6#F?F)F)*,%\"MGF)F*F)F+F)F*F)&F-6# %\"GGF)" }{TEXT -1 4 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(m[i]*`.`*g*`.`*x[i],i = 1 .. k) = M*`.`*g*`.`*x[G]; " "6#/-%$SumG6$*,&%\"mG6#%\"iG\"\"\"%\".GF,%\"gGF,F-F,&%\"xG6#F+F,/F+; F,%\"kG*,%\"MGF,F-F,F.F,F-F,&F06#%\"GGF," }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Another way to say this is, that for the purpose of calculating moments, " }{TEXT 259 54 "the mass of the object acts from the centre of gravity" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 " If we know where the centre of gravity of the object is located, and t he total mass M of the object, we can calculate the moment about an ax is from the product " }{XPPEDIT 18 0 "M*g*`.`*x[G];" "6#**%\"MG\"\"\"% \"gGF%%\".GF%&%\"xG6#%\"GGF%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x [G]" "6#&%\"xG6#%\"GG" }{TEXT -1 70 " is the perpendicular distance of the centre of gravity from the axis." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 191 "Conversely, if we can calculate the \+ total moment of the object by adding up the moments of the separate pi eces, and we also know the total mass M of the object, we can calculat e the distance " }{XPPEDIT 18 0 "x[G]" "6#&%\"xG6#%\"GG" }{TEXT -1 42 " of the centre of gravity G from the axis." }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(m[ i]*g*x[i],i=1..k) = M*g*x[G]" "6#/-%$SumG6$*(&%\"mG6#%\"iG\"\"\"%\"gGF ,&%\"xG6#F+F,/F+;F,%\"kG*(%\"MGF,F-F,&F/6#%\"GGF," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 5 "gives" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "x[G]=Sum(m[i]*g*x[i],i=1..k)/M*g" "6#/&%\"xG6#%\"GG* (-%$SumG6$*(&%\"mG6#%\"iG\"\"\"%\"gGF1&F%6#F0F1/F0;F1%\"kGF1%\"MG!\"\" F2F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "We can divide the top and bottom on the right hand side by g to get " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[G]= Sum(m[i]*x[i],i=1..k)/M" "6#/ &%\"xG6#%\"GG*&-%$SumG6$*&&%\"mG6#%\"iG\"\"\"&F%6#F0F1/F0;F1%\"kGF1%\" MG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 26 "Calculating the distances " }{XPPEDIT 18 0 "x[G]" "6#&%\"xG6#%\"GG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[G]" "6#&%\"yG6 #%\"GG" }{TEXT -1 92 " of the centre of gravity G from two perpendicul ar axes, determines its location completely." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "A plane object which has mass, \+ but no thickness, is called a" }{TEXT 259 7 " lamina" }{TEXT -1 222 ". Of course a lamima cannot exist in real life. It is a mathematical id ealisation just as the concepts of point and particle are mathematical idealisations. A point has position and no size, and a particle is a \+ point mass." }}{PARA 0 "" 0 "" {TEXT -1 71 "The object in the precedin g discussion could be thought of as a lamina." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 14 "uniform lamina" }{TEXT -1 74 " has its mass unifirmly distributed so that equ al areas have equal masses." }}{PARA 0 "" 0 "" {TEXT -1 16 "We use the term " }{TEXT 259 7 "density" }{TEXT -1 55 " in connection with lamin as to represent the notion of " }{TEXT 259 18 "mass per unit area" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 8 "centroid" }{TEXT -1 120 " of a plane s hape, such as a triangle or square, is the centre of gravity of a unif orm lamina which has the given shape." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "The centroid of a triangle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 68 "Consider a triangle located in the coordinate plane with \+ vertices at" }{XPPEDIT 18 0 " ``(0,2)" "6#-%!G6$\"\"!\"\"#" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(6,0)" "6#-%!G6$\"\"'\"\"!" }{TEXT -1 4 " and " }{XPPEDIT 18 0 "``(0,-2)" "6#-%!G6$\"\"!,$\"\"#!\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plots[polygonplot]([[0,2],[6,0],[0,-2]],\ncolor=COLOR(RGB,.9,. 9,.95),labels=[`x`,`y`]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 354 261 261 {PLOTDATA 2 "6%-%)POLYGONSG6#7%7$$\"\"!F)$\"\"#F)7$$\"\"'F)F(7$F($!\"# F)-%+AXESLABELSG6$%\"xG%\"yG-%&COLORG6&%$RGBG$\"\"*!\"\"F;$\"#&*F1" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "To find the centroid of this tr iangle we consider a uniform triangular lamina, with density " } {XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 41 ", having the shape of the given triangle." }}{PARA 0 "" 0 "" {TEXT -1 77 "Since the area of the triangle is 12 square units, the mass of the lamina is " }{XPPEDIT 18 0 "12*rho" "6#*&\"#7\"\"\"%$rhoGF%" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 381 302 302 {PLOTDATA 2 "6*-%)POLYGONSG 6%7'7$$\"#E!\"\"$!+LLLL6!\"*7$F($\"+LLLL6F-7$$\"#FF*$\"+++++6F-7$F2$!+ ++++6F-F'-%&COLORG6&%$RGBG$\"\")F*F=$\"#&)!\"#-%&STYLEG6#%,PATCHNOGRID G-%'CURVESG6%7&7$$\"\"!FL$\"\"#FL7$$\"\"'FLFK7$FK$FAFLFJ-F:6&F<$\"\"\" F*$\"\"&F*FL-%*THICKNESSG6#FW-F$6$FI-F:6&F<$\"\"*F*F[o$\"#&*FA-FG6%7'7 $$\"33+++++++E!#<$!3%******HLLL8\"Feo7$Fco$\"3%******HLLL8\"Feo7$$\"3; +++++++FFeo$\"33+++++++6Feo7$F\\p$!33+++++++6FeoFbo-%'COLOURG6&F " 0 "" {MPLTEXT 1 0 123 "Int((4-2*x/3),x=0..6);\nA : = value(%);\nW := A*rho*g;\nrho*g*Int((4-2*x/3)*x,x=0..6);\nM[x]:=valu e(%);\nx[G]='M[x]/W';\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$,&\"\"%\"\"\"*(\"\"#F(\"\"$!\"\"%\"xGF(F,/F-;\"\"!\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG,$*(\"#7\"\"\"%$rhoGF(%\"gGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%$rhoG\"\"\"%\"gGF%-%$IntG6$*&,&\"\"%F%*(\"\"#F%\" \"$!\"\"%\"xGF%F0F%F1F%/F1;\"\"!\"\"'F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"MG6#%\"xG,$*(\"#C\"\"\"%$rhoGF+%\"gGF+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"xG6#%\"GG*&&%\"MG6#F%\"\"\"%\"WG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#%\"GG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 67 ": The centroid of a triangle lie s at the intersection point of the " }{TEXT 259 7 "medians" }{TEXT -1 783 " of the triangle. A triangular lamina will balance on a knife edg e located at a median. To see why, imagine the triangle sliced into na rrow strips parallel to the side of the triangle which the median bise cts. Then each strip balances at its mid-point, and the collection of \+ all such mid-points \"form\" the median, as we pass to the limit by co nsidering the number of strips to tend to infinity and their width to \+ tend to 0. Since this holds for each of the three medians, the single \+ balance point for the triangular lamina, which is its centre of gravit y, lies on each of the three medians, and so must lie at their point o f intersection. Incidently, this shows that the three medians intersec t at a single point. Vector methods can be used to show that this inte rsection point lies " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\" " }{TEXT -1 153 " of the way along the median from the associated vert ex of the triangle to the bisection point where the median meets the o pposite side of the triangle. " }}{PARA 0 "" 0 "" {TEXT -1 102 "In the picture the mid-point K of the strip PQ lies on the median MC, where \+ M is the mid-point of AB. " }}{PARA 0 "" 0 "" {TEXT -1 55 "The three m edians intersect at the centroid G and GC = " }{XPPEDIT 18 0 "2/3" "6# *&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 " MC. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 454 306 306 {PLOTDATA 2 "67-%)POLYGONSG6$7'7 $$\"$k#!\"#$\"$o$F*7$$\"$s#F*$\"$k$F*7$$\"$3\"F*$\"#OF*7$$\"#'*F*$\"#K F*F'-%&COLORG6&%$RGBG$\"\")!\"\"F@$\"#&)F*-%'CURVESG6%7'7$$\"38++++++S E!#<$\"3;++++++!o$FL7$$\"3>++++++?FFL$\"38++++++SOFL7$$\"33++++++!3\"F L$\"3')*************f$!#=7$$\"3k*************f*FY$\"31+++++++KFYFI-F=6 &F?$\"\"\"FB$\"\"&FB\"\"!-%*THICKNESSG6#F\\o-FF6%7$7$$F\\oF_o$\"\"#F_o 7$$\"\"'F_oFho-%'COLOURG6&F?F_oF_oF_o-%*LINESTYLEG6#\"\"$-FF6%7$7$$Fcp F_oFgo7$Fho$\"\"%F_oF]pF`p-FF6%7$7$$F_oF_oF`q7$FjpFhpF]pF`p-F$6$7%F_qF ipFjo-F=6&F?$\"\"*FBFgq$\"#&*F*-FF6%FdqFinF`o-FF6&7$7$$\"33++++++]=FLF ho7$$\"3_mmmmmmmEFLFho-%'SYMBOLG6#%'CIRCLEGF]p-%&STYLEG6#%&POINTG-FF6& F_r-Fgr6#%(DIAMONDGF]pFjr-FF6&F_r-Fgr6#%&CROSSGF]pFjr-%%TEXTG6%7$$FBFB F\\tQ\"A6\"F]p-Fis6%7$$\"$%>F*$\"#UFBQ\"BF^tF]p-Fis6%7$$\"$:'F*FhoQ\"C F^tF]p-Fis6%7$$\"#\")F*$\"$0#F*Q\"MF^tF]p-Fis6%7$$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "The centroid of a region in the coordinate plane" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 106 "In the calculation of the centroid of a plane shape, we usuall y avoid referring to a corresponding lamina." }}{PARA 0 "" 0 "" {TEXT -1 76 "We use the total area of the shape in place of the total mass, \+ and consider " }{TEXT 259 15 "moments of area" }{TEXT -1 25 " instead \+ of true moments." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Suppose that we have a region in the coordinate plane des cribed as the region under a graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG -%\"fG6#%\"xG" }{TEXT -1 32 " between the two vertical lines " } {XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "Here we assume that the graph is above the " }{TEXT 299 1 "x" } {TEXT -1 10 " axis for " }{TEXT 296 1 "x" }{TEXT -1 9 " between " } {TEXT 297 1 "a" }{TEXT -1 5 " and " }{TEXT 298 1 "b" }{TEXT -1 36 ", o r at least does not go below the " }{TEXT 300 1 "x" }{TEXT -1 24 " axi s in this interval. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 506 392 392 {PLOTDATA 2 "6B-%'CURVESG6%7S7$$\"\"!F)$\"\"#F)7$$\"3s**** **\\i9Rl!#>$\"3RWRo@!Q@+#!#<7$$\"3/++vVA)GA\"!#=$\"30&R5\\?xu+#F27$$\" 3+++]Peui=F6$\"3\"yhv-6\\t,#F27$$\"3A++]i3&o]#F6$\"3![3Mi]@9.#F27$$\"3 %)***\\(oX*y9$F6$\"3mG)z5?Y&\\?F27$$\"3z***\\P9CAu$F6$\"3GF.r27-q?F27$ $\"3!)***\\P*zhdVF6$\"3')os*GfS*\\F6$\"3d%R`h8.Z7# F27$$\"3$)***\\(=$f%GcF6$\"3s=T^rxRe@F27$$\"3Q+++Dy,\"G'F6$\"3kQ)*eCfD (>#F27$$\"33++]77Jpmq&oV#F27$$\"3,++]siL-5F2$\"3*p f%e,!RB]#F27$$\"3-+++!R5'f5F2$\"3)f(zH*3(QhDF27$$\"3)***\\P/QBE6F2$\"3 )G\\e5H,Uj#F27$$\"3!******\\\"o?&=\"F2$\"3$>AhrfdBq#F27$$\"31+]Pa&4*\\ 7F2$\"3wb*eq%p8\"y#F27$$\"33+]7j=_68F2$\"3cNus)zW+'GF27$$\"33++vVy!eP \"F2$\"3Q7?Y6OUYHF27$$\"34+](=WU[V\"F2$\"3_1H];kQHIF27$$\"3)****\\7B>& )\\\"F2$\"3UXI@V*zF7$F27$$\"3)***\\P>:mk:F2$\"3*=W1^$G3CKF27$$\"3'*** \\iv&QAi\"F2$\"3K9H7)**GeJ$F27$$\"31++vtLU%o\"F2$\"38mw,^5k=MF27$$\"3! ******\\Nm'[F2$\"3EH*G*p\") \\\")QF27$$\"3z*****\\@80+#F2$\"3==[phl-,SF27$$\"31++]7,Hl?F2$\"3iJ^RC ;rKTF27$$\"3()**\\P4w)R7#F2$\"3Cj#*Q#ohcD%F27$$\"3;++]x%f\")=#F2$\"3wO l[\\4-%R%F27$$\"3!)**\\P/-a[AF2$\"3Z#QX`_mz_%F27$$\"3/+](=Yb;J#F2$\"3] f#3s[v=n%F27$$\"3')****\\i@OtBF2$\"3:QL>xRU;[F27$$\"3')**\\PfL'zV#F2$ \"37+++!*>=+DF2$\"3.\"=g:*\\XD^F27$$\"3-++DE&4Qc#F2 $\"3?D^WV'flG&F27$$\"3=+]P%>5pi#F2$\"3%3lbYeG.X&F27$$\"39+++bJ*[o#F2$ \"37FH)oiDVg&F27$$\"33++Dr\"[8v#F2$\"3kf.sz$e\\y&F27$$\"3++++Ijy5GF2$ \"3[MuX'*)f-&fF27$$\"31+]P/)fT(GF2$\"3^pCa!H(RIhF27$$\"31+]i0j\"[$HF2$ \"3m'=\")QPtlI'F27$$\"\"$F)$\"3++++++++lF2-%'COLOURG6&%$RGBG$\"*++++\" !\")F(F(-%*THICKNESSG6#F+-F$6$7$7$$\"3++++++++]F6F(7$Fg[l$\"3+++++++D@ F2-Fjz6&F\\[lF)F)F)-F$6$7$7$$\"3++++++++DF2F(7$Fb\\l$\"3+++++++D^F2F\\ \\l-F$6$7$7$$\"33++++++!f\"F2F(7$F[]l$\"37+++++0kKF2F\\\\l-F$6$7$7$$\" 3#*************\\;F2F(7$Fd]l$\"31+++++DhLF2F\\\\l-%)POLYGONSG6%7'7$$\" $f\"!\"#F(7$F^^l$\"+++0kK!\"*7$$\"$i\"F`^l$\"+++?7LFd^l7$$\"$l\"F`^l$ \"+++DhLFd^l7$F[_lF(-%&COLORG6&F\\[l$\"\")!\"\"Fc_l$\"#$)F`^l-%&STYLEG 6#%,PATCHNOGRIDG-Fj]l6%7U7$$\"\"&Fe_lF(7$$\"+++++]!#5$\"++++D@Fd^l7$$ \"+$3VfV&Fe`l$\"+'QZx9#Fd^l7$$\"+H[D:eFe`l$\"+Wf3p@Fd^l7$$\"+e0$=C'Fe` l$\"+WA![>#Fd^l7$$\"+3RBrmFe`l$\"+4o_AAFd^l7$$\"+zjf)4(Fe`l$\"+`.&>D#F d^l7$$\"+'4;[\\(Fe`l$\"+U8'3G#Fd^l7$$\"+j'y]!zFe`l$\"+V8X7BFd^l7$$\"+' zs$H$)Fe`l$\"+cA*oM#Fd^l7$$\"+7iI_()Fe`l$\"+?V,$Q#Fd^l7$$\"+<_M(=*Fe`l $\"+hl.ACFd^l7$$\"+3y_q&*Fe`l$\"+8](zX#Fd^l7$$\"+l+>+5Fd^l$\"+$3!>+DFd ^l7$$\"+vW]V5Fd^l$\"+&z]Wa#Fd^l7$$\"+NfC&3\"Fd^l$\"+q$z))e#Fd^l7$$\"+! =^J7\"Fd^l$\"+'GM2j#Fd^l7$$\"+#=C#o6Fd^l$\"+pQP#o#Fd^l7$$\"+FpS17Fd^l$ \"+O)3xs#Fd^l7$$\"+OD#3D\"Fd^l$\"+4&yAy#Fd^l7$$\"+xy8!H\"Fd^l$\"+qyAKG Fd^l7$$\"+OIFL8Fd^l$\"+&\\3)))GFd^l7$$\"+4zMu8Fd^l$\"+(3;W%HFd^l7$$\"+ H_?<9Fd^l$\"+J`B/IFd^l7$$\"+G;cc9Fd^l$\"+))eygIFd^l7$$\"+@G,*\\\"Fd^l$ \"+>(>N7$Fd^l7$$\"+!o2Ja\"Fd^l$\"+b1f!>$Fd^l7$$\"+%Q#\\\"e\"Fd^l$\"+!3 f0D$Fd^l7$$\"+;*[Hi\"Fd^l$\"+#f\")pJ$Fd^l7$$\"+qvxl;Fd^l$\"+cuS(Q$Fd^l 7$$\"+_qn2Fd^l$\"+t@^NQFd^l7$$\"+&)Hxe>Fd^l$\"+.eR=RFd^l7$$\" +.o-**>Fd^l$\"+zS0)*RFd^l7$$\"+TO5T?Fd^l$\"+P?0$3%Fd^l7$$\"+U9C#3#Fd^l $\"+6Z'y;%Fd^l7$$\"+1*3`7#Fd^l$\"+t*o%eUFd^l7$$\"+$*zym@Fd^l$\"+/^[ZVF d^l7$$\"+^j?4AFd^l$\"+]jHSWFd^l7$$\"+jMF^AFd^l$\"+-h6MXFd^l7$$\"+q(G** G#Fd^l$\"+')o)=i%Fd^l7$$\"+9@BMBFd^l$\"+\"y>Vs%Fd^l7$$\"+`v&QP#Fd^l$\" +U)*f<[Fd^l7$$\"+Ol5;CFd^l$\"+(R&y=\\Fd^l7$$\"+/UacCFd^l$\"+7ZI<]Fd^l7 $$\"+++++DFd^l$\"++++D^Fd^l7$$\"#DFe_lF(-Fa_l6&F\\[l$\"\"*Fe_lF]`m$\"# $*F`^lFh_l-F$6&7#7$$\"36++++++?;F2$\"3!***********4c;F2-%'SYMBOLG6#%'C IRCLEGF\\\\l-Fi_l6#%&POINTG-F$6&Fc`m-Fj`m6#%(DIAMONDGF\\\\lF]am-F$6&Fc `m-Fj`m6#%&CROSSGF\\\\lF]am-F$6%7$7$Fe`m$\"3\")**********>7LF27$$\"3#) *************H#F2F^bmF\\\\l-%*LINESTYLEGFb[l-F$6%7$Fd`m7$F*Fg`mF\\\\lF cbm-F$6%7$7$Fe`mF(7$Fe`m$!3%**************\\\"F6F\\\\lFcbm-F$6&7$7$F]` m$\"+++5c;Fd^l7$Ff^lFdcm7%7$$\"+++Si:Fd^l$\"+++5@Fd^l7$F`emFh^l7%7$ $\"++++v@Fd^l$\"++?r.KFd^lFdem7$$\"++++DAFd^lFiemFh_lF\\\\l-F$6&7$7$F` em$\"+++5c8Fd^l7$F`emF(7%7$F\\fm$\"*+!)[3\"Fd^lFdfm7$FgemFgfmFh_lF\\\\ l-F$6&7$7$$\"#>Fe_l$\"+++0y6Fd^l7$F^gmFdcm7%7$$\"++++v=Fd^l$\"++DR%e\" Fd^lFbgm7$$\"++++D>Fd^lFggmFh_lF\\\\l-F$6&7$7$F^gm$\"+++]!y%Fe`l7$F^gm F(7%7$Fjgm$\"*+]2<(Fe`lFbhm7$FegmFehmFh_lF\\\\l-%%TEXTG6%7$Fez$\"#fFe_ lQ)y~=~f(x)6\"Fiz-Fihm6%7$Ff^l$!\"$Fe_lQ\"xF_imF\\\\l-Fihm6%7$$Fe_lFe_ l$\"#jFe_lQ\"yF_imF\\\\l-Fihm6%7$F``l$!#:F`^lQ&x~=~aF_imF\\\\l-Fihm6%7 $Fi_mF`jmQ&x~=~bF_imF\\\\l-Fihm6%7$F^gm$\"\"\"F)Q%f(x)F_imF\\\\l-Fihm6 %7$F^gmF_`mQ#__F_imF\\\\l-Fihm6%7$$\"#\")F`^l$\"# " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1 " }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 375 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 100 "Find the coordinates of the centroid of the pl ane region in the first quadrant bounded by the curve " }{XPPEDIT 18 0 "y=4-x^2" "6#/%\"yG,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 9 " and the " }{TEXT 377 1 "x" }{TEXT -1 5 " and " }{TEXT 378 1 "y" }{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 376 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "f := x -> 4-x^2:\nplot s[polygonplot]([[0,0],op(op(1,op(1,plot(f(x),x=0..2))))],\n color=C OLOR(RGB,.9,.7,.8),labels=[`x`,`y`],tickmarks=[3,5]);\n" }}{PARA 13 " " 1 "" {GLPLOT2D 271 280 280 {PLOTDATA 2 "6&-%)POLYGONSG6#7T7$$\"\"!F) F(7$F($\"\"%F)7$$\"+L3VfV!#6$\"+O&*4)*R!\"*7$$\"+#H[D:)F0$\"+'f`L*RF37 $$\"+e0$=C\"!#5$\"+p&yX)RF37$$\"+3RBr;F<$\"+s(p?(RF37$$\"+zjf)4#F<$\"+ K*ef&RF37$$\"+'4;[\\#F<$\"+E*ex$RF37$$\"+i'y]!HF<$\"+!=0c\"RF37$$\"+'z s$HLF<$\"+oF:*)QF37$$\"+7iI_PF<$\"+\")>?fQF37$$\"+<_M(=%F<$\"++9mCQF37 $$\"+3y_qXF<$\"+cF5\"z$F37$$\"+]1!>+&F<$\"+**)4)\\PF37$$\"+]Z/NaF<$\"+ ')Gg/PF37$$\"+]$fC&eF<$\"+'>([dOF37$$\"+'z6:B'F<$\"+2Eo6OF37$$\"+<=C#o 'F<$\"+VkZ`NF37$$\"+n#pS1(F<$\"+a#*)4]$F37$$\"+i`A3vF<$\"+>bEOMF37$$\" +n(y8!zF<$\"+O@ovLF37$$\"+j.tK$)F<$\"+Zgl0LF37$$\"+)3zMu)F<$\"+Md^NKF3 7$$\"+#H_?<*F<$\"+oXteJF37$$\"+zihl&*F<$\"+_)*)\\3$F37$$\"+3#G,***F<$ \"+%Qt>+$F37$$\"+!o2J/\"F3$\"+pj#>\"HF37$$\"+%Q#\\\"3\"F3$\"+CUPIGF37$ $\"+;*[H7\"F3$\"+Kd)*QFF37$$\"+qvxl6F3$\"+dE'4k#F37$$\"+_qn27F3$\"+Ph^ TDF37$$\"+cp@[7F3$\"+IW&>W#F37$$\"+2'HKH\"F3$\"+#=dvK#F37$$\"+xanL8F3$ \"+B(48A#F37$$\"+v+'oP\"F3$\"+MjD/@F37$$\"+S<*fT\"F3$\"+$Rn\\*>F37$$\" +&)Hxe9F3$\"+y8)>(=F37$$\"+.o-*\\\"F3$\"+W'=Hv\"F37$$\"+TO5T:F3$\"+n&* *\\i\"F37$$\"+U9C#e\"F3$\"+??^'\\\"F37$$\"+1*3`i\"F3$\"+f4Pe8F37$$\"+$ *zym;F3$\"+&y<=A\"F37$$\"+^j?4F3$\"+F3$\"+)yZ$> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=4-x^2" "6#/-%\"fG6#%\"xG,&\"\"% \"\"\"*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "T he area of the region is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 2);" "6#/%\"AG-%$IntG6$-%\"fG6#% \"xG/F+;\"\"!\"\"#" }{XPPEDIT 18 0 " ``= Int(4-x^2,x = 0 .. 2)" "6#/%! G-%$IntG6$,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"/F,;\"\"!F-" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=4*x-x^3/3" "6#/%! G,&*&\"\"%\"\"\"%\"xGF(F(*&F)\"\"$F+!\"\"F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([2,``],[0,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7 $\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 8-8/3" "6#/%!G,&\"\")\"\"\"*&F&F'\"\"$!\"\"F*" } {XPPEDIT 18 0 "``=16/3" "6#/%!G*&\"#;\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The ( area) moment of the region about the " }{TEXT 379 1 "y" }{TEXT -1 9 " \+ axis is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = In t(x*f(x),x = 0 .. 2);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%\"fG6# F,F-/F,;\"\"!\"\"#" }{XPPEDIT 18 0 "``= Int(x*(4-x^2),x = 0 .. 2)" "6# /%!G-%$IntG6$*&%\"xG\"\"\",&\"\"%F**$F)\"\"#!\"\"F*/F);\"\"!F." } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=I nt(4*x-x^3,x = 0 .. 2)" "6#/%!G-%$IntG6$,&*&\"\"%\"\"\"%\"xGF+F+*$F,\" \"$!\"\"/F,;\"\"!\"\"#" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*x^2-x^4/4" "6#/%!G,&*&\"\"#\"\"\"*$%\"xGF'F (F(*&F*\"\"%F,!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2,`` ],[0,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F(" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=8-4" "6#/%!G,&\"\" )\"\"\"\"\"%!\"\"" }{XPPEDIT 18 0 "``=4" "6#/%!G\"\"%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The m oment of the region about the " }{TEXT 380 1 "x" }{TEXT -1 10 " axis i s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int(f(x )^2/2,x = 0 .. 2);" "6#/&%\"MG6#%\"xG-%$IntG6$*&-%\"fG6#F'\"\"#F/!\"\" /F';\"\"!F/" }{XPPEDIT 18 0 "``= Int((4-x^2)^2/2,x = 0 .. 2)" "6#/%!G- %$IntG6$*&,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"F.F.F//F-;\"\"!F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((16- 8*x^2+x^4)/2,x = 0 .. 2);" "6#/%!G-%$IntG6$*&,(\"#;\"\"\"*&\"\")F+*$% \"xG\"\"#F+!\"\"*$F/\"\"%F+F+F0F1/F/;\"\"!F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Int(8-4*x^2+x^4/2,x \+ = 0 .. 2);" "6#/%!G-%$IntG6$,(\"\")\"\"\"*&\"\"%F**$%\"xG\"\"#F*!\"\"* &F.F,F/F0F*/F.;\"\"!F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 8*x-4*x^3/3+x^5/10;" "6#/%!G,(*&\"\")\"\"\" %\"xGF(F(*(\"\"%F(*$F)\"\"$F(F-!\"\"F.*&F)\"\"&\"#5F.F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[0, ``])" "6#-%*PIECEWISEG6$7$\" \"#%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 16-32/3+16/5;" "6#/%!G,(\"#;\"\" \"*&\"#KF'\"\"$!\"\"F+*&F&F'\"\"&F+F'" }{XPPEDIT 18 0 "`` = 16-10;" "6 #/%!G,&\"#;\"\"\"\"#5!\"\"" }{XPPEDIT 18 0 "2/3+3;" "6#,&*&\"\"#\"\"\" \"\"$!\"\"F&F'F&" }{XPPEDIT 18 0 "1/5=9-2/3+1/5" "6#/*&\"\"\"F%\"\"&! \"\",(\"\"*F%*&\"\"#F%\"\"$F'F'*&F%F%F&F'F%" }{XPPEDIT 18 0 "``=8" "6# /%!G\"\")" }{XPPEDIT 18 0 "1/3+1/5=8" "6#/,&*&\"\"\"F&\"\"$!\"\"F&*&F& F&\"\"&F(F&\"\")" }{XPPEDIT 18 0 "8/15=128/15" "6#/*&\"\")\"\"\"\"#:! \"\"*&\"$G\"F&F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 381 1 "x" }{TEXT -1 33 " coo rdinate of the centroid is " }{XPPEDIT 18 0 "conjugate(x) = M[y]/A; " "6#/-%*conjugateG6#%\"xG*&&%\"MG6#%\"yG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4/``(16/3);" "6#/%!G*&\"\"%\"\"\"-F$6#*&\"# ;F'\"\"$!\"\"F-" }{XPPEDIT 18 0 "``=4*``(3/16)" "6#/%!G*&\"\"%\"\"\"-F $6#*&\"\"$F'\"#;!\"\"F'" }{XPPEDIT 18 0 "``=3/4" "6#/%!G*&\"\"$\"\"\" \"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 382 1 "y" }{TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(y) = M[x]/A;" "6#/-% *conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`` = ``(128/15)/``(16/3);" "6#/%!G*&-F$6#*&\"$G\"\"\"\" \"#:!\"\"F*-F$6#*&\"#;F*\"\"$F,F," }{XPPEDIT 18 0 "`` = ``(128/15)*``( 3/16);" "6#/%!G*&-F$6#*&\"$G\"\"\"\"\"#:!\"\"F*-F$6#*&\"\"$F*\"#;F,F* " }{XPPEDIT 18 0 "`` = 8/5;" "6#/%!G*&\"\")\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at th e point" }{XPPEDIT 18 0 "``(3/4,8/5);" "6#-%!G6$*&\"\"$\"\"\"\"\"%!\" \"*&\"\")F(\"\"&F*" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "f := x -> 4-x^2:\nInt(f(x ),x=0..2);\nA := value(%);\nInt(x*f(x),x=0..2);\nMy := value(%);\nInt( f(x)^2/2,x=0..2);\nexpand(%);\nMx := value(%);\nxG := My/A;\nyG := Mx/ A;\nprint(`The centroid is located at the point .. `,``(xG,yG));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&\"\"%\"\"\"*$)%\"xG\"\"#F(! \"\"/F+;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG#\"#;\"\"$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&\"\"%F(*$)F' \"\"#F(!\"\"F(/F';\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#!\"\",&\"\"% \"\"\"*$)%\"xGF(F,F)F(F,/F/;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&#\"\"\"\"\"#F&-%$IntG6$,(\"#;F&*&\"\")F&)%\"xGF'F&!\"\"*$)F0\"\"% F&F&/F0;\"\"!F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\"$G\"\" #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\"\"$\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#yGG#\"\")\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%IThe~centroid~is~located~at~the~point~..~G-%!G6$#\"\"$\"\"%#\" \")\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can draw a picture which shows the centroid of the region." }} {PARA 0 "" 0 "" {TEXT -1 65 "Can you imagine the region balancing hori zontally at this point?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "f := x -> 4-x^2:\nxG := 3/4;\nyG := 8/5;\np1 := plots[polygonplot ]([[0,0],op(op(1,op(1,plot(f(x),x=0..2))))],\n color=COLOR(RGB,.9,. 7,.8)):\np2 := plot([[[xG,yG]]$3],style=point,symbol=[circle,diamond,c ross],color=black):\nplots[display]([p1,p2],labels=[`x`,`y`],tickmarks =[3,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\"\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG#\"\")\"\"&" }}{PARA 13 "" 1 "" {GLPLOT2D 226 223 223 {PLOTDATA 2 "6)-%)POLYGONSG6$7T7$$\"\"!F)F(7$F($ \"\"%F)7$$\"+L3VfV!#6$\"+O&*4)*R!\"*7$$\"+#H[D:)F0$\"+'f`L*RF37$$\"+e0 $=C\"!#5$\"+p&yX)RF37$$\"+3RBr;F<$\"+s(p?(RF37$$\"+zjf)4#F<$\"+K*ef&RF 37$$\"+'4;[\\#F<$\"+E*ex$RF37$$\"+i'y]!HF<$\"+!=0c\"RF37$$\"+'zs$HLF<$ \"+oF:*)QF37$$\"+7iI_PF<$\"+\")>?fQF37$$\"+<_M(=%F<$\"++9mCQF37$$\"+3y _qXF<$\"+cF5\"z$F37$$\"+]1!>+&F<$\"+**)4)\\PF37$$\"+]Z/NaF<$\"+')Gg/PF 37$$\"+]$fC&eF<$\"+'>([dOF37$$\"+'z6:B'F<$\"+2Eo6OF37$$\"+<=C#o'F<$\"+ VkZ`NF37$$\"+n#pS1(F<$\"+a#*)4]$F37$$\"+i`A3vF<$\"+>bEOMF37$$\"+n(y8!z F<$\"+O@ovLF37$$\"+j.tK$)F<$\"+Zgl0LF37$$\"+)3zMu)F<$\"+Md^NKF37$$\"+# H_?<*F<$\"+oXteJF37$$\"+zihl&*F<$\"+_)*)\\3$F37$$\"+3#G,***F<$\"+%Qt>+ $F37$$\"+!o2J/\"F3$\"+pj#>\"HF37$$\"+%Q#\\\"3\"F3$\"+CUPIGF37$$\"+;*[H 7\"F3$\"+Kd)*QFF37$$\"+qvxl6F3$\"+dE'4k#F37$$\"+_qn27F3$\"+Ph^TDF37$$ \"+cp@[7F3$\"+IW&>W#F37$$\"+2'HKH\"F3$\"+#=dvK#F37$$\"+xanL8F3$\"+B(48 A#F37$$\"+v+'oP\"F3$\"+MjD/@F37$$\"+S<*fT\"F3$\"+$Rn\\*>F37$$\"+&)Hxe9 F3$\"+y8)>(=F37$$\"+.o-*\\\"F3$\"+W'=Hv\"F37$$\"+TO5T:F3$\"+n&**\\i\"F 37$$\"+U9C#e\"F3$\"+??^'\\\"F37$$\"+1*3`i\"F3$\"+f4Pe8F37$$\"+$*zym;F3 $\"+&y<=A\"F37$$\"+^j?4F3$\"+F3$\"+)yZ$> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 257 "" 0 "" {TEXT 258 8 "Ques tion" }{TEXT 265 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 100 "Find the coordi nates of the centroid of the plane region in the first quadrant bounde d by the curve " }{XPPEDIT 18 0 "y = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\" \"" }{TEXT -1 7 " , the " }{TEXT 341 1 "x" }{TEXT -1 20 " axis and the lines " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 266 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f := x -> 1/x:\nplots[polygonplot]([[1,0],op(op(1,op (1,plot(f(x),x=1..3)))),[3,0]],\n color=COLOR(RGB,.6,.7,1),labels=[ `x`,`y`],view=[0..4,0..1.2],\n tickmarks=[5,3]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 440 174 174 {PLOTDATA 2 "6'-%)POLYGONSG6#7U7$$\"\"\"\" \"!$F*F*7$F(F(7$$\"+3VfV5!\"*$\"+pwE#e*!#57$$\"+$[D:3\"F0$\"+\\\"*>Y#* F37$$\"+cI=C6F0$\"+Z%[`*))F37$$\"+\"RBr;\"F0$\"+gV2o&)F37$$\"+Q'f)47F0 $\"+8;Ul#)F37$$\"+5;[\\7F0$\"+v!>L+)F37$$\"+my]!H\"F0$\"+)G())[xF37$$ \"+!GPHL\"F0$\"+k%GA](F37$$\"+@1Bv8F0$\"+iw]rsF37$$\"+AXt=9F0$\"+[]`[q F37$$\"+\"y_qX\"F0$\"+f$pJ'oF37$$\"+l+>+:F0$\"+S?#em'F37$$\"+vW]V:F0$ \"+0KwykF37$$\"+NfC&e\"F0$\"+k%p\"3jF37$$\"+!=^Ji\"F0$\"+zb&3;'F37$$\" +#=C#o;F0$\"+\"=)R%*fF37$$\"+FpS1F0$\"+'oDf@&F37$$\"+G;cc>F0$\"+5p+6^F37$$\"+@G,* *>F0$\"+n\"pC+&F37$$\"+!o2J/#F0$\"+JY]%*[F37$$\"+%Q#\\\"3#F0$\"+*HXU![ F37$$\"+;*[H7#F0$\"+N*G/r%F37$$\"+qvxl@F0$\"+F$zsh%F37$$\"+_qn2AF0$\"+ D!['HXF37$$\"+cp@[AF0$\"+*HpzW%F37$$\"+2'HKH#F0$\"+YBmgVF37$$\"+xanLBF 0$\"+^f3&G%F37$$\"+v+'oP#F0$\"+d7B2UF37$$\"+S<*fT#F0$\"+**p3RTF37$$\"+ &)HxeCF0$\"+]#pq1%F37$$\"+.o-*\\#F0$\"+Axb,SF37$$\"+TO5TDF0$\"+qzHNRF3 7$$\"+U9C#e#F0$\"+nXgsQF37$$\"+1*3`i#F0$\"+jb24QF37$$\"+$*zymEF0$\"+#R H)\\PF37$$\"+^j?4FF0$\"+zy6\"p$F37$$\"+jMF^FF0$\"+)\\!oMOF37$$\"+q(G** y#F0$\"++4K%e$F37$$\"+9@BMGF0$\"+JDHGNF37$$\"+`v&Q(GF0$\"+&eV'zMF37$$ \"+Ol5;HF0$\"+%**H#HMF37$$\"+/UacHF0$\"+TsK#Q$F37$$\"\"$F*$\"+LLLLLF37 $FezF+-%+AXESLABELSG6$%\"xG%\"yG-%&COLORG6&%$RGBG$\"\"'!\"\"$\"\"(Fe[l F)-%*AXESTICKSG6$\"\"&Ffz-%%VIEWG6$;F+$\"\"%F*;F+$\"#7Fe[l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/x;" "6#/- %\"fG6#%\"xG*&\"\"\"F)F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The area of the region is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 1 .. 3);" "6#/%\"AG-%$IntG6$ -%\"fG6#%\"xG/F+;\"\"\"\"\"$" }{XPPEDIT 18 0 "``= Int(1/x,x = 1 .. 3) " "6#/%!G-%$IntG6$*&\"\"\"F)%\"xG!\"\"/F*;F)\"\"$" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(x);" "6#/%!G- %#lnG6#%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[1, `` ]);" "6#-%*PIECEWISEG6$7$\"\"$%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(3)-ln(1);" "6#/%!G, &-%#lnG6#\"\"$\"\"\"-F'6#F*!\"\"" }{XPPEDIT 18 0 "`` = ln(3);" "6#/%!G -%#lnG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " } {TEXT 342 1 "y" }{TEXT -1 9 " axis is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*f(x),x = 1 .. 3);" "6#/&%\"MG6#%\"y G-%$IntG6$*&%\"xG\"\"\"-%\"fG6#F,F-/F,;F-\"\"$" }{XPPEDIT 18 0 "``= In t(x*``(1/x),x = 1 .. 3)" "6#/%!G-%$IntG6$*&%\"xG\"\"\"-F$6#*&F*F*F)!\" \"F*/F);F*\"\"$" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(1,x = 1 .. 3);" "6#/%!G-%$IntG6$\"\"\"/%\"xG;F (\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x;" "6#/%!G%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE ([3, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"$%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3-1;" "6 #/%!G,&\"\"$\"\"\"F'!\"\"" }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " }{TEXT 343 1 "x" } {TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 1 .. 3);" "6#/&%\"MG6#%\"xG-%$IntG6$*&-% \"fG6#F'\"\"#F/!\"\"/F';\"\"\"\"\"$" }{XPPEDIT 18 0 "``= Int(``(1/2)*` `(1/x)^2,x = 1 .. 3)" "6#/%!G-%$IntG6$*&-F$6#*&\"\"\"F,\"\"#!\"\"F,*$- F$6#*&F,F,%\"xGF.F-F,/F3;F,\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(1/(2*x^2),x = 1 .. 3);" "6#/%! G-%$IntG6$*&\"\"\"F)*&\"\"#F)*$%\"xGF+F)!\"\"/F-;F)\"\"$" }{XPPEDIT 18 0 " ``=Int(x^(-2)/2,x=1..3)" "6#/%!G-%$IntG6$*&)%\"xG,$\"\"#!\"\"\" \"\"F,F-/F*;F.\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = x^(-1)/(-2);" "6#/%!G*&)%\"x G,$\"\"\"!\"\"F),$\"\"#F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE ([3, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"$%!G7$\"\"\"F(" }{XPPEDIT 18 0 "`` = -1/(2*x);" "6#/%!G,$*&\"\"\"F'*&\"\"#F'%\"xGF'!\"\"F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[1, ``]);" "6#-%*PIEC EWISEG6$7$\"\"$%!G7$\"\"\"F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1 /6-(-1/2)" "6#/%!G,&*&\"\"\"F'\"\"'!\"\"F),$*&F'F'\"\"#F)F)F)" } {XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 344 1 "x" }{TEXT -1 33 " coordinate of the centroid is " } {XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"M G6#%\"yG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/ln(3) ;" "6#/%!G*&\"\"#\"\"\"-%#lnG6#\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 345 1 "y" }{TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "c onjugate(y) = M[x]/A;" "6#/-%*conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"% \"AG!\"\"" }{XPPEDIT 18 0 "`` = ``(1/3)/ln(3);" "6#/%!G*&-F$6#*&\"\"\" F)\"\"$!\"\"F)-%#lnG6#F*F+" }{XPPEDIT 18 0 "`` = 1/(3*ln(3));" "6#/%!G *&\"\"\"F&*&\"\"$F&-%#lnG6#F(F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is locate d at the point" }{XPPEDIT 18 0 "``(2/ln(3), 1/(3*ln(3)));" "6#-%!G6$*& \"\"#\"\"\"-%#lnG6#\"\"$!\"\"*&F(F(*&F,F(-F*6#F,F(F-" }{TEXT -1 2 " \+ " }{TEXT 383 1 "~" }{TEXT -1 30 " (1.820478453, 0.3034130755). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "f := x -> 1/x:\nInt(f(x),x=1..3);\nA := value(%);\nInt(x*f(x),x=1 ..3);\nMy := value(%);\nInt(f(x)^2/2,x=1..3);\nMx := value(%);\nxG := \+ My/A;\nyG := Mx/A;\nxGf := evalf(evalf(xG,13)):\nyGf := evalf(evalf(yG ,13)):\nprint(`The centroid is located at the point .. `,``(xG,yG)*`~` *``(xGf,yGf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'% \"xG!\"\"/F(;F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%#lnG6# \"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$\"\"\"/%\"xG;F&\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"\"F(*&\"\"#F()%\"xGF*F(!\"\"F(/F,;F(\" \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG,$*&\"\"#\"\"\"-%#lnG6#\"\"$!\"\"F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*&#\"\"\"\"\"$F(*&F(F(-%#lnG6# F)!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%IThe~centroid~is~locat ed~at~the~point~..~G*(-%!G6$,$*&\"\"#\"\"\"-%#lnG6#\"\"$!\"\"F+,$*&#F+ F/F+*&F+F+F,F0F+F+F+%\"|irGF+-F&6$$\"+`%y/#=!\"*$\"+b28MI!#5F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can dr aw a picture which shows the centroid of the region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "f := x - > 1/x:\nxG := 2/ln(3);\nyG := 1/3*1/ln(3);\np1 := plots[polygonplot]([ [1,0],op(op(1,op(1,plot(f(x),x=1..3)))),[3,0]],\n color=COLOR(RGB,. 6,.7,1)):\np2 := plot([[[xG,yG]]$3],style=point,symbol=[circle,diamond ,cross],color=black):\nplots[display]([p1,p2],labels=[`x`,`y`],view=[0 ..4,0..1.2],\n tickmarks=[5,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG,$*&\"\"#\"\"\"-%#lnG6#\"\"$!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*&#\"\"\"\"\"$F(*&F(F(-%#lnG6#F)!\"\"F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 496 200 200 {PLOTDATA 2 "6)-%)POLYGONSG6$7U 7$$\"\"\"\"\"!$F*F*7$F(F(7$$\"+3VfV5!\"*$\"+pwE#e*!#57$$\"+$[D:3\"F0$ \"+\\\"*>Y#*F37$$\"+cI=C6F0$\"+Z%[`*))F37$$\"+\"RBr;\"F0$\"+gV2o&)F37$ $\"+Q'f)47F0$\"+8;Ul#)F37$$\"+5;[\\7F0$\"+v!>L+)F37$$\"+my]!H\"F0$\"+) G())[xF37$$\"+!GPHL\"F0$\"+k%GA](F37$$\"+@1Bv8F0$\"+iw]rsF37$$\"+AXt=9 F0$\"+[]`[qF37$$\"+\"y_qX\"F0$\"+f$pJ'oF37$$\"+l+>+:F0$\"+S?#em'F37$$ \"+vW]V:F0$\"+0KwykF37$$\"+NfC&e\"F0$\"+k%p\"3jF37$$\"+!=^Ji\"F0$\"+zb &3;'F37$$\"+#=C#o;F0$\"+\"=)R%*fF37$$\"+FpS1F0$\"+'oDf@&F37$$\"+G;cc>F0$\"+5p+6^F 37$$\"+@G,**>F0$\"+n\"pC+&F37$$\"+!o2J/#F0$\"+JY]%*[F37$$\"+%Q#\\\"3#F 0$\"+*HXU![F37$$\"+;*[H7#F0$\"+N*G/r%F37$$\"+qvxl@F0$\"+F$zsh%F37$$\"+ _qn2AF0$\"+D!['HXF37$$\"+cp@[AF0$\"+*HpzW%F37$$\"+2'HKH#F0$\"+YBmgVF37 $$\"+xanLBF0$\"+^f3&G%F37$$\"+v+'oP#F0$\"+d7B2UF37$$\"+S<*fT#F0$\"+**p 3RTF37$$\"+&)HxeCF0$\"+]#pq1%F37$$\"+.o-*\\#F0$\"+Axb,SF37$$\"+TO5TDF0 $\"+qzHNRF37$$\"+U9C#e#F0$\"+nXgsQF37$$\"+1*3`i#F0$\"+jb24QF37$$\"+$*z ymEF0$\"+#RH)\\PF37$$\"+^j?4FF0$\"+zy6\"p$F37$$\"+jMF^FF0$\"+)\\!oMOF3 7$$\"+q(G**y#F0$\"++4K%e$F37$$\"+9@BMGF0$\"+JDHGNF37$$\"+`v&Q(GF0$\"+& eV'zMF37$$\"+Ol5;HF0$\"+%**H#HMF37$$\"+/UacHF0$\"+TsK#Q$F37$$\"\"$F*$ \"+LLLLLF37$FezF+-%&COLORG6&%$RGBG$\"\"'!\"\"$\"\"(F`[lF)-%'CURVESG6&7 #7$$\"3juOD`%y/#=!#<$\"3(3zAavIT.$!#=-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6& F][lF*F*F*-%&STYLEG6#%&POINTG-Fd[l6&Ff[l-F_\\l6#%(DIAMONDGFb\\lFe\\l-F d[l6&Ff[l-F_\\l6#%&CROSSGFb\\lFe\\l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6 #%(DEFAULTG-%*AXESTICKSG6$\"\"&Ffz-%%VIEWG6$;F+$\"\"%F*;F+$\"#7F`[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" } {TEXT 267 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 110 "Find the coordinates \+ of the centroid of the triangular plane region in the first quadrant b ounded by the line " }{XPPEDIT 18 0 "y=x/2" "6#/%\"yG*&%\"xG\"\"\"\"\" #!\"\"" }{TEXT -1 6 ", the " }{TEXT 346 1 "x" }{TEXT -1 19 " axis and \+ the line " }{XPPEDIT 18 0 "x = 4" "6#/%\"xG\"\"%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solutio n" }{TEXT 335 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "plots[polygonplot]([[0,0],[ 4,2],[4,0]],\n color=COLOR(RGB,1,.85,.6),labels=[`x`,`y`],t ickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 284 154 154 {PLOTDATA 2 "6&-%)POLYGONSG6#7%7$$\"\"!F)F(7$$\"\"%F)$\"\"#F)7$F+F(-%&COLORG6&%$ RGBG\"\"\"$\"#&)!\"#$\"\"'!\"\"-%*AXESTICKSG6$\"\"&\"\"$-%+AXESLABELSG 6$%\"xG%\"yG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x/2" "6#/-%\"fG6#%\"xG*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 26 "The area of the region is:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 4);" "6#/% \"AG-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!\"\"%" }{XPPEDIT 18 0 "`` = Int(x/2 ,x = 0 .. 4)" "6#/%!G-%$IntG6$*&%\"xG\"\"\"\"\"#!\"\"/F);\"\"!\"\"%" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x ^2/4" "6#/%!G*&%\"xG\"\"#\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([4,``],[0,``])" "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"!F(" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=4 " "6#/%!G\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 101 "That \+ the area of the triangle is 4 can also be seen by using the formula fo r the area of a triangle. " }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment \+ of the region about the " }{TEXT 347 1 "y" }{TEXT -1 10 " axis is: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*f(x),x \+ = 0 .. 4);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%\"fG6#F,F-/F,;\" \"!\"\"%" }{XPPEDIT 18 0 "``= Int(x^2/2,x = 0 .. 4)" "6#/%!G-%$IntG6$* &%\"xG\"\"#F*!\"\"/F);\"\"!\"\"%" }{TEXT -1 2 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x^3/6" "6#/%!G*&%\"xG\"\"$\"\"'!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4,``],[0,``])" "6#-%*PI ECEWISEG6$7$\"\"%%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=32 /3" "6#/%!G*&\"#K\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region \+ about the " }{TEXT 348 1 "x" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 0 .. 4);" "6#/&%\"MG6#%\"xG-%$IntG6$*&-%\"fG6#F'\"\"#F/!\"\"/F';\"\"!\"\"%" } {XPPEDIT 18 0 "``= Int((x/2)^2/2,x = 0 .. 4)" "6#/%!G-%$IntG6$*&*&%\"x G\"\"\"\"\"#!\"\"F,F,F-/F*;\"\"!\"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(x^2/8,x = 0 .. 4);" "6#/% !G-%$IntG6$*&%\"xG\"\"#\"\")!\"\"/F);\"\"!\"\"%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = x^3/24;" "6#/%!G*&%\"xG\"\"$\"#C!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[0, ``])" "6#-%*PIECEWISEG6$7$ \"\"%%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=64/24" "6#/%!G*&\"#k\"\"\"\"#C!\"\"" }{XPPEDIT 18 0 "``=8/3" "6#/%!G*&\"\")\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 349 1 "x" }{TEXT -1 32 " coordinate of the centroid is " }{XPPEDIT 18 0 " conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"MG6#%\"yG\"\"\"% \"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(32/3)/4;" "6#/%!G*& -F$6#*&\"#K\"\"\"\"\"$!\"\"F*\"\"%F," }{XPPEDIT 18 0 "`` = 8/3;" "6#/% !G*&\"\")\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 350 1 "y" }{TEXT -1 32 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(y) \+ = M[x]/A;" "6#/-%*conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"%\"AG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(8/3)/4;" "6#/%!G*&-F$6#*&\"\") \"\"\"\"\"$!\"\"F*\"\"%F," }{XPPEDIT 18 0 "``=2/3" "6#/%!G*&\"\"#\"\" \"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at the point" } {XPPEDIT 18 0 "``(8/3,2/3);" "6#-%!G6$*&\"\")\"\"\"\"\"$!\"\"*&\"\"#F( F)F*" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "f := x -> x/2:\nInt(f(x),x=0..4); \nA := value(%);\nInt(x*f(x),x=0..4);\nMy := value(%);\nInt(f(x)^2/2,x =0..4);\nexpand(%);\nMx := value(%);\nxG := My/A;\nyG := Mx/A;\nprint( `The centroid is located at the point .. `,``(xG,yG));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#!\"\"%\"xG\"\"\"F+/F*;\"\"!\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#!\"\"%\"xGF(\"\"\"/F*;\"\"!\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG#\"#K\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\")!\"\"%\"xG\"\"#\"\"\"/F*;\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\")F&-%$IntG6$*$)%\"xG \"\"#F&/F-;\"\"!\"\"%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\" \")\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\"\")\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%IThe~centroid~is~located~at~the~point~..~G-%!G6$#\"\") \"\"$#\"\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can draw a picture which shows the centroid of the reg ion.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "xG := 8/3;\nyG := 2/3;\np1 := plots[polygonplot]([[0,0],[4,2],[4,0]],\n \+ color=COLOR(RGB,1,.85,.6)):\np2 := plot([[[xG,yG]]$3],style=poi nt,symbol=[circle,diamond,cross],color=black):\nplots[display]([p1,p2] ,labels=[`x`,`y`],tickmarks=[5,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#xGG#\"\")\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG#\"\"#\"\" $" }}{PARA 13 "" 1 "" {GLPLOT2D 344 164 164 {PLOTDATA 2 "6)-%)POLYGONS G6$7%7$$\"\"!F)F(7$$\"\"%F)$\"\"#F)7$F+F(-%&COLORG6&%$RGBG\"\"\"$\"#&) !\"#$\"\"'!\"\"-%'CURVESG6&7#7$$\"3_mmmmmmmE!#<$\"3Immmmmmmm!#=-%'SYMB OLG6#%'CIRCLEG-%'COLOURG6&F3F)F)F)-%&STYLEG6#%&POINTG-F<6&F>-FG6#%(DIA MONDGFJFM-F<6&F>-FG6#%&CROSSGFJFM-%*AXESTICKSG6$\"\"&\"\"$-%+AXESLABEL SG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$FboFbo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 91 ": The centroid of a triangle lies at the in tersection point of the medians of the triangle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "p1 := plots [polygonplot]([[0,0],[4,2],[4,0]],\n color=COLOR (RGB,1,.85,.6)):\np2 := plot([[[xG,yG]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\np3 := plot([[[2,0],[4,2] ],[[0,0],[4,1]],[[4,0],[2,1]]],\n color=brown,linestyle=2): \nplots[display]([p1,p2,p3],labels=[`x`,`y`],tickmarks=[5,3]);" }} {PARA 13 "" 1 "" {GLPLOT2D 319 162 162 {PLOTDATA 2 "6,-%)POLYGONSG6$7% 7$$\"\"!F)F(7$$\"\"%F)$\"\"#F)7$F+F(-%&COLORG6&%$RGBG\"\"\"$\"#&)!\"#$ \"\"'!\"\"-%'CURVESG6&7#7$$\"3_mmmmmmmE!#<$\"3Immmmmmmm!#=-%'SYMBOLG6# %'CIRCLEG-%'COLOURG6&F3F)F)F)-%&STYLEG6#%&POINTG-F<6&F>-FG6#%(DIAMONDG FJFM-F<6&F>-FG6#%&CROSSGFJFM-F<6%7$7$F-F(F*-FK6&F3$\")#)eqk!\")$\"))eq k\"F]oF^o-%*LINESTYLEG6#F.-F<6%7$F'7$F+$F4F)FinF`o-F<6%7$F/7$F-FgoFinF `o-%*AXESTICKSG6$\"\"&\"\"$-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAU LTG-%%VIEWG6$FipFip" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 4" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 268 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 96 "Find the coordinates of the centroid of t he trapezoid in the first quadrant bounded by the line " }{XPPEDIT 18 0 "y = x/2+2;" "6#/%\"yG,&*&%\"xG\"\"\"\"\"#!\"\"F(F)F(" }{TEXT -1 6 " , the " }{TEXT 352 1 "x" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "x = 4" "6#/%\"xG\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 336 2 ": " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plots[polygonplot]([[0,0],[0,2],[4,4],[4,0]],\n \+ color=COLOR(RGB,.8,.7,.6),labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 232 210 210 {PLOTDATA 2 "6%-%)POLYGONSG6#7&7$$\"\"!F )F(7$F($\"\"#F)7$$\"\"%F)F.7$F.F(-%+AXESLABELSG6$%\"xG%\"yG-%&COLORG6& %$RGBG$\"\")!\"\"$\"\"(F<$\"\"'F<" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x/2+2" "6#/-%\"fG6#%\"xG,&*&F' \"\"\"\"\"#!\"\"F*F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The area of the region is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 4);" "6#/%\"AG-%$IntG6$-%\"fG6#% \"xG/F+;\"\"!\"\"%" }{XPPEDIT 18 0 "``= Int(x/2+2,x = 0 .. 4)" "6#/%!G -%$IntG6$,&*&%\"xG\"\"\"\"\"#!\"\"F+F,F+/F*;\"\"!\"\"%" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x^2/4+2*x" "6# /%!G,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&F(F+F'F+F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([4,``],[0,``])" "6#-%*PIECEWISEG6$7$\"\"%%!G7 $\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=4+8" "6#/%!G,&\"\"%\"\"\"\"\")F'" }{XPPEDIT 18 0 "`` =12" "6#/%!G\"#7" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 155 "This area could also be calculated by us ing the formula for the area of a trapezoid or by dividing the region \+ into an upper triangle and a lower rectangle." }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " }{TEXT 353 1 "y" } {TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*f(x),x = 0 .. 4);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"x G\"\"\"-%\"fG6#F,F-/F,;\"\"!\"\"%" }{XPPEDIT 18 0 "``= Int(x^2/2+2*x,x = 0 .. 4)" "6#/%!G-%$IntG6$,&*&%\"xG\"\"#F+!\"\"\"\"\"*&F+F-F*F-F-/F* ;\"\"!\"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=x^3/6+x^2" "6#/%!G,&*&%\"xG\"\"$\"\"'!\"\"\"\"\"*$F' \"\"#F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[0, ``])" " 6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=32/3+16" "6#/%!G,&*&\"#K\"\"\" \"\"$!\"\"F(\"#;F(" }{XPPEDIT 18 0 "``=80/3" "6#/%!G*&\"#!)\"\"\"\"\"$ !\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " }{TEXT 354 1 "x " }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 0 .. 4);" "6#/&%\"MG6#%\"xG-%$I ntG6$*&-%\"fG6#F'\"\"#F/!\"\"/F';\"\"!\"\"%" }{XPPEDIT 18 0 "``= Int(` `(1/2)*(x/2+2)^2,x = 0 .. 4)" "6#/%!G-%$IntG6$*&-F$6#*&\"\"\"F,\"\"#! \"\"F,*$,&*&%\"xGF,F-F.F,F-F,F-F,/F2;\"\"!\"\"%" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(``(1/2)*(x^2/4 +2*x+4),x=0..4)" "6#/%!G-%$IntG6$*&-F$6#*&\"\"\"F,\"\"#!\"\"F,,(*&%\"x GF-\"\"%F.F,*&F-F,F1F,F,F2F,F,/F1;\"\"!F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(x^2/8+x+2,x=0..4)" " 6#/%!G-%$IntG6$,(*&%\"xG\"\"#\"\")!\"\"\"\"\"F*F.F+F./F*;\"\"!\"\"%" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x ^3/24+x^2/2+2*x" "6#/%!G,(*&%\"xG\"\"$\"#C!\"\"\"\"\"*&F'\"\"#F-F*F+*& F-F+F'F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[0, ``]) " "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=8/3+8+8" "6#/%!G,(*&\"\")\"\" \"\"\"$!\"\"F(F'F(F'F(" }{XPPEDIT 18 0 "``=56/3" "6#/%!G*&\"#c\"\"\"\" \"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 355 1 "x" }{TEXT -1 32 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"MG6#%\"yG \"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(80/3)/12;" " 6#/%!G*&-F$6#*&\"#!)\"\"\"\"\"$!\"\"F*\"#7F," }{XPPEDIT 18 0 "`` = 80/ 36;" "6#/%!G*&\"#!)\"\"\"\"#O!\"\"" }{XPPEDIT 18 0 "``=20/9" "6#/%!G*& \"#?\"\"\"\"\"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 356 1 "y" }{TEXT -1 32 " \+ coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(y) = M[x]/A ;" "6#/-%*conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(56/3)/12;" "6#/%!G*&-F$6#*&\"#c\"\"\"\" \"$!\"\"F*\"#7F," }{XPPEDIT 18 0 "`` = 56/36;" "6#/%!G*&\"#c\"\"\"\"#O !\"\"" }{XPPEDIT 18 0 "``=14/9" "6#/%!G*&\"#9\"\"\"\"\"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at the point" }{XPPEDIT 18 0 "``(20/9,14/9 );" "6#-%!G6$*&\"#?\"\"\"\"\"*!\"\"*&\"#9F(F)F*" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "f := x -> x/2+2:\nInt(f(x), x=0..4);\nA := value(%);\nInt(x*f(x),x=0..4);\nMy := value(%);\nInt(f( x)^2/2,x=0..4);\nexpand(%);\nMx := value(%);\nxG := My/A;\nyG := Mx/A; \nprint(`The centroid is located at the point .. `,``(xG,yG));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"#!\"\"%\"xG\"\"\"F+F( F+/F*;\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&*&\"\"#!\"\"F' F(F(F+F(F(/F';\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG#\"# !)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#!\"\",&*& F(F)%\"xG\"\"\"F-F(F-F(F-/F,;\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\")F&-%$IntG6$,(*$)%\"xG\"\"#F&F&*&F'F&F.F&F&\"#;F& /F.;\"\"!\"\"%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\"#c\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\"#?\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG#\"#9\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %IThe~centroid~is~located~at~the~point~..~G-%!G6$#\"#?\"\"*#\"#9F)" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can d raw a picture which shows the centroid of the region.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "xG := 20/9;\nyG := 14/9;\np1 := pl ots[polygonplot]([[0,0],[0,2],[4,4],[4,0]],\n co lor=COLOR(RGB,.8,.7,.6)):\np2 := plot([[[xG,yG]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nplots[display]([ p1,p2],labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\" #?\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG#\"#9\"\"*" }}{PARA 13 "" 1 "" {GLPLOT2D 250 182 182 {PLOTDATA 2 "6(-%)POLYGONSG6$7&7$$\" \"!F)F(7$F($\"\"#F)7$$\"\"%F)F.7$F.F(-%&COLORG6&%$RGBG$\"\")!\"\"$\"\" (F7$\"\"'F7-%'CURVESG6&7#7$$\"3KAAAAAAAA!#<$\"3ebbbbbbb:FC-%'SYMBOLG6# %'CIRCLEG-%'COLOURG6&F4F)F)F)-%&STYLEG6#%&POINTG-F=6&F?-FG6#%(DIAMONDG FJFM-F=6&F?-FG6#%&CROSSGFJFM-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFA ULTG-%%VIEWG6$F]oF]o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "There is another wa y to find the centroid of this region, making use of the result of the previous example." }}{PARA 0 "" 0 "" {TEXT -1 86 "Divide the region i nto an upper triangle and a lower rectangle by the horizontal line " } {XPPEDIT 18 0 "y = 2" "6#/%\"yG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "p1 := plo ts[polygonplot]([[0,0],[0,2],[4,4],[4,0]],\n col or=COLOR(RGB,.8,.7,.6)):\np2 := plot([[[8/3,8/3],[2,1]]$3],style=point ,\n symbol=[circle,diamond,cross],color=black):\np3 := p lot([[[2,2],[4,4]],[[0,2],[4,3]],[[4,2],[2,3]]],\n color=bro wn,linestyle=2):\np4 := plot([[0,2],[4,2]],color=black,linestyle=3):\n plots[display]([p1,p2,p3,p4],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 235 196 196 {PLOTDATA 2 "6,-%)POLYGONSG6$7&7$$\"\"!F)F(7$F($ \"\"#F)7$$\"\"%F)F.7$F.F(-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"(F7$\"\"'F7- %'CURVESG6&7$7$$\"3_mmmmmmmE!# " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "The upper tria ngle has the same shape as the triangle in the previous question, but \+ it is moved up 2 units so that it will fit on top of the rectangle." } }{PARA 0 "" 0 "" {TEXT -1 57 "The centroid of the triangle therefore \+ lies at the point" }{XPPEDIT 18 0 "``(8/3,8/3);" "6#-%!G6$*&\"\")\"\" \"\"\"$!\"\"*&F'F(F)F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 100 "The centroid of the lower rectangle is at the centre of the recta ngle, which is located at the point" }{XPPEDIT 18 0 " ``(2,1)" "6#-%!G 6$\"\"#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "The total moment of the region about the \+ " }{TEXT 420 1 "y" }{TEXT -1 50 " axis is the sum of the moments of th e two pieces." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 269 22 "moment of the triangle" }{TEXT -1 11 " about the " }{TEXT 357 1 "y" }{TEXT -1 8 " axis is" }}{PARA 256 "" 0 "" {TEXT -1 84 " \"area of triangle\" times \"perpendicular distanc e of centroid of triangle from the " }{TEXT 359 1 "y" }{TEXT -1 6 " ax is\"" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*`.`*`` (8/3);" "6#/%!G*(\"\"%\"\"\"%\".GF'-F$6#*&\"\")F'\"\"$!\"\"F'" } {XPPEDIT 18 0 "`` = 32/3;" "6#/%!G*&\"#K\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The \+ " }{TEXT 256 23 "moment of the rectangle" }{TEXT -1 11 " about the " } {TEXT 358 1 "y" }{TEXT -1 9 " axis is:" }}{PARA 256 "" 0 "" {TEXT -1 86 " \"area of rectangle\" times \"perpendicular distance of centroid of rectangle from the " }{TEXT 360 1 "y" }{TEXT -1 6 " axis\"" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``= 8*`.`*2" "6#/%!G *(\"\")\"\"\"%\".GF'\"\"#F'" }{XPPEDIT 18 0 " ``= 16" "6#/%!G\"#;" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "The total moment of the region about the y axis is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "M[y] = 32/3+16;" "6#/&%\"MG6#%\"yG,&*&\"#K\"\"\"\"\"$! \"\"F+\"#;F+" }{XPPEDIT 18 0 "`` = 80/3;" "6#/%!G*&\"#!)\"\"\"\"\"$!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Similarly, the total moment of the region about the " } {TEXT 361 1 "x" }{TEXT -1 50 " axis is the sum of the moments of the t wo pieces." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 256 22 "moment of the triangle" }{TEXT -1 11 " abou t the " }{TEXT 362 1 "x" }{TEXT -1 9 " axis is:" }}{PARA 256 "" 0 "" {TEXT -1 84 " \"area of triangle\" times \"perpendicular distance of \+ centroid of triangle from the " }{TEXT 363 1 "x" }{TEXT -1 7 " axis\" \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*`.`*``(8/3 );" "6#/%!G*(\"\"%\"\"\"%\".GF'-F$6#*&\"\")F'\"\"$!\"\"F'" }{XPPEDIT 18 0 "``=32/3" "6#/%!G*&\"#K\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 256 23 "moment of the rectang le" }{TEXT -1 11 " about the " }{TEXT 364 1 "x" }{TEXT -1 9 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 86 " \"area of rectangle\" times \"per pendicular distance of centroid of rectangle from the " }{TEXT 365 1 " y" }{TEXT -1 7 " axis\" " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=8*`.`*1" "6#/%!G*(\"\")\"\"\"%\".GF'F'F'" }{XPPEDIT 18 0 "``=8" "6#/%!G\"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "The total moment of the region about the " }{TEXT 366 1 "x" } {TEXT -1 9 " axis is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "M[x] = 32/3+8;" "6#/&%\"MG6#%\"xG,&*&\"#K\"\"\"\"\"$!\"\"F+\"\") F+" }{XPPEDIT 18 0 "`` = 56/3;" "6#/%!G*&\"#c\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "Then the coordinates of the c entroid can be calculated as before. " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG* &&%\"MG6#%\"yG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ` `(80/3)/12;" "6#/%!G*&-F$6#*&\"#!)\"\"\"\"\"$!\"\"F*\"#7F," }{XPPEDIT 18 0 "``=20/9" "6#/%!G*&\"#?\"\"\"\"\"*!\"\"" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "conjugate(y) = M[x] /A;" "6#/-%*conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(56/3)/12;" "6#/%!G*&-F$6#*&\"#c\"\"\" \"\"$!\"\"F*\"#7F," }{XPPEDIT 18 0 "``=14/9" "6#/%!G*&\"#9\"\"\"\"\"*! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 397 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 100 "Find the coordinates of the centroid of the plane region in the f irst quadrant bounded by the curve " }{XPPEDIT 18 0 "y = 5*x-x^2;" "6# /%\"yG,&*&\"\"&\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 9 " and the " }{TEXT 399 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 398 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "f := x -> 5*x-x^2:\nplots[polygonplot]([[0,0],op(op(1,op(1,plot(f(x),x= 0..5)))),[5,0]],\n color=COLOR(RGB,1,.7,.6),labels=[`x`,`y`],view=[ 0..5.5,0..6.5],\n tickmarks=[5,7]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 337 278 278 {PLOTDATA 2 "6'-%)POLYGONSG6#7U7$$\"\"!F)F(F'7$$ \"+3x&)*3\"!#5$\"+f&40L&F-7$$\"+tq8Q?F-$\"+#4&Gv(*F-7$$\"+'RwX5$F-$\"+ _U!fX\"!\"*7$$\"+rZ3yTF-$\"+i%yW\">F:7$$\"+[4\\Y_F-$\"+,)))zM#F:7$$\"+ S-/PiF-$\"+5M^HFF:7$$\"+cmpisF-$\"+,2)Q5$F:7$$\"+!*>VB$)F-$\"+%z?*oMF: 7$$\"+Jbw!Q*F-$\"+Y^R5QF:7$$\"+/j$o/\"F:$\"+t_JQTF:7$$\"+_>jU6F:$\"+#) >b2WF:7$$\"+j^Z]7F:$\"+!o(o)o%F:7$$\"+)=h(e8F:$\"+tRdZ\\F:7$$\"+Q[6j9F :$\"+g\"p[<&F:7$$\"+\\z(yb\"F:$\"+TgSi`F:7$$\"+a/cq;F:$\"+S+.ibF:7$$\" +T>'eF:7$$\"+#pW`(>F:$\"+2otufF: 7$$\"+\"f#=$3#F:$\"+ZKEwgF:7$$\"+s(pe=#F:$\"++AK^hF:7$$\"+tI,$H#F:$\"+ 7k:2iF:7$$\"+qSS\"R#F:$\"+Cp?QiF:7$$\"+_?`(\\#F:$\"+4R**\\iF:7$$\"+*>p xg#F:$\"++eQQiF:7$$\"+f4t.FF:$\"+'p$\\3iF:7$$\"+!Hst!GF:$\"+wA_bhF:7$$ \"+DRW9HF:$\"+LiBygF:7$$\"+JE>>IF:$\"+7!R/)fF:7$$\"+\"RU07$F:$\"+Tr#\\ 'eF:7$$\"+>S2LKF:$\"+$[-Er&F:7$$\"+#p)=MLF:$\"+F#HTb&F:7$$\"+)=]@W$F:$ \"+CINi`F:7$$\"+\\$z*RNF:$\"+aHWo^F:7$$\"+jC$pk$F:$\"+EfaM\\F:7$$\"+2q cZPF:$\"+ild$p%F:7$$\"+.\"fF&QF:$\"+4G/?WF:7$$\"+/OgbRF:$\"+[\"=78%F:7 $$\"+mAFjSF:$\"+C)zh!QF:7$$\"+$)*pp;%F:$\"+v5@rMF:7$$\"+xe,tUF:$\"++ZT 1JF:7$$\"+dO=yVF:$\"+\\hUAFF:7$$\"+D>#[Z%F:$\"+l$y+N#F:7$$\"+&G!e&e%F: $\"+t[N+>F:7$$\"+$)Qk%o%F:$\"+.6Lx9F:7$$\"+TjE!z%F:$\"+*3!o/5F:7$$\"+4 0O\"*[F:$\"+U\"\\RJ&F-7$$\"\"&F)F(Faz-%+AXESLABELSG6$%\"xG%\"yG-%&COLO RG6&%$RGBG\"\"\"$\"\"(!\"\"$\"\"'F`[l-%*AXESTICKSG6$FczF_[l-%%VIEWG6$; F($\"#bF`[l;F($\"#lF`[l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let \+ " }{XPPEDIT 18 0 "f(x) = 5*x-x^2;" "6#/-%\"fG6#%\"xG,&*&\"\"&\"\"\"F'F +F+*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The \+ area of the region is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 5);" "6#/%\"AG-%$IntG6$-%\"fG6#%\"xG/F+;\" \"!\"\"&" }{XPPEDIT 18 0 "`` = Int(5*x-x^2,x = 0 .. 5);" "6#/%!G-%$Int G6$,&*&\"\"&\"\"\"%\"xGF+F+*$F,\"\"#!\"\"/F,;\"\"!F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 5*x^2/2-x^3/3; " "6#/%!G,&*(\"\"&\"\"\"*$%\"xG\"\"#F(F+!\"\"F(*&F*\"\"$F.F,F," } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([5, ``],[0, ``]);" "6#-%*PIEC EWISEG6$7$\"\"&%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 125/2-125/3;" "6#/%!G,&*&\"$D\"\" \"\"\"\"#!\"\"F(*&F'F(\"\"$F*F*" }{XPPEDIT 18 0 "`` = 125/6;" "6#/%!G* &\"$D\"\"\"\"\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 6 ": The " }{TEXT 404 1 "x" }{TEXT -1 82 " coordinate of the centroid lies on the vertic al line of symmetry of the parabola " }{XPPEDIT 18 0 "y=5*x-x^2" "6#/% \"yG,&*&\"\"&\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 20 ", which is t he line " }{XPPEDIT 18 0 "x=5/2" "6#/%\"xG*&\"\"&\"\"\"\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "Hence we do not really \+ need to find the moment of the region about the " }{TEXT 400 1 "y" } {TEXT -1 28 " axis, but we do so anyway. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*f (x),x = 0 .. 5);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%\"fG6#F,F-/ F,;\"\"!\"\"&" }{XPPEDIT 18 0 "`` = Int(x*(5*x-x^2),x = 0 .. 5);" "6#/ %!G-%$IntG6$*&%\"xG\"\"\",&*&\"\"&F*F)F*F**$F)\"\"#!\"\"F*/F);\"\"!F- " }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = Int(5*x^2-x^3,x = 0 .. 5);" "6#/%!G-%$IntG6$,&*&\"\"&\"\"\"*$%\"x G\"\"#F+F+*$F-\"\"$!\"\"/F-;\"\"!F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 5*x^3/3-x^4/4;" "6#/%!G,&*(\"\" &\"\"\"*$%\"xG\"\"$F(F+!\"\"F(*&F*\"\"%F.F,F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([5, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"&% !G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 625/3-625/4;" "6#/%!G,&*&\"$D'\"\"\"\"\"$!\"\"F(*& F'F(\"\"%F*F*" }{XPPEDIT 18 0 "`` = 625/12;" "6#/%!G*&\"$D'\"\"\"\"#7! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " }{TEXT 401 1 "x" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 0 .. 5);" "6#/&%\"MG6#%\"xG-%$I ntG6$*&-%\"fG6#F'\"\"#F/!\"\"/F';\"\"!\"\"&" }{XPPEDIT 18 0 "`` = Int( (5*x-x^2)^2/2,x = 0 .. 5);" "6#/%!G-%$IntG6$*&,&*&\"\"&\"\"\"%\"xGF,F, *$F-\"\"#!\"\"F/F/F0/F-;\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((25*x^2-10*x^3+x^4)/2,x = 0 .. 5);" "6#/%!G-%$IntG6$*&,(*&\"#D\"\"\"*$%\"xG\"\"#F,F,*&\"#5F,*$F.\"\" $F,!\"\"*$F.\"\"%F,F,F/F4/F.;\"\"!\"\"&" }{XPPEDIT 18 0 "`` = Int(25*x ^2/2-5*x^3+x^4/2,x = 0 .. 5);" "6#/%!G-%$IntG6$,(*(\"#D\"\"\"*$%\"xG\" \"#F+F.!\"\"F+*&\"\"&F+*$F-\"\"$F+F/*&F-\"\"%F.F/F+/F-;\"\"!F1" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 25*x^3/6-5*x^4/4+x^5/10;" "6#/%!G,(*(\"#D\"\"\"*$% \"xG\"\"$F(\"\"'!\"\"F(*(\"\"&F(*$F*\"\"%F(F1F-F-*&F*F/\"#5F-F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([5, ``],[0, ``]);" "6#-%*PIEC EWISEG6$7$\"\"&%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 5^5/6-5^5/4+5^5/10;" "6#/%!G,(*&\" \"&F'\"\"'!\"\"\"\"\"*&F'F'\"\"%F)F)*&F'F'\"#5F)F*" }{XPPEDIT 18 0 "`` =5^5*(1/6-1/4+1/10)" "6#/%!G*&\"\"&F&,(*&\"\"\"F)\"\"'!\"\"F)*&F)F)\" \"%F+F+*&F)F)\"#5F+F)F)" }{XPPEDIT 18 0 "``=5^5*(10-15+6)/60" "6#/%!G* (\"\"&F&,(\"#5\"\"\"\"#:!\"\"\"\"'F)F)\"#gF+" }{XPPEDIT 18 0 "``=5^5/6 0" "6#/%!G*&\"\"&F&\"#g!\"\"" }{XPPEDIT 18 0 "``=5^4/12" "6#/%!G*&\"\" &\"\"%\"#7!\"\"" }{XPPEDIT 18 0 "``=625/12" "6#/%!G*&\"$D'\"\"\"\"#7! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 402 1 "x" }{TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjug ateG6#%\"xG*&&%\"MG6#%\"yG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(625/12)/``(125/6);" "6#/%!G*&-F$6#*&\"$D'\"\"\"\"#7!\"\" F*-F$6#*&\"$D\"F*\"\"'F,F," }{XPPEDIT 18 0 "``=``(625/12)*``(6/125)" " 6#/%!G*&-F$6#*&\"$D'\"\"\"\"#7!\"\"F*-F$6#*&\"\"'F*\"$D\"F,F*" } {XPPEDIT 18 0 "``=5/2" "6#/%!G*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 403 1 "x" }{TEXT -1 33 " coordinate of the centroid is " } {XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"M G6#%\"yG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(625/ 12)/``(125/6);" "6#/%!G*&-F$6#*&\"$D'\"\"\"\"#7!\"\"F*-F$6#*&\"$D\"F* \"\"'F,F," }{XPPEDIT 18 0 "``=5/2" "6#/%!G*&\"\"&\"\"\"\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at the point" }{XPPEDIT 18 0 "``( 5/2,5/2);" "6#-%!G6$*&\"\"&\"\"\"\"\"#!\"\"*&F'F(F)F*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "f := x -> 5*x-x^2:\nInt(f(x),x=0..5);\nA := value(%);\nInt(x* f(x),x=0..5);\nMy := value(%);\nInt(f(x)^2/2,x=0..5);\nMx := value(%); \nxG := My/A;\nyG := Mx/A;\nprint(`The centroid is located at the poin t .. `,``(xG,yG));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\" \"&\"\"\"%\"xGF)F)*$)F*\"\"#F)!\"\"/F*;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG#\"$D\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*&%\"xG\"\"\",&*&\"\"&F(F'F(F(*$)F'\"\"#F(!\"\"F(/F';\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG#\"$D'\"#7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#!\"\",&*&\"\"&\"\"\"%\"xGF-F-*$)F.F (F-F)F(F-/F.;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\"$D'\" #7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\"\"&\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#yGG#\"\"&\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%IThe~centroid~is~located~at~the~point~..~G-%!G6$#\"\"&\"\"#F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can draw a picture which shows the centroid of the region. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 374 "f := x -> 5*x-x^2:\nxG := 2.5:\nyG := 2.5:\np1 := plots[polygonplot]([[0,0 ],op(op(1,op(1,plot(f(x),x=0..5)))),[5,0]],\n color=COLOR(RGB,1,.7, .6)):\np2 := plot([[[xG,yG]]$3],style=point,symbol=[circle,diamond,cro ss],color=black):\np3 := plot([[2.5,0],[2.5,6.25]],color=black,linesty le=2):\nplots[display]([p1,p2,p3],labels=[`x`,`y`],view=[0..5.5,0..6.5 ],\n tickmarks=[5,7]);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 302 302 {PLOTDATA 2 "6*-%)POLYGONSG6$7U7$$\"\"!F)F(F'7$$\"+3x&)*3\"!#5$\"+ f&40L&F-7$$\"+tq8Q?F-$\"+#4&Gv(*F-7$$\"+'RwX5$F-$\"+_U!fX\"!\"*7$$\"+r Z3yTF-$\"+i%yW\">F:7$$\"+[4\\Y_F-$\"+,)))zM#F:7$$\"+S-/PiF-$\"+5M^HFF: 7$$\"+cmpisF-$\"+,2)Q5$F:7$$\"+!*>VB$)F-$\"+%z?*oMF:7$$\"+Jbw!Q*F-$\"+ Y^R5QF:7$$\"+/j$o/\"F:$\"+t_JQTF:7$$\"+_>jU6F:$\"+#)>b2WF:7$$\"+j^Z]7F :$\"+!o(o)o%F:7$$\"+)=h(e8F:$\"+tRdZ\\F:7$$\"+Q[6j9F:$\"+g\"p[<&F:7$$ \"+\\z(yb\"F:$\"+TgSi`F:7$$\"+a/cq;F:$\"+S+.ibF:7$$\"+T>'eF:7$$\"+#pW`(>F:$\"+2otufF:7$$\"+\"f#=$3#F:$ \"+ZKEwgF:7$$\"+s(pe=#F:$\"++AK^hF:7$$\"+tI,$H#F:$\"+7k:2iF:7$$\"+qSS \"R#F:$\"+Cp?QiF:7$$\"+_?`(\\#F:$\"+4R**\\iF:7$$\"+*>pxg#F:$\"++eQQiF: 7$$\"+f4t.FF:$\"+'p$\\3iF:7$$\"+!Hst!GF:$\"+wA_bhF:7$$\"+DRW9HF:$\"+Li BygF:7$$\"+JE>>IF:$\"+7!R/)fF:7$$\"+\"RU07$F:$\"+Tr#\\'eF:7$$\"+>S2LKF :$\"+$[-Er&F:7$$\"+#p)=MLF:$\"+F#HTb&F:7$$\"+)=]@W$F:$\"+CINi`F:7$$\"+ \\$z*RNF:$\"+aHWo^F:7$$\"+jC$pk$F:$\"+EfaM\\F:7$$\"+2qcZPF:$\"+ild$p%F :7$$\"+.\"fF&QF:$\"+4G/?WF:7$$\"+/OgbRF:$\"+[\"=78%F:7$$\"+mAFjSF:$\"+ C)zh!QF:7$$\"+$)*pp;%F:$\"+v5@rMF:7$$\"+xe,tUF:$\"++ZT1JF:7$$\"+dO=yVF :$\"+\\hUAFF:7$$\"+D>#[Z%F:$\"+l$y+N#F:7$$\"+&G!e&e%F:$\"+t[N+>F:7$$\" +$)Qk%o%F:$\"+.6Lx9F:7$$\"+TjE!z%F:$\"+*3!o/5F:7$$\"+40O\"*[F:$\"+U\" \\RJ&F-7$$\"\"&F)F(Faz-%&COLORG6&%$RGBG\"\"\"$\"\"(!\"\"$\"\"'F[[l-%'C URVESG6&7#7$$\"3++++++++D!# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "E xample 6" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 337 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 70 "Find the coordinates of the centroid \+ of the sector of the unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$ %\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 56 " which lies in the first \+ quadrant and is bounded by the " }{TEXT 421 1 "x" }{TEXT -1 5 " and " }{TEXT 422 1 "y" }{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 338 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "First we draw a picture of the region." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "u := evalf(Pi/60,15):\nsector := [[0,0],seq([cos(i*u),sin(i*u)],i =0..30)]:\nplots[polygonplot](sector,color=COLOR(RGB,.8,.7,1),\n l abels=[`x`,`y`],tickmarks=[6,6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 197 187 187 {PLOTDATA 2 "6&-%)POLYGONSG6#7B7$$\"\"!F)F(7$$\"\"\"F)F(7$$\"+ [`H')**!#5$\"+CcfL_!#67$$\"+a*=_%**F0$\"+KYGX5F07$$\"+1M)o()*F0$\"+^YM k:F07$$\"+2gZ\"y*F0$\"+3p6z?F07$$\"+j#e#f'*F0$\"+^/>)e#F07$$\"+j^c5&*F 0$\"+W*p,4$F07$$\"+lU!eL*F0$\"+&\\zOe$F07$$\"+wXXN\"*F0$\"+JkOnSF07$$ \"+U_15*)F0$\"+(*\\!*RXF07$$\"+QSDg')F0$\"+++++]F07$$\"+zcq'Q)F0$\"+^. RYaF07$$\"+W*p,4)F0$\"+BD&y(eF07$$\"+9'f9x(F0$\"+6R?$H'F07$$\"+b#[9V(F 0$\"+jgI\"p'F07$$\"+7y1rqF0F`p7$F]pF[p7$FhoFfo7$FcoFao7$$\"+].RYaF0F\\ o7$$\"+-+++]F0$\"+PSDg')F07$$\"+&*\\!*RXF0$\"+V_15*)F07$$\"+MkOnSF0$\" +vXXN\"*F07$$\"+%\\zOe$F0FN7$$\"+[*p,4$F0$\"+i^c5&*F07$FFFD7$$\"+/p6z? F0$\"+3gZ\"y*F07$$\"+_YMk:F0F:7$$\"+HYGX5F0F57$$\"+[cfL_F3$\"+Z`H')**F 07$$!+3Q.^?!#>F+-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"(FjsF,-%*AXESTICKSG6$ \"\"'F`t-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = sqrt(1-x^2);" "6#/-%\"fG6#%\" xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 26 "The area of the region is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. 1);" "6#/%\"AG-%$IntG6$ -%\"fG6#%\"xG/F+;\"\"!\"\"\"" }{XPPEDIT 18 0 "``= Int(sqrt(1-x^2),x = \+ 0 .. 1)" "6#/%!G-%$IntG6$-%%sqrtG6#,&\"\"\"F,*$%\"xG\"\"#!\"\"/F.;\"\" !F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "This integral can be evaluated using the substitution " }{XPPEDIT 18 0 "x = sin*theta; " "6#/%\"xG*&%$sinG\"\"\"%&thetaGF'" }{TEXT -1 87 ", but this is not n ecessary, because the region is a sector of a circle with radius 1. " }}{PARA 0 "" 0 "" {TEXT -1 17 "It is clear that " }{XPPEDIT 18 0 "A=Pi /4" "6#/%\"AG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Since the sector is sy mmetrical about the line " }{XPPEDIT 18 0 "y = x" "6#/%\"yG%\"xG" } {TEXT -1 37 ", the centroid must lie on this line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region \+ about the " }{TEXT 351 1 "y" }{TEXT -1 8 " axis is" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*f(x),x = 0 .. 1);" "6#/& %\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%\"fG6#F,F-/F,;\"\"!F-" }{XPPEDIT 18 0 "``= Int(x*sqrt(1-x^2),x = 0 .. 1)" "6#/%!G-%$IntG6$*&%\"xG\"\"\" -%%sqrtG6#,&F*F**$F)\"\"#!\"\"F*/F);\"\"!F*" }{TEXT -1 3 ", " }} {PARA 0 "" 0 "" {TEXT -1 39 "and the moment of the region about the " }{TEXT 392 1 "x" }{TEXT -1 9 " axis is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 0 .. 1);" "6#/&%\"MG6#% \"xG-%$IntG6$*&-%\"fG6#F'\"\"#F/!\"\"/F';\"\"!\"\"\"" }{XPPEDIT 18 0 " ``= Int((1-x^2)/2,x = 0 .. 1)" "6#/%!G-%$IntG6$*&,&\"\"\"F**$%\"xG\"\" #!\"\"F*F-F./F,;\"\"!F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Because sector is symmetrical abou t the line " }{XPPEDIT 18 0 "y = x" "6#/%\"yG%\"xG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "M[x] = M[y]" "6#/&%\"MG6#%\"xG&F%6#%\"yG" }{TEXT -1 27 ", so we need only find one." }}{PARA 0 "" 0 "" {TEXT -1 28 "Evaluatin g the integral for " }{XPPEDIT 18 0 "M[x]" "6#&%\"MG6#%\"xG" }{TEXT -1 53 " involves less work than evaluating the integral for " } {XPPEDIT 18 0 "M[y]" "6#&%\"MG6#%\"yG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x]=Int((1-x^2)/2,x = 0 .. 1)" "6 #/&%\"MG6#%\"xG-%$IntG6$*&,&\"\"\"F-*$F'\"\"#!\"\"F-F/F0/F';\"\"!F-" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=I nt(1/2-x^2/2,x = 0 .. 1)" "6#/%!G-%$IntG6$,&*&\"\"\"F*\"\"#!\"\"F**&% \"xGF+F+F,F,/F.;\"\"!F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/2-x^3/6;" "6#/%!G,&*&%\"xG\"\"\"\"\"#!\" \"F(*&F'\"\"$\"\"'F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, \+ ``],[0, ``])" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2-1/6;" "6#/ %!G,&*&\"\"\"F'\"\"#!\"\"F'*&F'F'\"\"'F)F)" }{XPPEDIT 18 0 "`` = 1/3; " "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The integral for " } {XPPEDIT 18 0 "M[y]" "6#&%\"MG6#%\"yG" }{TEXT -1 48 " can also be eval uated as a check. Substituting " }{XPPEDIT 18 0 "u=1-x^2" "6#/%\"uG,& \"\"\"F&*$%\"xG\"\"#!\"\"" }{TEXT -1 18 " helps with this. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y]=Int(x*sqrt(1-x^2),x = \+ 0 .. 1)" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F-F-*$F,\" \"#!\"\"F-/F,;\"\"!F-" }{TEXT -1 10 " ... " }{XPPEDIT 18 0 "PIECE WISE([u=1-x^2,`x =`*0*` implies u =`*1],[du=-2*x*dx,`x =`*1*` implies u =`*0],[``(-1/2)*du=x*dx,``])" "6#-%*PIECEWISEG6%7$/%\"uG,&\"\"\"F* *$%\"xG\"\"#!\"\"**%$x~=GF*\"\"!F*%.~implies~~u~=GF*F*F*7$/%#duG,$*(F- F*F,F*%#dxGF*F.**F0F*F*F*F2F*F1F*7$/*&-%!G6#,$*&F*F*F-F.F.F*F5F**&F,F* F8F*F>" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sqrt(u),u = 1 .. 0);" "6#-%$IntG6$-%%sqrtG6#%\"uG/F );\"\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(u^(1/2),u = 0 .. 1);" "6#-%$IntG6$)%\"uG*&\"\"\"F) \"\"#!\"\"/F';\"\"!F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 78 " ( because changing the order of the limits changes the sign of the int egral ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/3; " "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2) " "6#)%\"xG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PI ECEWISE([1, ``],[0, ``])" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F(" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 13 "as expected. " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 367 1 "x" }{TEXT -1 32 " coordinate of the centroid is " } {XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"M G6#%\"yG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1/3) /``(Pi/4);" "6#/%!G*&-F$6#*&\"\"\"F)\"\"$!\"\"F)-F$6#*&%#PiGF)\"\"%F+F +" }{XPPEDIT 18 0 "`` = ``(1/3)*``(4/Pi);" "6#/%!G*&-F$6#*&\"\"\"F)\" \"$!\"\"F)-F$6#*&\"\"%F)%#PiGF+F)" }{XPPEDIT 18 0 "`` = 4/(3*Pi);" "6# /%!G*&\"\"%\"\"\"*&\"\"$F'%#PiGF'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 368 1 "y " }{TEXT -1 32 " coordinate of the centroid is " }{XPPEDIT 18 0 "conj ugate(y) = 4/(3*Pi);" "6#/-%*conjugateG6#%\"yG*&\"\"%\"\"\"*&\"\"$F*%# PiGF*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at the point" } {XPPEDIT 18 0 "``(4/(3*Pi),4/(3*Pi));" "6#-%!G6$*&\"\"%\"\"\"*&\"\"$F( %#PiGF(!\"\"*&F'F(*&F*F(F+F(F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "f := x -> s qrt(1-x^2):\nInt(f(x),x=0..1);\nA := value(%);\nInt(x*f(x),x=0..1);\nM y := value(%);\nInt(f(x)^2/2,x=0..1);\nMx := value(%);\nxG := My/A;\ny G := Mx/A;\nprint(`The centroid is located at the point .. `,``(xG,yG) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*$)%\"xG\" \"#F(!\"\"#F(F,/F+;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$ *&\"\"%!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ *&%\"xG\"\"\",&F(F(*$)F'\"\"#F(!\"\"#F(F,/F';\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$,&#\"\"\"\"\"#F(*&F)!\"\"%\"xGF)F+/F,;\"\"!F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#MxG#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG,$*(\"\"%\"\"\"\"\"$!\"\"%#PiGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*(\"\"%\"\"\"\"\"$!\"\"%#PiGF*F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%IThe~centroid~is~located~at~the~point~..~G-%!G6 $,$*(\"\"%\"\"\"\"\"$!\"\"%#PiGF,F*F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 62 "We can draw a picture which shows the centroid of the region.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "u := evalf(Pi/60,15):\nxG := 4/(3*Pi);\nyG := xG;\nsector := [[0, 0],seq([cos(i*u),sin(i*u)],i=0..30)]:\np1 := plots[polygonplot](sector ,color=COLOR(RGB,.8,.7,1)):\np2 := plot([[[xG,yG]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\np3 := plot([[0 ,0],[1/sqrt(2),1/sqrt(2)]],linestyle=2,\n color=COLOR(RGB,.6 ,.1,.8)):\nplots[display]([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#xGG,$*(\"\"%\"\"\"\"\"$!\"\"%#PiGF*F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*(\"\"%\"\"\"\"\"$!\"\"%#PiGF* F(" }}{PARA 13 "" 1 "" {GLPLOT2D 262 212 212 {PLOTDATA 2 "6)-%)POLYGON SG6$7B7$$\"\"!F)F(7$$\"\"\"F)F(7$$\"+[`H')**!#5$\"+CcfL_!#67$$\"+a*=_% **F0$\"+KYGX5F07$$\"+1M)o()*F0$\"+^YMk:F07$$\"+2gZ\"y*F0$\"+3p6z?F07$$ \"+j#e#f'*F0$\"+^/>)e#F07$$\"+j^c5&*F0$\"+W*p,4$F07$$\"+lU!eL*F0$\"+& \\zOe$F07$$\"+wXXN\"*F0$\"+JkOnSF07$$\"+U_15*)F0$\"+(*\\!*RXF07$$\"+QS Dg')F0$\"+++++]F07$$\"+zcq'Q)F0$\"+^.RYaF07$$\"+W*p,4)F0$\"+BD&y(eF07$ $\"+9'f9x(F0$\"+6R?$H'F07$$\"+b#[9V(F0$\"+jgI\"p'F07$$\"+7y1rqF0F`p7$F ]pF[p7$FhoFfo7$FcoFao7$$\"+].RYaF0F\\o7$$\"+-+++]F0$\"+PSDg')F07$$\"+& *\\!*RXF0$\"+V_15*)F07$$\"+MkOnSF0$\"+vXXN\"*F07$$\"+%\\zOe$F0FN7$$\"+ [*p,4$F0$\"+i^c5&*F07$FFFD7$$\"+/p6z?F0$\"+3gZ\"y*F07$$\"+_YMk:F0F:7$$ \"+HYGX5F0F57$$\"+[cfL_F3$\"+Z`H')**F07$$!+3Q.^?!#>F+-%&COLORG6&%$RGBG $\"\")!\"\"$\"\"(FjsF,-%'CURVESG6&7#7$$\"3)e(Qy:=8WU!#=Fbt-%'SYMBOLG6# %'CIRCLEG-%'COLOURG6&FgsF)F)F)-%&STYLEG6#%&POINTG-F^t6&F`t-Fft6#%(DIAM ONDGFitF\\u-F^t6&F`t-Fft6#%&CROSSGFitF\\u-F^t6%7$F'7$$\"3sva'=\"y1rqFd tF^v-Fes6&Fgs$\"\"'Fjs$F,FjsFhs-%*LINESTYLEG6#\"\"#-%+AXESLABELSG6%%\" xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$FawFaw" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl e 7" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 411 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 101 "Find the coordinates of the centroid of \+ the plane region in the first quadrant bounded by the curve " } {XPPEDIT 18 0 "y = 2/sqrt(4-x^2);" "6#/%\"yG*&\"\"#\"\"\"-%%sqrtG6#,& \"\"%F'*$%\"xGF&!\"\"F/" }{TEXT -1 6 ", the " }{TEXT 416 1 "x" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "x = sqrt(3);" "6#/%\"xG-%% sqrtG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 412 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "f := x -> 2/sqrt(4-x^2):\nplots[polygonplot]([[sqrt(3),0],[0,0],op(op(1,op (1,plot(f(x),x=0..sqrt(3)))))],\n color=COLOR(RGB,1,.2,.8),labels=[ `x`,`y`],view=[0..2,0..2],\n tickmarks=[3,3]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 267 228 228 {PLOTDATA 2 "6'-%)POLYGONSG6#7W7$$\"+330K(e^#F?$\"+qr+35 F*7$$\"+=9K$)GF?$\"+unb55F*7$$\"+,Df\\KF?$\"+`qY85F*7$$\"+ItMEOF?$\"+R Z&o,\"F*7$$\"+*=$>eRF?$\"+o)y,-\"F*7$$\"+GIxJVF?$\"+$>9V-\"F*7$$\"+Ao) oq%F?$\"+y'**)G5F*7$$\"+p%y$o]F?$\"+=\\uL5F*7$$\"+;vk'R&F?$\"+d:_Q5F*7 $$\"+l6*py&F?$\"+)4)oW5F*7$$\"+OMm$\\37\"F*7$$\"+z()*fO*F?$\" +1GxJ6F*7$$\"+yG-D(*F?$\"+B;SW6F*7$$\"+!*Hf45F*$\"+p&G%e6F*7$$\"+2!ze/ \"F*$\"+2(*>t6F*7$$\"+$f()43\"F*$\"+@)o&)=\"F*7$$\"+#pp*>6F*$\"+_k*p? \"F*7$$\"+Vo*\\:\"F*$\"+h?!\\A\"F*7$$\"+-eR#>\"F*$\"+l?eX7F*7$$\"+<[GE 7F*$\"+;q'eE\"F*7$$\"+jWLj7F*$\"+5P#**G\"F*7$$\"+#H&>)H\"F*$\"+o)pXJ\" F*7$$\"+.\\jM8F*$\"+r-pU8F*7$$\"+$Gh-P\"F*$\"+cR$GP\"F*7$$\"+,)evS\"F* $\"+Z3i29F*7$$\"+W2[V9F*$\"+iXuW9F*7$$\"+>h@![\"F*$\"+ka,([\"F*7$$\"+2 tk;:F*$\"+:)GS`\"F*7$$\"+&yB,b\"F*$\"+Occ#e\"F*7$$\"+2;\\)e\"F*$\"+#)) Gek\"F*7$$\"+V#3Gi\"F*$\"+,j!4r\"F*7$$\"+OpRf;F*$\"+5YV\"z\"F*7$$\"+gp !pn\"F*$\"++F&\\$=F*7$$\"+$)pT%p\"F*$\"+hbJ#)=F*7$$\"+&*QB8F*7$$\"+230KF*-%*AXESTICKSG6$\"\"$Fj[l-%&COLORG6&%$RG BGF0$\"\"#!\"\"$\"\")Fa\\l-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F+$F`\\ lF,F\\]l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f (x) = 2/sqrt(4-x^2);" "6#/-%\"fG6#%\"xG*&\"\"#\"\"\"-%%sqrtG6#,&\"\"%F **$F'F)!\"\"F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The are a of the region is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. sqrt(3));" "6#/%\"AG-%$IntG6$-%\"fG6#%\"xG/F+ ;\"\"!-%%sqrtG6#\"\"$" }{XPPEDIT 18 0 "`` = Int(2/sqrt(4-x^2),x = 0 .. sqrt(3));" "6#/%!G-%$IntG6$*&\"\"#\"\"\"-%%sqrtG6#,&\"\"%F**$%\"xGF)! \"\"F2/F1;\"\"!-F,6#\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*arcsin(x/2)" "6#/%!G*&\"\"#\"\"\"-%'arcs inG6#*&%\"xGF'F&!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sq rt(3), ``],[0, ``])" "6#-%*PIECEWISEG6$7$-%%sqrtG6#\"\"$%!G7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2*arcsin(sqrt(3)/2)" "6#/%!G*&\"\"#\"\"\"-%'arcsinG6#*&-%%sqrtG6#\"\"$ F'F&!\"\"F'" }{XPPEDIT 18 0 "`` = 2*``(Pi/3);" "6#/%!G*&\"\"#\"\"\"-F$ 6#*&%#PiGF'\"\"$!\"\"F'" }{XPPEDIT 18 0 "`` = 2*Pi/3;" "6#/%!G*(\"\"# \"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The integral for the moment of the region about the " }{TEXT 419 1 "y" }{TEXT -1 44 " axis can be found \+ with of the substitution " }{XPPEDIT 18 0 "u=4-x^2" "6#/%\"uG,&\"\"%\" \"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*f(x),x = 0 .. sqrt(3));" "6#/&%\"MG 6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%\"fG6#F,F-/F,;\"\"!-%%sqrtG6#\"\"$" } {XPPEDIT 18 0 "`` = Int(2*x/sqrt(4-x^2),x = 0 .. sqrt(3));" "6#/%!G-%$ IntG6$*(\"\"#\"\"\"%\"xGF*-%%sqrtG6#,&\"\"%F**$F+F)!\"\"F2/F+;\"\"!-F- 6#\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(2*x/sqrt(4-x^2),x = 0 . . sqrt(3))" "6#/%!G-%$IntG6$*(\"\"#\"\"\"%\"xGF*-%%sqrtG6#,&\"\"%F**$F +F)!\"\"F2/F+;\"\"!-F-6#\"\"$" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = 4-x^2, `x =`*0*` implies u =`*4],[du = -2*x*dx, `x =` *sqrt(3)*` implies u =`*1],[-du = 2*x*dx, ``]);" "6#-%*PIECEWISEG6%7$ /%\"uG,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"**%$x~=GF+\"\"!F+%.~implies~~u~=GF +F*F+7$/%#duG,$*(F.F+F-F+%#dxGF+F/**F1F+-%%sqrtG6#\"\"$F+F3F+F+F+7$/,$ F6F/*(F.F+F-F+F9F+%!G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -Int(1/sqrt(u),u = 4 .. 1);" "6#/%!G,$-%$IntG 6$*&\"\"\"F*-%%sqrtG6#%\"uG!\"\"/F.;\"\"%F*F/" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(u^(-1/2),u = 1 .. 4);" "6#/%!G-%$IntG6$)%\"uG,$*&\"\"\"F,\"\"#!\"\"F./F);F,\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 78 "( because changing the \+ order of the limits changes the sign of the integral ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=2*u^(1/2)" "6#/%!G*&\"\"#\"\"\")%\"uG*&F'F'F&!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\" \"%%!G7$\"\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*sqrt(u);" "6# /%!G*&\"\"#\"\"\"-%%sqrtG6#%\"uGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PI ECEWISE([4, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"\"F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4-2;" "6#/%!G,&\"\"%\"\"\"\"\"#!\" \"" }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 35 "The moment of the region about the " }{TEXT 413 1 "x" } {TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 0 .. sqrt(3));" "6#/&%\"MG6#%\"xG-%$IntG 6$*&-%\"fG6#F'\"\"#F/!\"\"/F';\"\"!-%%sqrtG6#\"\"$" }{XPPEDIT 18 0 "`` = Int(``(1/2)*``(4/(4-x^2)),x = 0 .. sqrt(3));" "6#/%!G-%$IntG6$*&-F$ 6#*&\"\"\"F,\"\"#!\"\"F,-F$6#*&\"\"%F,,&F2F,*$%\"xGF-F.F.F,/F5;\"\"!-% %sqrtG6#\"\"$" }{XPPEDIT 18 0 " ``=Int(2/(4-x^2),x = 0 .. sqrt(3))" "6 #/%!G-%$IntG6$*&\"\"#\"\"\",&\"\"%F**$%\"xGF)!\"\"F//F.;\"\"!-%%sqrtG6 #\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The integrand \+ " }{XPPEDIT 18 0 "2/(4-x^2) = 2/((2+x)*(2-x));" "6#/*&\"\"#\"\"\",&\" \"%F&*$%\"xGF%!\"\"F+*&F%F&*&,&F%F&F*F&F&,&F%F&F*F+F&F+" }{TEXT -1 47 " has a partial fraction expansion of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2/((2+x)*(2-x)) = A/(2+x)+B/(2-x);" "6# /*&\"\"#\"\"\"*&,&F%F&%\"xGF&F&,&F%F&F)!\"\"F&F+,&*&%\"AGF&,&F%F&F)F&F +F&*&%\"BGF&,&F%F&F)F+F+F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "It is sufficient to find numbers " }{TEXT 417 1 "A" }{TEXT -1 5 " and " }{TEXT 418 1 "B" }{TEXT -1 11 " such that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2 = A*(2-x)+B*(2+x);" "6#/\"\"#,&* &%\"AG\"\"\",&F$F(%\"xG!\"\"F(F(*&%\"BGF(,&F$F(F*F(F(F(" }{TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 15 "is an identity." }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"# " }{TEXT -1 14 ", (i) becomes " }{XPPEDIT 18 0 "2 = 4*B;" "6#/\"\"#*& \"\"%\"\"\"%\"BGF'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "B = 1/2; " "6#/%\"BG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 11 ", and when " }{XPPEDIT 18 0 "x=-2" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 14 ", (i) becomes " } {XPPEDIT 18 0 "2 = 4*A;" "6#/\"\"#*&\"\"%\"\"\"%\"AGF'" }{TEXT -1 10 " , so that " }{XPPEDIT 18 0 "A = 1/2;" "6#/%\"AG*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 " 2/(4-x^2) = 1/2;" "6#/*&\"\"#\"\"\",&\"\"%F&*$%\"xGF%!\"\"F+*&F&F&F%F+ " }{XPPEDIT 18 0 "``(1/(2+x)+1/(2-x))" "6#-%!G6#,&*&\"\"\"F(,&\"\"#F(% \"xGF(!\"\"F(*&F(F(,&F*F(F+F,F,F(" }{TEXT -1 20 ", which means that: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = 1/2;" "6#/ &%\"MG6#%\"xG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int (``(1/(2+x)+1/(2-x)),x = 0 .. sqrt(3))" "6#-%$IntG6$-%!G6#,&*&\"\"\"F+ ,&\"\"#F+%\"xGF+!\"\"F+*&F+F+,&F-F+F.F/F/F+/F.;\"\"!-%%sqrtG6#\"\"$" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` (ln(abs(2+x))-ln(abs(2-x)))" "6#-%!G6#,&-%#lnG6#-%$absG6#,&\"\"#\"\"\" %\"xGF/F/-F(6#-F+6#,&F.F/F0!\"\"F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PI ECEWISE([sqrt(3), ``],[0, ``])" "6#-%*PIECEWISEG6$7$-%%sqrtG6#\"\"$%!G 7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/2 ;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs ((2+x)/(2-x)))" "6#-%#lnG6#-%$absG6#*&,&\"\"#\"\"\"%\"xGF,F,,&F+F,F-! \"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sqrt(3), ``],[0, `` ]);" "6#-%*PIECEWISEG6$7$-%%sqrtG6#\"\"$%!G7$\"\"!F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "ln((2+sqrt(3))/(2-sqrt(3)))" "6#-%#lnG6#*&,&\"\"#\" \"\"-%%sqrtG6#\"\"$F)F),&F(F)-F+6#F-!\"\"F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 414 1 "x" }{TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"MG6#%\"yG \"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/``(2*Pi/3);" "6#/%!G*&\"\"#\"\"\"-F$6#*(F&F'%#PiGF'\"\"$!\"\"F-" }{XPPEDIT 18 0 "`` = 3/Pi;" "6#/%!G*&\"\"$\"\"\"%#PiG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 415 1 "y" }{TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "c onjugate(y) = M[x]/A;" "6#/-%*conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"% \"AG!\"\"" }{XPPEDIT 18 0 "`` = ``(1/2)*ln((2+sqrt(3))/(2-sqrt(3)))/`` (2*Pi/3);" "6#/%!G*(-F$6#*&\"\"\"F)\"\"#!\"\"F)-%#lnG6#*&,&F*F)-%%sqrt G6#\"\"$F)F),&F*F)-F26#F4F+F+F)-F$6#*(F*F)%#PiGF)F4F+F+" }{XPPEDIT 18 0 "`` = 3/(4*Pi);" "6#/%!G*&\"\"$\"\"\"*&\"\"%F'%#PiGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((2+sqrt(3))/(2-sqrt(3)));" "6#-%#lnG6#*&,& \"\"#\"\"\"-%%sqrtG6#\"\"$F)F),&F(F)-F+6#F-!\"\"F1" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The cent roid is located at the point " }{XPPEDIT 18 0 "``(3/Pi,3/(4*Pi)*ln((2+ sqrt(3))/(2-sqrt(3))));" "6#-%!G6$*&\"\"$\"\"\"%#PiG!\"\"*(F'F(*&\"\"% F(F)F(F*-%#lnG6#*&,&\"\"#F(-%%sqrtG6#F'F(F(,&F3F(-F56#F'F*F*F(" } {TEXT -1 2 " " }{TEXT 423 1 "~" }{TEXT -1 33 " ( 0.9549296586,0.6288 010774 ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 315 "f := x -> 2/sqrt(4-x^2):\nInt(f(x),x=0..sqrt(3)); \nA := value(%);\nInt(x*f(x),x=0..sqrt(3));\nMy := value(%);\nInt(f(x) ^2/2,x=0..sqrt(3));\nMx := combine(value(%));\nxG := My/A;\nyG := Mx/A ;\nxGf := evalf(evalf(xG,13)):\nyGf := evalf(evalf(yG,13)):\nprint(`Th e centroid is located at the point .. `,``(xG,yG)*`~`*``(xGf,yGf));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#\"\"\",&\"\"%F)*$)% \"xGF(F)!\"\"#F/F(F)/F.;\"\"!*$\"\"$#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*(\"\"#\"\"\"\"\"$!\"\"%#PiGF(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#\"\"\"%\"xGF),&\"\"%F)*$)F*F(F )!\"\"#F/F(F)/F*;\"\"!*$\"\"$#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#MyG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"#\"\" \",&\"\"%F)*$)%\"xGF(F)!\"\"F/F)/F.;\"\"!*$\"\"$#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG,$*&#\"\"\"\"\"#F(-%#lnG6#*&,&F)F(*$\"\"$#F( F)!\"\"F(,&F)F(F/F(F2F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG,$* &\"\"$\"\"\"%#PiG!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*& #\"\"$\"\"%\"\"\"*&-%#lnG6#*&,&\"\"#F**$F(#F*F1!\"\"F*,&F1F*F2F*F4F*%# PiGF4F*F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%IThe~centroid~is~located ~at~the~point~..~G*(-%!G6$,$*&\"\"$\"\"\"%#PiG!\"\"F+,$*&#F*\"\"%F+*&- %#lnG6#*&,&\"\"#F+*$F*#F+F8F-F+,&F8F+F9F+F-F+F,F-F+F-F+%\"|irGF+-F&6$$ \"+'e'H\\&*!#5$\"+u2,)G'FAF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "We can draw a picture which shows the cen troid of the region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 361 "f := x -> 2/sqrt(4-x^2):\nxG := 3/Pi;\ny G := 3/(4*Pi)*ln((2+sqrt(3))/(2-sqrt(3)));\np1 := plots[polygonplot]([ [sqrt(3),0],[0,0],op(op(1,op(1,plot(f(x),x=0..sqrt(3)))))],\n color =COLOR(RGB,1,.2,.8)):\np2 := plot([[[xG,yG]]$3],style=point,symbol=[ci rcle,diamond,cross],color=black):\nplots[display]([p1,p2],labels=[`x`, `y`],view=[0..2,0..2],\n tickmarks=[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG,$*&\"\"$\"\"\"%#PiG!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*&#\"\"$\"\"%\"\"\"*&%#PiG!\"\"-%#lnG6#*&,&\"\" #F**$F(#F*F3F*F*,&F3F*F4F-F-F*F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 337 249 249 {PLOTDATA 2 "6)-%)POLYGONSG6$7W7$$\"+330K(e^#F?$\"+qr+35F*7$$\"+=9K$)GF?$ \"+unb55F*7$$\"+,Df\\KF?$\"+`qY85F*7$$\"+ItMEOF?$\"+RZ&o,\"F*7$$\"+*=$ >eRF?$\"+o)y,-\"F*7$$\"+GIxJVF?$\"+$>9V-\"F*7$$\"+Ao)oq%F?$\"+y'**)G5F *7$$\"+p%y$o]F?$\"+=\\uL5F*7$$\"+;vk'R&F?$\"+d:_Q5F*7$$\"+l6*py&F?$\"+ )4)oW5F*7$$\"+OMm$\\37\"F*7$$\"+z()*fO*F?$\"+1GxJ6F*7$$\"+yG -D(*F?$\"+B;SW6F*7$$\"+!*Hf45F*$\"+p&G%e6F*7$$\"+2!ze/\"F*$\"+2(*>t6F* 7$$\"+$f()43\"F*$\"+@)o&)=\"F*7$$\"+#pp*>6F*$\"+_k*p?\"F*7$$\"+Vo*\\: \"F*$\"+h?!\\A\"F*7$$\"+-eR#>\"F*$\"+l?eX7F*7$$\"+<[GE7F*$\"+;q'eE\"F* 7$$\"+jWLj7F*$\"+5P#**G\"F*7$$\"+#H&>)H\"F*$\"+o)pXJ\"F*7$$\"+.\\jM8F* $\"+r-pU8F*7$$\"+$Gh-P\"F*$\"+cR$GP\"F*7$$\"+,)evS\"F*$\"+Z3i29F*7$$\" +W2[V9F*$\"+iXuW9F*7$$\"+>h@![\"F*$\"+ka,([\"F*7$$\"+2tk;:F*$\"+:)GS` \"F*7$$\"+&yB,b\"F*$\"+Occ#e\"F*7$$\"+2;\\)e\"F*$\"+#))Gek\"F*7$$\"+V# 3Gi\"F*$\"+,j!4r\"F*7$$\"+OpRf;F*$\"+5YV\"z\"F*7$$\"+gp!pn\"F*$\"++F& \\$=F*7$$\"+$)pT%p\"F*$\"+hbJ#)=F*7$$\"+&*QB8F*7$$\"+230K< F*$\"+(*******>F*-%&COLORG6&%$RGBGF0$\"\"#!\"\"$\"\")F]\\l-%'CURVESG6& 7#7$$\"3=?P^&e'H\\&*!#=$\"3=SZ=u2,)G'Fg\\l-%'SYMBOLG6#%'CIRCLEG-%'COLO URG6&Fj[lF,F,F,-%&STYLEG6#%&POINTG-Fa\\l6&Fc\\l-F[]l6#%(DIAMONDGF^]lFa ]l-Fa\\l6&Fc\\l-F[]l6#%&CROSSGF^]lFa]l-%*AXESTICKSG6$\"\"$Fb^l-%+AXESL ABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F+$F\\\\lF,F__l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" } {TEXT 384 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 116 "Find the coordinates o f the centroid of the plane region in the first quadrant bounded by th e section of the curve " }{XPPEDIT 18 0 "y = cos*x;" "6#/%\"yG*&%$cos G\"\"\"%\"xGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0<=x" "6#1\"\"! %\"xG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT -1 10 ", and the " }{TEXT 386 1 "x" }{TEXT -1 6 " axis." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solutio n" }{TEXT 385 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "f := x -> cos(x):\nplots[polygonplot]([[0, 0],op(op(1,op(1,plot(f(x),x=0..Pi/2))))],\n color=COLOR(RGB,.6,1,.7 ),labels=[`x`,`y`],view=[0..1.8,0..1.1],\n tickmarks=[4,3]);\n" }} {PARA 13 "" 1 "" {GLPLOT2D 316 211 211 {PLOTDATA 2 "6'-%)POLYGONSG6#7T 7$$\"\"!F)F(7$F($\"\"\"F)7$$\"+j*))QU$!#6$\"+l!RT***!#57$$\"+Uk*HS'F0$ \"+@y]z**F37$$\"+yVJ`(*F0$\"+ATZ_**F37$$\"+SSe78F3$\"+l(zR\"**F37$$\"+ RPB[;F3$\"+7NZk)*F37$$\"+wRUf>F3$\"+)GY'3)*F37$$\"+TMk\"G#F3$\"+EC$3u* F37$$\"+uK)[h#F3$\"+zG1g'*F37$$\"+-W0ZHF3$\"+6u()o&*F37$$\"+OKt)G$F3$ \"+Y$oSY*F37$$\"+RTo*e$F3$\"+osfi$*F37$$\"+wN[GRF3$\"+oRAQ#*F37$$\"+cT noUF3$\"+1)F3$\"+\"Gew#oF37$$\"+-8-%\\)F3$\"+e>K /mF37$$\"+V,i>))F3$\"+\"=xjN'F37$$\"+1c*f:*F3$\"+mF:$4'F37$$\"+rL2&[*F 3$\"+hf'*GeF37$$\"+IIZ.)*F3$\"+!zStc&F37$$\"+d,q:5!\"*$\"+H.Dq_F37$$\" +oiYZ5Fbu$\"+lEn(*\\F37$$\"+sLQ\"3\"Fbu$\"+y$y5q%F37$$\"+4t676Fbu$\"+- 0kFWF37$$\"+@wrX6Fbu$\"+X[#R7%F37$$\"+'*GLx6Fbu$\"+u[*Q$QF37$$\"+n*z.@ \"Fbu$\"+D(Qm_$F37$$\"+?&*oU7Fbu$\"+!39DA$F37$$\"+FY^w7Fbu$\"+P]_+HF37 $$\"+EA448Fbu$\"+O+F(e#F37$$\"+EvSU8Fbu$\"+xX3kAF37$$\"+fpWv8Fbu$\"++F 4T>F37$$\"+mn!eS\"Fbu$\"+Q.UU;F37$$\"+JDgS9Fbu$\"+4IE)H\"F37$$\"+yUsr9 Fbu$\"+de+\"*)*F07$$\"+^l!\\]\"Fbu$\"+Q4@%e'F07$$\"+@imO:Fbu$\"+()zM7M F07$$\"+Cjzq:Fbu$\"+Fm*[9$!#=-%&COLORG6&%$RGBG$\"\"'!\"\"F,$\"\"(Fa[l- %*AXESTICKSG6$\"\"%\"\"$-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F($\"#=Fa [l;F($\"#6Fa[l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = cos*x;" "6#/-%\"fG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The area of the region is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = Int(f(x),x = 0 .. Pi/2);" "6#/%\"AG-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!*&%#PiG\"\"\"\"\"#!\" \"" }{XPPEDIT 18 0 "``= Int(cos*x,x = 0 .. Pi/2)" "6#/%!G-%$IntG6$*&%$ cosG\"\"\"%\"xGF*/F+;\"\"!*&%#PiGF*\"\"#!\"\"" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin*x;" "6#/%!G* &%$sinG\"\"\"%\"xGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F ," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = sin(Pi/2)-0;" "6#/%!G,&-%$sinG6#*&%#PiG\"\"\"\"\"#!\"\"F+\"\"!F- " }{XPPEDIT 18 0 "`` = 1;" "6#/%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The moment " } {XPPEDIT 18 0 "M[y]" "6#&%\"MG6#%\"yG" }{TEXT -1 25 " of the region ab out the " }{TEXT 387 1 "y" }{TEXT -1 59 " axis can be found using the \+ integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v -Int(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\" \"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = In t(x*f(x),x = 0 .. Pi/2);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\"-%\"f G6#F,F-/F,;\"\"!*&%#PiGF-\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(x*cos*x,x = 0 .. Pi/2);" "6 #/%!G-%$IntG6$*(%\"xG\"\"\"%$cosGF*F)F*/F);\"\"!*&%#PiGF*\"\"#!\"\"" } {TEXT -1 5 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = x, v = sin*x],[du/dx = 1, dv/dx = cos*x]);" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%\"vG*&%$sinG \"\"\"F)F.7$/*&%#duGF.%#dxG!\"\"F./*&%#dvGF.F3F4*&%$cosGF.F)F." } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$ *&%\"uG\"\"\"-F$6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{XPPEDIT 18 0 " ``= u*v -Int(v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$ 6#*&%#duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ x*sin*x;" "6#/%!G*(%\"xG\"\"\"%$sinGF'F&F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``])" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"\" \"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(sin*x,x = 0 .. Pi/2);" "6#,$-%$IntG6$*&%$sinG\"\"\"%\"xGF)/F*;\"\"!*&%#PiGF)\" \"#!\"\"F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = x*sin*x;" "6#/%!G*(%\"xG\"\"\"%$sinGF'F&F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``])" "6#-%*PI ECEWISEG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "``+cos*x;" "6#,&%!G\"\"\"*&%$cosGF%%\"xGF%F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``])" "6#-%*PIECEWIS EG6$7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi/2-1" "6#/%!G,&*&%#PiG \"\"\"\"\"#!\"\"F(F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the \+ " }{TEXT 388 1 "x" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int(f(x)^2/2,x = 0 .. Pi/2);" "6 #/&%\"MG6#%\"xG-%$IntG6$*&-%\"fG6#F'\"\"#F/!\"\"/F';\"\"!*&%#PiG\"\"\" F/F0" }{XPPEDIT 18 0 "`` = Int(cos^2*x/2,x = 0..Pi/2)" "6#/%!G-%$IntG6 $*(%$cosG\"\"#%\"xG\"\"\"F*!\"\"/F+;\"\"!*&%#PiGF,F*F-" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(1/4+cos*2* x/4,x=0..Pi/2)" "6#/%!G-%$IntG6$,&*&\"\"\"F*\"\"%!\"\"F***%$cosGF*\"\" #F*%\"xGF*F+F,F*/F0;\"\"!*&%#PiGF*F/F," }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x/4+sin(2*x)/8" "6#/%!G,&*&% \"xG\"\"\"\"\"%!\"\"F(*&-%$sinG6#*&\"\"#F(F'F(F(\"\")F*F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/2, ``],[0, ``])" "6#-%*PIECEWISEG6 $7$*&%#PiG\"\"\"\"\"#!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Pi/8" "6#/%!G*&%#PiG\"\"\"\" \")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 389 1 "x" }{TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjugate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"MG6#%\"yG \"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi/2-1;" "6#/%! G,&*&%#PiG\"\"\"\"\"#!\"\"F(F(F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 390 1 "y" } {TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjug ate(y) = M[x]/A;" "6#/-%*conjugateG6#%\"yG*&&%\"MG6#%\"xG\"\"\"%\"AG! \"\"" }{XPPEDIT 18 0 "`` = Pi/8;" "6#/%!G*&%#PiG\"\"\"\"\")!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at the point" }{XPPEDIT 18 0 "``(Pi/2-1,Pi/8);" "6#-%!G6$,&*&%#PiG\" \"\"\"\"#!\"\"F)F)F+*&F(F)\"\")F+" }{TEXT -1 2 " " }{TEXT 391 1 "~" } {TEXT -1 31 " (0.5707963268, 0.3926990817). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "f := x -> cos(x): \nInt(f(x),x=0..Pi/2);\nA := value(%);\nInt(x*f(x),x=0..Pi/2);\nMy := \+ value(%);\nInt(f(x)^2/2,x=0..Pi/2);\nMx := value(%);\nxG := My/A;\nyG \+ := Mx/A;\nxGf := evalf(evalf(xG,13)):\nyGf := evalf(evalf(yG,13)):\npr int(`The centroid is located at the point .. `,``(xG,yG)*`~`*``(xGf,yG f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#%\"xG/F);\" \"!,$*&\"\"#!\"\"%#PiG\"\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" AG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$c osG6#F'F(/F';\"\"!,$*&\"\"#!\"\"%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG,&*&\"\"#!\"\"%#PiG\"\"\"F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&#\"\"\"\"\"#F)*$)-%$cosG6#%\"xGF*F)F)F) /F0;\"\"!,$*&F*!\"\"%#PiGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Mx G,$*&\"\")!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG ,&*&\"\"#!\"\"%#PiG\"\"\"F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# yGG,$*&\"\")!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%ITh e~centroid~is~located~at~the~point~..~G*(-%!G6$,&*&\"\"#!\"\"%#PiG\"\" \"F-F-F+,$*&\"\")F+F,F-F-F-%\"|irGF--F&6$$\"+oK'zq&!#5$\"+<3*p#RF6F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can draw a picture which shows the centroid of the region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 313 "f := x -> cos(x):\nxG := Pi/2-1;\nyG := Pi/8;\np1 := plots[polygonplot]([[0, 0],op(op(1,op(1,plot(f(x),x=0..Pi/2))))],\n color=COLOR(RGB,.6,1,.7 )):\np2 := plot([[[xG,yG]]$3],style=point,symbol=[circle,diamond,cross ],color=black):\nplots[display]([p1,p2],labels=[`x`,`y`],view=[0..1.8, 0..1.1],\n tickmarks=[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#xGG,&*&\"\"#!\"\"%#PiG\"\"\"F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*&\"\")!\"\"%#PiG\"\"\"F*" }}{PARA 13 "" 1 "" {GLPLOT2D 337 249 249 {PLOTDATA 2 "6)-%)POLYGONSG6$7T7$$\"\"!F)F(7$F($\"\"\"F)7$ $\"+j*))QU$!#6$\"+l!RT***!#57$$\"+Uk*HS'F0$\"+@y]z**F37$$\"+yVJ`(*F0$ \"+ATZ_**F37$$\"+SSe78F3$\"+l(zR\"**F37$$\"+RPB[;F3$\"+7NZk)*F37$$\"+w RUf>F3$\"+)GY'3)*F37$$\"+TMk\"G#F3$\"+EC$3u*F37$$\"+uK)[h#F3$\"+zG1g'* F37$$\"+-W0ZHF3$\"+6u()o&*F37$$\"+OKt)G$F3$\"+Y$oSY*F37$$\"+RTo*e$F3$ \"+osfi$*F37$$\"+wN[GRF3$\"+oRAQ#*F37$$\"+cTnoUF3$\"+1)F3$\"+\"Gew#oF37$$\"+-8-%\\)F3$\"+e>K/mF37$$\"+V,i>))F3$\"+\"=x jN'F37$$\"+1c*f:*F3$\"+mF:$4'F37$$\"+rL2&[*F3$\"+hf'*GeF37$$\"+IIZ.)*F 3$\"+!zStc&F37$$\"+d,q:5!\"*$\"+H.Dq_F37$$\"+oiYZ5Fbu$\"+lEn(*\\F37$$ \"+sLQ\"3\"Fbu$\"+y$y5q%F37$$\"+4t676Fbu$\"+-0kFWF37$$\"+@wrX6Fbu$\"+X [#R7%F37$$\"+'*GLx6Fbu$\"+u[*Q$QF37$$\"+n*z.@\"Fbu$\"+D(Qm_$F37$$\"+?& *oU7Fbu$\"+!39DA$F37$$\"+FY^w7Fbu$\"+P]_+HF37$$\"+EA448Fbu$\"+O+F(e#F3 7$$\"+EvSU8Fbu$\"+xX3kAF37$$\"+fpWv8Fbu$\"++F4T>F37$$\"+mn!eS\"Fbu$\"+ Q.UU;F37$$\"+JDgS9Fbu$\"+4IE)H\"F37$$\"+yUsr9Fbu$\"+de+\"*)*F07$$\"+^l !\\]\"Fbu$\"+Q4@%e'F07$$\"+@imO:Fbu$\"+()zM7MF07$$\"+Cjzq:Fbu$\"+Fm*[9 $!#=-%&COLORG6&%$RGBG$\"\"'!\"\"F,$\"\"(Fa[l-%'CURVESG6&7#7$$\"3el*[zE jzq&Fjz$\"3STs)p\"3*p#RFjz-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&F^[lF)F)F)- %&STYLEG6#%&POINTG-Fe[l6&Fg[l-F^\\l6#%(DIAMONDGFa\\lFd\\l-Fe[l6&Fg[l-F ^\\l6#%&CROSSGFa\\lFd\\l-%*AXESTICKSG6$\"\"%\"\"$-%+AXESLABELSG6%%\"xG %\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#=Fa[l;F($\"#6Fa[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ex ample 9" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 339 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 83 "Find the coordinates of the centroid of the plane region the bounded by the curves " }{XPPEDIT 18 0 "y = sqrt (4-x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = sqrt(1-x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)* $%\"xG\"\"#!\"\"" }{TEXT -1 9 " and the " }{TEXT 369 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 262 8 "Solution" }{TEXT 340 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "Firs t we draw a picture of the region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 470 "u := evalf(Pi/60,15):\nleft := [seq([2*cos(i*u),2*sin(i*u)],i=45..60),\n seq([cos((60-i)*u),sin( (60-i)*u)],i=0..30),\n seq([2*cos(i*u),2*sin(i*u)],i=30..44)]:\np1 := plots[polygonplot](left,color=COLOR(RGB,.9,.6,.5)):\nright := [seq([2 *cos(i*u),2*sin(i*u)],i=15..30),\n seq([cos((30-i)*u),sin((30-i)*u)] ,i=0..30),\n seq([2*cos(i*u),2*sin(i*u)],i=0..14)]:\np2 := plots[pol ygonplot](right,color=COLOR(RGB,.9,.6,.5)):\nplots[display]([p1,p2],la bels=[`x`,`y`],tickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 300 162 162 {PLOTDATA 2 "6&-%)POLYGONSG6$7jn7$$!+iN@99!\"*$\"+jN@99F*7$$!+ ^'*G'[\"F*$\"+77EQ8F*7$$!+B>Ha:F*$\"+$yS'e7F*7$$!+*)R.=;F*$\"+/0dv6F*7 $$!+N6Mx;F*$\"+r!y#*3\"F*7$$!+330KF* $\"+#*)R.='FK7$$!+`;&=$>F*$\"+)*3Qw^FK7$$!+,_Hc>F*$\"+CQBeTFK7$$!+\"ow `(>F*$\"++$*oGJFK7$$!+\"zV!*)>F*$\"+u#p04#FK7$$!+q!fs*>F*$\"+D\">n/\"F K7$$!\"#\"\"!$!+I_8/#)!#>7$$!\"\"Fap$!+:w1-TFdp7$$!+[`H')**FK$\"+FcfL_ !#67$$!+`*=_%**FK$\"+PYGX5FK7$$!+1M)o()*FK$\"+]YMk:FK7$$!+2gZ\"y*FK$\" +7p6z?FK7$$!+j#e#f'*FK$\"+\\/>)e#FK7$$!+i^c5&*FK$\"+Y*p,4$FK7$$!+mU!eL *FK$\"+#\\zOe$FK7$$!+wXXN\"*FK$\"+KkOnSFK7$$!+W_15*)FK$\"+$*\\!*RXFK7$ $!+QSDg')FK$\"+++++]FK7$$!+xcq'Q)FK$\"+a.RYaFK7$$!+X*p,4)FK$\"+AD&y(eF K7$$!+8'f9x(FK$\"+8R?$H'FK7$$!+c#[9V(FK$\"+igI\"p'FK7$$!+6y1rqFK$\"+8y 1rqFK7$$!+mgI\"p'FK$\"+_#[9V(FK7$$!+5R?$H'FK$\"+:'f9x(FK7$$!+FD&y(eFK$ \"+T*p,4)FK7$$!+^.RYaFK$\"+zcq'Q)FK7$$!+(*******\\FK$\"+SSDg')FK7$$!+* *\\!*RXFK$\"+T_15*)FK7$$!+GkOnSFK$\"+yXXN\"*FK7$$!+)\\zOe$FK$\"+kU!eL* FK7$$!+U*p,4$FK$\"+j^c5&*FK7$$!+b/>)e#FK$\"+i#e#f'*FK7$$!+3p6z?FK$\"+2 gZ\"y*FK7$$!+YYMk:FK$\"+2M)o()*FK7$$!+MYGX5FK$\"+a*=_%**FK7$$!+*e&fL_F _q$\"+[`H')**FK7$$!+3Q.^?Fdp$\"\"\"Fap7$$!+;w1-TFdp$\"\"#Fap7$$!+=\">n /\"FK$\"+q!fs*>F*7$$!+o#p04#FK$\"+\"zV!*)>F*7$$!+#H*oGJFK$\"+\"ow`(>F* 7$$!+;QBeTFK$\"+,_Hc>F*7$$!+54Qw^FK$\"+_;&=$>F*7$$!+%))R.='FK$\"+LI6-> F*7$$!+'**et;(FK$\"+`3;n=F*7$$!+cGtM\")FK$\"+;44F=F*7$$!+)**4)z!*FK$\" +[I,#y\"F*7$$!+%*********FK$\"+330KHa:F*7$$!+87EQ8F*$\"+]'*G'[\"F*-% &COLORG6&%$RGBG$\"\"*Fgp$\"\"'Fgp$\"\"&Fgp-F$6$7jn7$$\"+iN@99F*Fj_l7$$ \"+87EQ8F*$\"+^'*G'[\"F*7$$\"+#yS'e7F*Fe^l7$$\"+00dv6F*$\"+*)R.=;F*7$$ \"+q!y#*3\"F*F[^l7$FD$\"+230KF*7$$\"+ -4Qw^FK$\"+`;&=$>F*7$$\"+3QBeTFK$\"+-_Hc>F*7$$\"+/$*oGJFKFc[l7$$\"+e#p 04#FKF^[l7$$\"+I\">n/\"FK$\"+p!fs*>F*FazF\\z7$$\"+[cfL_F_q$\"+Z`H')**F K7$$\"+HYGX5FKFey7$$\"+_YMk:FK$\"+1M)o()*FK7$$\"+/p6z?FK$\"+3gZ\"y*FK7 $$\"+^/>)e#FK$\"+j#e#f'*FK7$$\"+[*p,4$FK$\"+i^c5&*FK7$$\"+%\\zOe$FK$\" +lU!eL*FK7$$\"+MkOnSFK$\"+vXXN\"*FK7$$\"+&*\\!*RXFK$\"+V_15*)FK7$$\"+- +++]FK$\"+PSDg')FK7$$\"+].RYaFKFhv7$$\"+BD&y(eFK$\"+W*p,4)FK7$$\"+6R?$ H'FK$\"+9'f9x(FK7$$\"+jgI\"p'FK$\"+b#[9V(FK7$$\"+7y1rqFKFigl7$FfglFdgl 7$FaglF_gl7$F\\glFjfl7$Fhv$\"+^.RYaFK7$$\"+QSDg')FKF[t7$$\"+U_15*)FK$ \"+(*\\!*RXFK7$$\"+wXXN\"*FK$\"+JkOnSFK7$Feel$\"+&\\zOe$FK7$Fax$\"+W*p ,4$FK7$F[elFidl7$F[y$\"+3p6z?FK7$Fadl$\"+^YMk:FK7$Fey$\"+KYGX5FK7$Fjy$ \"+CcfL_F_q7$F_z$FapFap7$FdzFbjl7$FizF\\p7$F^[l$\"+k#p04#FK7$Fc[l$\"+- $*oGJFK7$Fh[l$\"+;QBeTFK7$FdblFbbl7$Fb\\l$\"+)))R.='FK7$Fg\\l$\"+!**et ;(FK7$Fgal$\"+iGtM\")FK7$Fa]l$\"+%**4)z!*FK7$Ff]lFD7$F[^lFj`l7$Fg`lFe` l7$Fe^lFb`l7$F_`lF]`lF\\_l-%*AXESTICKSG6$Fe_l\"\"$-%+AXESLABELSG6$%\"x G%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "By symmet ry, the centroid lies on the " }{TEXT 370 1 "y" }{TEXT -1 6 " axis." } }{PARA 0 "" 0 "" {TEXT -1 9 "The area " }{TEXT 394 1 "A" }{TEXT -1 74 " of the region can be computed using the formula for the area of a ci rcle." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 393 1 "A" }{TEXT -1 90 " = \"the area of a semi-circle of radius 2\" minus \"the area of a semi-circle of radius 1\"" }}{PARA 256 "" 0 "" {TEXT -1 5 " = " } {XPPEDIT 18 0 "Pi*`.`*2^2/2-Pi/2 = 3*Pi/2;" "6#/,&**%#PiG\"\"\"%\".GF' \"\"#F)F)!\"\"F'*&F&F'F)F*F**(\"\"$F'F&F'F)F*" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " } {TEXT 371 1 "x" }{TEXT -1 9 " axis is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x]" "6#&%\"MG6#%\"xG" }{TEXT -1 98 " = \"the m oment of the semi-circle of radius 2\" minus \"the moment of the sem i-circle of radius 1\"" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " } {XPPEDIT 18 0 "Int(sqrt(4-x^2)^2/2,x=-2..2)-Int(sqrt(1-x^2)^2/2,x=-1.. 1)" "6#,&-%$IntG6$*&-%%sqrtG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"F0F0F1/F/; ,$F0F1F0F--F%6$*&-F)6#,&F-F-*$F/F0F1F0F0F1/F/;,$F-F1F-F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "Int((4-x^2)/ 2,x=-2..2)-Int((1-x^2)/2,x=-1..1)" "6#,&-%$IntG6$*&,&\"\"%\"\"\"*$%\"x G\"\"#!\"\"F*F-F./F,;,$F-F.F-F*-F%6$*&,&F*F**$F,F-F.F*F-F./F,;,$F*F.F* F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(2-x^2/2,x = -2 .. 2)-Int(1/2-x^2/2,x = -1 .. 1);" "6#,&-%$In tG6$,&\"\"#\"\"\"*&%\"xGF(F(!\"\"F,/F+;,$F(F,F(F)-F%6$,&*&F)F)F(F,F)*& F+F(F(F,F,/F+;,$F)F,F)F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(2*x-x^3/6)" "6#/%!G-F$6#,&*&\"\"#\"\"\" %\"xGF*F**&F+\"\"$\"\"'!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEW ISE([2,``],[-2,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$,$F'!\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "- ``(x/2-x^3/6)" "6#,$-%!G6#,&*&%\"xG\" \"\"\"\"#!\"\"F**&F)\"\"$\"\"'F,F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 " PIECEWISE([1, ``],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$,$F'!\"\" F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(``(4-4/3)-(-4+4/3))-``(``(1/2-1 /6) - (-1/2+1/6))" "6#/%!G,&-F$6#,&-F$6#,&\"\"%\"\"\"*&F,F-\"\"$!\"\"F 0F-,&F,F0*&F,F-F/F0F-F0F--F$6#,&-F$6#,&*&F-F-\"\"#F0F-*&F-F-\"\"'F0F0F -,&*&F-F-F:F0F0*&F-F-F " 0 "" {MPLTEXT 1 0 160 "Int(sqrt(4-x^2),x=-2..2)-Int(sqrt(1-x^2),x=-1..1);\n A := value(%);\nInt((4-x^2)/2,x=-2..2)-Int((1-x^2)/2,x=-1..1);\nMx := \+ value(%);\nyG := Mx/A;\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$*$,&\"\"%\"\"\"*$)%\"xG\"\"#F*!\"\"#F*F./F-; !\"#F.F*-F%6$*$,&F*F*F+F/F0/F-;F/F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*(\"\"$\"\"\"\"\"#!\"\"%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,&\"\"#\"\"\"*&F(!\"\"%\"xGF(F+/F,;!\"#F(F)- F%6$,&#F)F(F)*&F(F+F,F(F+/F,;F+F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#MxG#\"#9\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG,$*(\"#G\" \"\"\"\"*!\"\"%#PiGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PU(H!** !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 100 "Find the coo rdinates of the centroid of the plane region in the first quadrant bou nded by the curve " }{XPPEDIT 18 0 "y = 8-x^3;" "6#/%\"yG,&\"\")\"\"\" *$%\"xG\"\"$!\"\"" }{TEXT -1 9 " and the " }{TEXT 373 1 "x" }{TEXT -1 5 " and " }{TEXT 374 1 "y" }{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(4/5,24/7)" "6#-%!G6$*&\"\"%\" \"\"\"\"&!\"\"*&\"#CF(\"\"(F*" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 46 "______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "__ ____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 78 "Find the coordinates of the centroid of t he plane region bounded by the curve " }{XPPEDIT 18 0 "y = sqrt(x);" " 6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 22 " , the vertical lines " } {XPPEDIT 18 0 "x = 1, x = 4" "6$/%\"xG\"\"\"/F$\"\"%" }{TEXT -1 9 " an d the " }{TEXT 405 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(93/35,45/56)" "6#-%!G6$*&\"#$*\"\"\" \"#N!\"\"*&\"#XF(\"#cF*" }{TEXT -1 1 " " }{TEXT 406 1 "~" }{TEXT -1 31 " (2.657142857, 0.8035714286) " }}}{PARA 0 "" 0 "" {TEXT -1 46 "_ _____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________ __________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 78 "Find the coordinates of the centroid of t he plane region bounded by the curve " }{XPPEDIT 18 0 "y = exp(-x);" " 6#/%\"yG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 7 " , the " }{TEXT 407 1 "x" }{TEXT -1 5 " and " }{TEXT 408 1 "y" }{TEXT -1 28 " axes and the verti cal line " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``((exp(1)-2)/(ex p(1)-1),(exp(1)+1)/(4*exp(1)))" "6#-%!G6$*&,&-%$expG6#\"\"\"F+\"\"#!\" \"F+,&-F)6#F+F+F+F-F-*&,&-F)6#F+F+F+F+F+*&\"\"%F+-F)6#F+F+F-" }{TEXT -1 1 " " }{TEXT 409 1 "~" }{TEXT -1 30 " (0.4180232931, 0.3419698603) \+ " }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 46 "______________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 115 "Find the coordinates of the centroid of the plane region in the first quadrant bounded by \+ the section of the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$si nG\"\"\"%\"xGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0<=x" "6#1\"\" !%\"xG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 9 " and the " }{TEXT 410 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(Pi/2,Pi/8);" "6#-%!G6$*&%#PiG\"\"\"\"\"#! \"\"*&F'F(\"\")F*" }{TEXT -1 1 " " }{TEXT 271 1 "~" }{TEXT -1 28 " (1. 570796327,0.3926990817) " }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________ __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_______________________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 101 "Find the coordinates of the centroid of the plane region in the first quadrant bounded by the curves " } {XPPEDIT 18 0 "y = sqrt(4-x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"%\"\"\"*$%\"x G\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = sqrt(1-x^2)" "6#/%\" yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 9 " and the " } {TEXT 395 1 "x" }{TEXT -1 5 " and " }{TEXT 396 1 "y" }{TEXT -1 6 " axe s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(28/(9 *Pi),28/(9*Pi))" "6#-%!G6$*&\"#G\"\"\"*&\"\"*F(%#PiGF(!\"\"*&F'F(*&F*F (F+F(F," }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 46 "_____________ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_______________________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "Code f or moment of mass picture" }}{PARA 0 "" 0 "" {TEXT -1 12 "OLD PICTURE \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1257 "p1 := plot([(1+t^6)^(- 1/4),t*(1+t^6)^(-1/4),t=-50..50],\n color=brown,thickness=2):\np 5 := PLOT(POLYGONS(op(1,op(1,p1))),STYLE(PATCHNOGRID),\n \+ COLOR(RGB,.9,.8,.7)):\np2 := plot([[0,-.15],[0,.15]],color=brown,thick ness=2):\np3 := plot([[-.5,-1],[-.5,1]],color=black):\np4 := plot([[[- .5,.6],[.4,.6]],[[-.5,.1],[.15,.1]],\n [[-.5,-.36],[.71,-.36]]],lin estyle=3,color=COLOR(RGB,0,.6,0)):\np6 := plot([[-.5,-.01],[.53,-.01]] ,\n linestyle=3,color=COLOR(RGB,.3,.2,.1)):\nd1 := plottools[disk] ([.43,.6],.025, color=brown):\nd2 := plottools[disk]([.19,.1],.025, co lor=brown):\nd3 := plottools[disk]([.74,-.36],.025, color=brown):\na1 \+ := plottools[arrow]([.43,.56],[.43,.2],.01,.05,.3,color=black):\na2 := plottools[arrow]([.19,.06],[.19,-.3],.01,.05,.3,color=black):\na3 := \+ plottools[arrow]([.74,-.4],[.74,-.76],.01,.05,.3,color=black):\na4 := \+ plottools[arrow]([.54,-.02],[.54,-1.2],.01,.05,.11,color=brown):\nt1 : = plots[textplot]([[-.03,.71,`x1`],[-.17,.19,`x2`],\n [-.1,-.27,`x3`] ],color=COLOR(RGB,0,.6,0)):\n t2 := plots[textplot]([[.55,.37,`m1.g`], [.31,-.14,`m2.g`],\n[.86,-.6,`m3.g`],[.65,-.98,`M.g`],[-.55,1.06,`A`], [-.55,-1,`B`],\n[.59,.05,`G`],[-.09,-.07,`xG`]],color=black):\nplots[d isplay](\{t1,t2,p1,p2,p3,p4,p5,p6,d1,d2,d3,a1,a2,a3,a4\},axes=none);; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2251 "n1 := 30: n2 := 20:\nh1 := evalf(1/n1): h2 := evalf(1/n2):\nd := evalf(Pi/n2): a := .2: b := .1: \np1 := plot([[-1,-1],[6,6]],color=black):\npts := [seq([1.6*(i*h2)^2- 1.6*i*h2+2.8,.8*(i*h2)^2],i=0..n2),\n seq([1.6*i*h2+6.2,-.8*(i*h2 )^2+1.6*i*h2+4.2],i=0..n2),\n seq([-6.5*(i*h1)^2+8.*i*h1+10.,-2.5 *(i*h1)^2+5],i=0..n1),\n seq([-1.5*(i*h1)^2-5.*i*h1+11.5,2.5*(i*h 1-1)^2],i=0..n1),[2.8,0]]:\np2 := plot(pts,color=brown):\np3 := plots[ polygonplot](pts,color=COLOR(RGB,.9,.8,.7)):\nd1 := plots[polygonplot] ([seq([8.4+a*cos(i*d),4.2+b*sin(i*d)],i=0..2*n2)],color=brown):\na1 := plottools[arrow]([8.4,.55],[8.4,-1],.04,.2,.17,color=black):\nb1 := p lot([[8.4,4.1],[8.4,.55]],thickness=3,color=COLOR(RGB,.8,.7,.7)):\nd2 \+ := plots[polygonplot]([seq([10.1+a*cos(i*d),2.7+b*sin(i*d)],i=0..2*n2) ],color=brown):\na2 := plottools[arrow]([10.1,1.36],[10.1,-1.5],.03,.2 ,.1,color=black):\nb2 := plot([[10.1,2.6],[10.1,1.36]],thickness=3,col or=COLOR(RGB,.8,.7,.7)):\nd3 := plots[polygonplot]([seq([5.9+a*cos(i*d ),1.5+b*sin(i*d)],i=0..2*n2)],color=brown):\na3 := plottools[arrow]([5 .9,.04],[5.9,-1.5],.03,.2,.17,color=black):\nb3 := plot([[5.9,1.4],[5. 9,.04]],thickness=3,color=COLOR(RGB,.8,.7,.7)):\np4 := plot([[[1.5,1.5 ],[5.7,1.5]],[[2.7,2.7],[9.9,2.7]],[[4.2,4.2],[8.2,4.2]]],\n linestyl e=3,color=COLOR(RGB,0,.6,0)):\np5 := plot([[3.1,3.1],[7.9,3.1]],linest yle=3,color=red):\np6 := plot([[7.9,3.1],[7.9,.36]],thickness=3,color= COLOR(RGB,.9,.7,.7)):\na4 := plottools[arrow]([7.9,.36],[7.9,-2.3],0,. 2,.17,arrow,thickness=3,color=red):\nt1 := plots[textplot]([[5.6,4.6,` x`],[6.9,2.4,`x`],\n [4,1.2,`x`]],font=[HELVETICA,10],color=COLOR(RGB ,0,.6,0)):\nt2 := plots[textplot]([[5.75,4.5,`1`],[7.05,2.3,`2`],\n [ 4.15,1.1,`3`]],font=[HELVETICA,8],color=COLOR(RGB,0,.6,0)):\nt3 := plo ts[textplot]([[8.9,-.3,`m .g`],[10.6,-.55,`m .g`],\n[6.35,-.8,`m .g `],[-1.1,-1.2,`A`],[6.2,6.1,`B`]],font=[HELVETICA,10],color=black):\nt 4 := plots[textplot]([[8.92,-.4,`1`],[10.62,-.65,`2`],\n[6.37,-.9,`3`] ],font=[HELVETICA,8],color=black):\nt5 := plots[textplot]([[6.1,3.5,`x `],[8.5,-1.7,`M.g`]],\n font=[HELVETICA,10],color=red):\nt6 := p lots[textplot]([6.25,3.4,`G`],font=[HELVETICA,8],color=red):\nplots[di splay]([d1,a1,b1,d2,a2,b2,d3,a3,b3,p1,p2,p3,p4,p5,a4,p6,\n t1,t2,t3, t4,t5,t6],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "bezier([[2.8,0],[2,0],[2.8,.8]],ani mate=false,info=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "bezier([[ 6.2,4.2],[7,5],[7.8,5]],animate=false,info=true):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "bezier([[10,5],[14,5],[11.5,2.5]],animate=false,inf o=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "bezier([[11.5,2.5],[9,0 ],[5,0]],animate=false,info=true):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "3-D moment of mass picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "OLD PICTURE " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 630 "with(plots): with(plottools):\nto3d := _X- >[op(1,_X),op(2,_X),0]:\np1 := plot([(1+t^6)^(-1/4),t*(1+t^6)^(-1/4),t =-30..30],\ncolor=brown,thickness=2):\ncrv := op(1,op(1,p1)):\nsc := m ap(to3d,crv):\nlam := polygonplot3d(sc,color=COLOR(RGB,.9,.8,.7)):\nax is := pointplot3d([[-.5,-1,0],[-.5,1,0]],style=line,color=black):\ndis t := pointplot3d([[-.5,0,0],[.54,0,0]],style=line,\n linestyle=2 ,color=black):\nar := arrow([.54,0,-.02],[.54,0,-1.2],.01,.05,.11,colo r=brown):\ntxt := textplot3d([[-.15,.05,.05,xG],[-.55,1.06,.05,`A`],\n [-.55,-1,.05,`B`],[.65,0,-.98,`M.g`]],color=black):\ndisplay(\{la m,axis,dist,ar,txt\},orientation=[-71,61]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "Code for triangle pictures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 526 "f := x -> 2-x/3:\nx1 := 2.6: x2 := 2.7: \ntrng := [[0,2],[6,0],[0,-2],[0,2]] :\nstrp := [[x1,-f(x1)],[x1,f(x1)],[x2,f(x2)],[x2,-f(x2)],[x1,-f(x1)]] :\np1 := plots[polygonplot](strp,color=COLOR(RGB,.8,.8,.85),style=patc hnogrid):\np2 := plot(trng,color=COLOR(RGB,.1,.5,0),thickness=1):\np3 \+ := plots[polygonplot](trng,color=COLOR(RGB,.9,.9,.95)):\np4 := plot(st rp,color=black,linestyle=1):\nt1 := plots[textplot]([2.47,-.17,`x`]): \nplots[display]([p1,p2,p3,p4,t1],tickmarks=[7,5],\n la bels=[``,``],view=[-.35..6.3,-2.2..2.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 868 "r1 := .16: r2 := \+ .18: r3 := .17:\nA1 := expand([2,4]+r1*[4,-2]):\nA2 := expand([2,4]+r2 *[4,-2]):\nB1 := expand(r1*[6,2]):\nB2 := expand(r2*[6,2]):\nC1 := exp and([1,2]+r3*[5,0]):\nC2 := [8/3,2]:\nstrp := [A1,A2,B2,B1,A1]:\ntrng \+ := [[0,0],[2,4],[6,2]]:\np1 := plots[polygonplot](strp,color=COLOR(RGB ,.8,.8,.85)):\np2 := plot(strp,color=COLOR(RGB,.1,.5,0),thickness=1): \np3 := plot([[[1,2],[6,2]],[[3,1],[2,4]],[[0,0],[4,3]]],\n color =black,linestyle=3):\np4 := plots[polygonplot](trng,color=COLOR(RGB,.9 ,.9,.95)):\np5 := plot(trng,color=COLOR(RGB,.1,.5,0),thickness=1):\np6 := plot([[C1,C2]$3],style=point,symbol=[circle,diamond,cross],\n \+ color=black):\nt1 := plots[textplot]([[-.1,-.1,`A`],[1.94,4. 2,`B`],\n [6.15,2,`C`],[.81,2.05,`M`],[1.7,2.2,`K`],\n [2.74,3. 82,`Q`],[1,.18,`P`],[2.87,1.87,`G`]],color=black): \nplots[display]([p 1,p2,p3,p4,p5,p6,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "Code for moment of area picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1600 "f := x -> 2+x^2/2:\nx1 := .5: x2 := 2.5: x3 := 1.59 : x4 := 1.65: x5 := 1.62:\np1 := plot(f(x),x=0..3,color=red,thickness= 2):\np2 := plot([[[x1,0],[x1,f(x1)]],[[x2,0],[x2,f(x2)]],\n [[x3,0] ,[x3,f(x3)]],[[x4,0],[x4,f(x4)]]],color=black):\np3 := plots[polygonpl ot]([[x3,0],[x3,f(x3)],[x5,f(x5)],[x4,f(x4)],[x4,0]],\n color=COLO R(RGB,.8,.8,.83),style=patchnogrid):\npp := plot(f(x),x=x1..x2):\np4 : = plots[polygonplot]([[x1,0],op(op(1,op(1,pp))),[x2,0]],\n color=C OLOR(RGB,.9,.9,.93),style=patchnogrid):\np5 := plot([[[x5,f(x5)/2]]$3] ,style=point,\n symbol=[circle,diamond,cross],color=black):\np6 := plot([[[x5,f(x5)],[2.3,f(x5)]],\n [[x5,f(x5)/2],[2,f(x5)/2]],[[x5,0 ],[x5,-.15]]],color=black,linestyle=2):\np7 := plottools[arrow]([.9,f( x5)/2],[x5,f(x5)/2],0,.13,.08,arrow,color=black):\np8 := plottools[arr ow]([.72,f(x5)/2],[0,f(x5)/2],0,.13,.08,arrow,color=black):\np9 := plo ttools[arrow]([2.2,f(x5)/2+.3],[2.2,f(x5)],0,.05,.08,arrow,color=black ):\np10 := plottools[arrow]([2.2,f(x5)/2-.3],[2.2,0],0,.05,.08,arrow,c olor=black):\np11 := plottools[arrow]([1.9,f(x5)/4+.35],[1.9,f(x5)/2], 0,.05,.15,arrow,color=black):\np12 := plottools[arrow]([1.9,f(x5)/4-.3 5],[1.9,0],0,.05,.15,arrow,color=black):\nt1 := plots[textplot]([3,5.9 ,`y = f(x)`],color=red):\nt2 := plots[textplot]([[x5,-.3,`x`],[-.1,6.3 ,`y`],\n [x1,-.15,`x = a`],[x2,-.15,`x = b`],[1.9,1,`f(x)`],\n [1 .9,.93,`__`],[.81,1.7,`x`],[2.2,f(x5)/2,`f(x)`]],color=black):\nt3 := \+ plots[textplot]([1.9,.66,`2`],font=[HELVETICA,9],color=black):\nplots[ display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,t1,t2,t3],\n tickma rks=[0,0],view=[-.2..3,-.3..6.5]);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }