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Pappus' theorem " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter S tone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 2 3.3.2007" }}{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 68 "The centroid of a region in the coordinate plane between two graphs " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 103 "Suppose that we ha ve a plane region in the coordinate plane described as the region betw een two graphs " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 37 ", and 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 33 "Here we assume t hat the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 23 " is above the graph of " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"y G-%\"gG6#%\"xG" }{TEXT -1 5 " for " }{TEXT 264 1 "x" }{TEXT -1 9 " bet ween " }{TEXT 265 1 "a" }{TEXT -1 5 " and " }{TEXT 266 1 "b" }{TEXT -1 45 ", or at least does not go below the graph of " }{XPPEDIT 18 0 " y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 60 " in this interval. In th e picture both graphs are above the " }{TEXT 300 1 "x" }{TEXT -1 33 " \+ axis, but this is not required. 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lF^`nF[jlFdel-F$6&7$7$Fg_n$\"#tF\\hl7$Fg_nF_[l7%7$Fa`n$\"*+++%eF][mFi` n7$F\\`nF\\anF[jlFdel-%%TEXTG6%7$$\"#:Ffhl$\"$0#F\\hlQ)y~=~f(x)6\"Fhz- F`an6%7$$\"$c\"F\\hl$\"\"'FfhlQ)y~=~g(x)FhanFidl-F`an6%7$$FfhlFfhl$\"$ a#F\\hlQ\"yFhanFdel-F`an6%7$$\"$$fF\\hl$!\"&F\\hlQ\"xFhanFdel-F`an6%7$ F_elF]cnQ&x~=~aFhanFdel-F`an6%7$FjelF]cnQ&x~=~bFhanFdel-F`an6%7$Fg_n$ \"##*F\\hlQ,f(x)~+~g(x)FhanFdel-F`an6%7$Fg_n$\"#*)F\\hlQ(_______FhanFd el-F`an6%7$$\"+++++ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 305 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 80 "Find \+ the coordinates of the centroid of the plane region bounded by the cur ves " }{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=1/x" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" } {TEXT -1 16 ", and the lines " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 261 8 "Solution" }{TEXT 306 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 568 "f := x -> sqrt(x):\ng := x -> 1/x:\np1 := plot([f(x),g(x)],x= 0..5,y=0..2.3,color=[red,blue],thickness=2):\na := 1: b := 4:\npp := p lot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)): \npp := plot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1,o p(1,pp)):\np2 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gp ts[i-1]],i=2..25)],\n style=patchnogrid,color=COLOR(RGB,.95, .95,.95)):\np3 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]],\n \+ [[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],linestyle=[1$2,2$2],color=black): \nplots[display]([p1,p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 497 268 268 {PLOTDATA 2 "6+-%'CURVESG6%7W7$$\"\"!F)F(7$$\"3ALL$3FWYs#!#>$\"3!3 \\Z6l\\1l\"!#=7$$\"3WmmmT&)G\\aF-$\"3s*>zL7rVL#F07$$\"3m****\\7G$R<)F- $\"3!pH$Gh!4!fGF07$$\"3GLLL3x&)*3\"F0$\"3f\")\\H-$*H,LF07$$\"3))**\\i! 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)R:F^im7$Fh^n$\"+X![t@%FdimFc^n7&Fg^n7$$\"+&>q0]#F^im$\"+J\">8e\"F^im7 $Fa_n$\"+'*y3**RFdimF\\_n7&F`_n7$$\"+DM^IEF^im$\"+id)=i\"F^im7$Fj_n$\" +Q#R:!QFdimFe_n7&Fi_n7$$\"+0ytbFF^im$\"+2:/g;F^im7$Fc`n$\"+aAzGOFdimF^ `n7&Fb`n7$$\"+RNXpGF^im$\"+Wh%Rp\"F^im7$F\\an$\"+:T)\\[$FdimFg`n7&F[an 7$$\"+XDn/IF^im$\"+I\"*RLJF^im$\"+'eJhw\"F^im7$F^bn$\"+'))Gf?$FdimFian7&F]bn7$$\"+4wY_KF^im$ \"+**)fM!=F^im7$Fgbn$\"+_')euIFdimFbbn7&Ffbn7$$\"+IOTqLF^im$\"+J'oe$=F ^im7$F`cn$\"+&)[*p'HFdimF[cn7&F_cn7$$\"+4\">)*\\$F^im$\"+[.yq=F^im7$Fi cn$\"+J0HdGFdimFdcn7&Fhcn7$$\"+EP/BOF^im$\"+k%HM!>F^im7$Fbdn$\"+:-6gFF dimF]dn7&Fadn7$$\"+)o:;v$F^im$\"+(z3p$>F^im7$F[en$\"+G#=bm#FdimFfdn7&F jdn7$$\"+%)[opQF^im$\"+k9:n>F^im7$Fden$\"+Z(*=%e#FdimF_en7&Fcen7$$\"\" %F)$Fe\\lF)7$F]fn$\"+++++DFdimFhen-%&COLORG6&F^\\l$\"#&*!\"#FffnFffn-% &STYLEG6#%,PATCHNOGRIDG-F$6%7$FhhmFhhm-%*LINESTYLEG6#Fjhm-F\\\\l6&F^\\ lF)F)F)-F$6%7$7$F]fn$\"3++++++++DF0F\\fnF`gnFcgn-F$6%7$7$FihmF(Fhhm-Fa gnFd\\lFcgn-F$6%7$7$F]fnF(FhgnF_hnFcgn-%+AXESLABELSG6%Q\"x6\"Q\"yFhhn- %%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fg[l;F($\"#B!\"\"" 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" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{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(x);" "6#/-%\"fG6#%\"xG-% %sqrtG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)=1/x" "6#/-%\"gG6# %\"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)-g(x)),x = 1 .. 4);" "6#/%\"AG-%$IntG6$- %!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F/!\"\"/F/;F0\"\"%" }{XPPEDIT 18 0 " `` = Int(``(sqrt(x)-1/x),x = 1 .. 4);" "6#/%!G-%$IntG6$-F$6#,&-%%sqrtG 6#%\"xG\"\"\"*&F/F/F.!\"\"F1/F.;F/\"\"%" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(x^(1/2)-x^(-1),x = 1 .. 4);" "6#/%!G-%$IntG6$,&)%\"xG*&\"\"\"F,\"\"#!\"\"F,)F*,$F,F.F./F*;F, \"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/3" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^(3/2)-ln*x;" "6#,&)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(*&%# lnGF(F%F(F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[1, ``] );" "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 16/3-ln*4-2/3;" "6#/%!G,(*&\"#;\"\"\"\"\"$!\"\"F(*&%#lnGF(\"\" %F(F**&\"\"#F(F)F*F*" }{XPPEDIT 18 0 "`` = 14/3-2*ln*2;" "6#/%!G,&*&\" #9\"\"\"\"\"$!\"\"F(*(\"\"#F(%#lnGF(F,F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of t he region about the " }{TEXT 307 1 "y" }{TEXT -1 9 " axis is:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = Int(x*(f(x)-g( x)),x = 1 .. 4);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\"\",&-%\"fG6#F,F --%\"gG6#F,!\"\"F-/F,;F-\"\"%" }{XPPEDIT 18 0 "`` = Int(x*(sqrt(x)-1/x ),x = 1 .. 4);" "6#/%!G-%$IntG6$*&%\"xG\"\"\",&-%%sqrtG6#F)F**&F*F*F)! \"\"F0F*/F);F*\"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = Int(x^(3/2)-1,x = 1 .. 4);" "6#/%!G-%$IntG6$,&) %\"xG*&\"\"$\"\"\"\"\"#!\"\"F-F-F//F*;F-\"\"%" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2/5" "6#/%!G*&\"\" #\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(5/2)-x" "6#,&)% \"xG*&\"\"&\"\"\"\"\"#!\"\"F(F%F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([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 "``=``(64/5-4)-(2/5-1)" "6#/%!G,&-F$6#,& *&\"#k\"\"\"\"\"&!\"\"F+\"\"%F-F+,&*&\"\"#F+F,F-F+F+F-F-" }{XPPEDIT 18 0 "``=62/5-3" "6#/%!G,&*&\"#i\"\"\"\"\"&!\"\"F(\"\"$F*" }{XPPEDIT 18 0 "``=47/5" "6#/%!G*&\"#Z\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The momen t of the region about the " }{TEXT 308 1 "x" }{TEXT -1 10 " axis is: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int((f(x)^ 2-g(x)^2)/2,x = 1 .. 4);" "6#/&%\"MG6#%\"xG-%$IntG6$*&,&*$-%\"fG6#F'\" \"#\"\"\"*$-%\"gG6#F'F1!\"\"F2F1F7/F';F2\"\"%" }{XPPEDIT 18 0 "`` = In t((sqrt(x)^2-(1/x)^2)/2,x = 1 .. 4);" "6#/%!G-%$IntG6$*&,&*$-%%sqrtG6# %\"xG\"\"#\"\"\"*$*&F0F0F.!\"\"F/F3F0F/F3/F.;F0\"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((x-x^(-2)) /2,x = 1 .. 4);" "6#/%!G-%$IntG6$*&,&%\"xG\"\"\")F*,$\"\"#!\"\"F/F+F.F //F*;F+\"\"%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "`` = Int(``(x/2-x^(-2)/2),x = 1 .. 4);" "6#/%!G-%$IntG6 $-F$6#,&*&%\"xG\"\"\"\"\"#!\"\"F-*&)F,,$F.F/F-F.F/F//F,;F-\"\"%" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x^2/4+x^(-1)/2;" "6#/%!G,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&)F',$F+F*F+F( F*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[1, ``])" "6#- %*PIECEWISEG6$7$\"\"%%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x^2/4+1/(2*x);" "6#/%!G,&*&%\"xG \"\"#\"\"%!\"\"\"\"\"*&F+F+*&F(F+F'F+F*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([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+1/8)-(1/4+1/2)" "6#/%!G,&-F$6 #,&\"\"%\"\"\"*&F*F*\"\")!\"\"F*F*,&*&F*F*F)F-F**&F*F*\"\"#F-F*F-" } {XPPEDIT 18 0 "``=33/8-3/4" "6#/%!G,&*&\"#L\"\"\"\"\")!\"\"F(*&\"\"$F( \"\"%F*F*" }{XPPEDIT 18 0 "``=27/8" "6#/%!G*&\"#F\"\"\"\"\")!\"\"" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 309 1 "x" } {TEXT -1 33 " coordinate of the centroid is " }{XPPEDIT 18 0 "conjug ate(x) = M[y]/A;" "6#/-%*conjugateG6#%\"xG*&&%\"MG6#%\"yG\"\"\"%\"AG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(47/5)/``(14/3-2*ln*2);" "6#/%!G*&-F$6#*&\"#Z\"\"\"\"\"&!\"\"F*-F$6#,&*&\"#9F*\"\"$F,F**(\"\"#F *%#lnGF*F4F*F,F," }{XPPEDIT 18 0 "`` = 47/(5*(14/3-2*ln*2));" "6#/%!G* &\"#Z\"\"\"*&\"\"&F',&*&\"#9F'\"\"$!\"\"F'*(\"\"#F'%#lnGF'F0F'F.F'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 310 1 "y" }{TEXT -1 33 " coordinate of the ce ntroid is " }{XPPEDIT 18 0 "conjugate(y) = M[x]/A;" "6#/-%*conjugate G6#%\"yG*&&%\"MG6#%\"xG\"\"\"%\"AG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(27/8)/``(14/3-2*ln*2);" "6#/%!G*&-F$6#*&\"#F\"\"\"\"\")!\" \"F*-F$6#,&*&\"#9F*\"\"$F,F**(\"\"#F*%#lnGF*F4F*F,F," }{XPPEDIT 18 0 " `` = 27/(8*(14/3-2*ln*2));" "6#/%!G*&\"#F\"\"\"*&\"\")F',&*&\"#9F'\"\" $!\"\"F'*(\"\"#F'%#lnGF'F0F'F.F'F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The centroid is located at the point" }{XPPEDIT 18 0 "``(47/(5*(14/3-2*ln*2)),27/(8*(14/3-2*l n*2)));" "6#-%!G6$*&\"#Z\"\"\"*&\"\"&F(,&*&\"#9F(\"\"$!\"\"F(*(\"\"#F( %#lnGF(F1F(F/F(F/*&\"#FF(*&\"\")F(,&*&F-F(F.F/F(*(F1F(F2F(F1F(F/F(F/" }{TEXT -1 2 " " }{TEXT 311 1 "~" }{TEXT -1 32 " ( 2.865528399, 1.028 846633 ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 329 "f := x -> sqrt(x):\ng := x -> 1/x:\nInt(f(x)-g(x), x=1..4);\nA := value(%);\nInt(x*(f(x)-g(x)),x=1..4);\nMy := value(%); \nInt((f(x)^2-g(x)^2)/2,x=1..4);\nexpand(%);\nMx := value(%);\nxG := M y/A;\nyG := Mx/A;\nxGf := evalf(evalf(xG,13)):\nyGf := evalf(evalf(yG, 13)):\nprint(`The centroid is located at the point .. `,``(xG,yG)*`~`* ``(xGf,yGf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*$%\"xG#\" \"\"\"\"#F**&F*F*F(!\"\"F-/F(;F*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&#\"#9\"\"$\"\"\"*&\"\"#F)-%#lnG6#F+F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\",&*$F'#F(\"\"#F(*&F(F(F'!\" \"F.F(/F';F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MyG#\"#Z\"\"& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"#!\"\"%\"xG\"\"\" F+*&F+F+*&F(F+)F*F(F+F)F)/F*;F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*&#\"\"\"\"\"#F&-%$IntG6$,&%\"xGF&*&F&F&*$)F,F'F&!\"\"F0/F,;F&\"\" %F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\"#F\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG,$*&#\"#Z\"\"&\"\"\"*&F*F*,&#\"#9\"\"$F** &\"\"#F*-%#lnG6#F1F*!\"\"F5F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# yGG,$*&#\"#F\"\")\"\"\"*&F*F*,&#\"#9\"\"$F**&\"\"#F*-%#lnG6#F1F*!\"\"F 5F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%IThe~centroid~is~located~at~ the~point~..~G*(-%!G6$,$*&#\"#Z\"\"&\"\"\"*&F-F-,&#\"#9\"\"$F-*&\"\"#F --%#lnG6#F4F-!\"\"F8F-F-,$*&#\"#F\"\")F-F.F-F-F-%\"|irGF--F&6$$\"+*RGb 'G!\"*$\"+Lm%)G5FCF-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can draw a picture which shows the centroid of the \+ region. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 261 8 "S olution" }{TEXT 295 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "f := x -> 6-x^2:\ng := x -> x:\np1 := plot([f(x),g(x)],x=0..2.6,color=[red,blue],thickness=2):\na := 0: \+ b := 2:\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,numpoints=25): \ngpts := op(1,op(1,pp)):\np2 := plots[polygonplot]([seq([fpts[i-1],fp ts[i],gpts[i],gpts[i-1]],i=2..25)],\n style=patchnogrid,colo r=COLOR(RGB,.95,.95,.95)):\np3 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)], [b,f(b)]],\n [[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],linestyle=[1$2,2$ 2],color=black):\nplots[display]([p1,p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 356 282 282 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)$\"\"'F)7 $$\"3'HLLL3gsm&!#>$\"33'y9j@)y'*f!#<7$$\"3smm\"zFJ)f5!#=$\"3!\\\\Bmdn( ))fF27$$\"3RLL$eszVh\"F6$\"3'G=35yPR(fF27$$\"3cLL$33/E<#F6$\"3?j%z]\"z z_fF27$$\"3Qmm\"HHv\"GFF6$\"30\">6dfqb#fF27$$\"3ELLeC4EVKF6$\"3K32v&e7 [*eF27$$\"3)****\\7E-mx$F6$\"3S8Kg`FPdeF27$$\"3eLLeMY=GVF6$\"3Xn&*ox\" oE\"eF27$$\"3E++Dw!)*z([F6$\"3O85oZ80idF27$$\"3Anmm\"y[NW&F6$\"3c`hhmx n.dF27$$\"32ML$3:'oTfF6$\"3]`*\\oljpk&F27$$\"3<+++X3Z-lF6$\"3G]#*4H(yr d&F27$$\"3g*****\\3wF6$\"3w)\\10O `6U&F27$$\"3BLLeM`'45)F6$\"3a\"zykgVPM&F27$$\"3kmmmhV\"po)F6$\"3I%pI() =v`C&F27$$\"3qmmmY+H$=*F6$\"3;Hz=R=nc^F27$$\"37++DrHpg(*F6$\"3G3*4ss)G Z]F27$$\"3emmmR#zr-\"F2$\"3E3yf4G!\\%\\F27$$\"39+]7Z\\D$3\"F2$\"3X8#H& >(el#[F27$$\"3>+]P\"G_m8\"F2$\"3>2]C\"f@!3ZF27$$\"3wmm\"zzmB>\"F2$\"3w afA>9EyXF27$$\"3xm;H;,`V7F2$\"3oLc()\\Gj`WF27$$\"3WLL3nmr)H\"F2$\"3]d* R'=]L8VF27$$\"3UL$eM)*RgN\"F2$\"3%HKiKcb6;%F27$$\"3$***\\())4SfS\"F2$ \"3[b`PQCLBSF27$$\"3]LLe!fL)f9F2$\"3?sX!y)e))oQF27$$\"3++++T3^::F2$\"3 %HZ63*oA.PF27$$\"33++Do,)*p:F2$\"3C,p)H!pO$F27$$\"33++v*[)>\"o\"F2$\"3!*=B1Q;dtJF27$$\"3immm>\"yPt\"F2$ \"3q<^wJM,%*HF27$$\"3?++](4=**y\"F2$\"3#*z>C/K>'z#F27$$\"3@L$e9E*yS=F2 $\"37%o**[*[\\6EF27$$\"3A++]!)[S'*=F2$\"34!y:#H&[OS#F27$$\"3]m;zV[t[>F 2$\"38pAf3DV-AF27$$\"3C+]iLZV.?F2$\"3)R#\\5o#\\i)>F27$$\"3#ommT(Q\"p0# F2$\"3?t1E9`5pA#F2$\"3q*)z&*oq&G1\"F27$$\"3wm;z,blw AF2$\"3qTthhsRo\")F67$$\"3:+++,u!pK#F2$\"3uYUrr%>]&eF67$$\"3immT[<]%Q# F2$\"3e'fyz69:9$F67$$\"3ILLL>[,OCF2$\"3#HxQ))**zJe'F/7$$\"39+]7(\\Q4\\ #F2$!3kVNfkfuZ?F67$$\"3B+]([Y2Na#F2$!3G-&[(QAI%p%F67$$\"33+++++++EF2$! 3u1++++++wF6-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F $6%7S7$F(F(7$F-F-7$F4F47$F:F:7$F?F?7$FDFD7$FIFI7$FNFN7$FSFS7$FXFX7$Fgn Fgn7$F\\oF\\o7$FaoFao7$FfoFfo7$F[pF[p7$F`pF`p7$FepFep7$FjpFjp7$F_qF_q7 $FdqFdq7$FiqFiq7$F^rF^r7$FcrFcr7$FhrFhr7$F]sF]s7$FbsFbs7$FgsFgs7$F\\tF \\t7$FatFat7$FftFft7$F[uF[u7$F`uF`u7$FeuFeu7$FjuFju7$F_vF_v7$FdvFdv7$F ivFiv7$F^wF^w7$FcwFcw7$FhwFhw7$F]xF]x7$FbxFbx7$FgxFgx7$F\\yF\\y7$FayFa y7$FfyFfy7$F[zF[z7$F`zF`z7$FezFez-Fjz6&F\\[lF(F(F][lF`[l-%)POLYGONSG6< 7&F'7$$\"+n;')=()!#6$\"+X\")R#*f!\"*7$F__lF__lFg[l7&F^_l7$$\"+e'40j\"! #5$\"+$Q9M(fFd_l7$Fh_lFh_lFe_l7&Fg_l7$$\"+<6m$[#Fj_l$\"+vUJQfFd_l7$F`` lF``lF]`l7&F_`l7$$\"+(>%Fj_l$\"+Hd$Q#eFd_l7$F^alF^alF[al7&F]al7$$\"+#>K'*)\\Fj_l$\"+1d. ^dFd_l7$FealFealFbal7&Fdal7$$\"+Dt:5eFj_l$\"+>2UicFd_l7$F\\blF\\blFial 7&F[bl7$$\"+#fX(emFj_l$\"+r5hcbFd_l7$FcblFcblF`bl7&Fbbl7$$\"+DCh/vFj_l $\"+Cz!oV&Fd_l7$FjblFjblFgbl7&Fibl7$$\"+L/pu$)Fj_l$\"+,ck)H&Fd_l7$Facl FaclF^cl7&F`cl7$$\"+Fd_l $\"+3%f*RBFd_l7$F\\ilF\\ilFihl7&F[il7$$Fc[lF)FcilFbilF`il-%&COLORG6&F \\[l$\"#&*!\"#FgilFgil-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$Fg[lF'-%*LINESTY LEG6#\"\"\"-Fjz6&F\\[lF)F)F)-F$6%7$FbilFbilFajlFejl-F$6%7$Fg[lFg[l-Fbj lFb[lFejl-F$6%7$7$FcilF(FbilF][mFejl-%+AXESLABELSG6%Q\"x6\"Q!Ff[m-%%FO NTG6#%(DEFAULTG-%%VIEWG6$;F($\"#E!\"\"F[\\m" 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" }}}}{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) = 6-x^2;" "6#/-%\"fG6#%\"xG,&\" \"'\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = x; " "6#/-%\"gG6#%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The graphs of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" } {TEXT -1 17 " intersect where " }{XPPEDIT 18 0 "6-x^2 = x;" "6#/,&\"\" '\"\"\"*$%\"xG\"\"#!\"\"F(" }{TEXT -1 16 ", that is where " }{XPPEDIT 18 0 "x^2+x-6 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"F&F(\"\"'!\"\"\"\"!" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "(x-2)*(x+3)=0" "6#/*&,&%\"xG\"\"\"\" \"#!\"\"F',&F&F'\"\"$F'F'\"\"!" }{TEXT -1 9 ", giving " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=-3" "6 #/%\"xG,$\"\"$!\"\"" }{TEXT -1 4 ". \n" }}{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)-g(x)),x = 0 .. 2);" "6#/%\"AG-%$IntG6$- %!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F/!\"\"/F/;\"\"!\"\"#" }{XPPEDIT 18 0 "`` = Int(``(6-x^2-x),x = 0 .. 2);" "6#/%!G-%$IntG6$-F$6#,(\"\"'\"\" \"*$%\"xG\"\"#!\"\"F.F0/F.;\"\"!F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=6*x-x^3/3-x^2/2" "6#/%!G,(*&\"\"' \"\"\"%\"xGF(F(*&F)\"\"$F+!\"\"F,*&F)\"\"#F.F,F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([2, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"#% !G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 12-8/3-2;" "6#/%!G,(\" #7\"\"\"*&\"\")F'\"\"$!\"\"F+\"\"#F+" }{XPPEDIT 18 0 "``=22/3" "6#/%!G *&\"#A\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " }{TEXT 296 1 "y" }{TEXT -1 9 " axis is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[y] = \+ Int(x*(f(x)-g(x)),x = 0 .. 2);" "6#/&%\"MG6#%\"yG-%$IntG6$*&%\"xG\"\" \",&-%\"fG6#F,F--%\"gG6#F,!\"\"F-/F,;\"\"!\"\"#" }{XPPEDIT 18 0 "`` = \+ Int(x*(6-x^2-x),x = 0 .. 2);" "6#/%!G-%$IntG6$*&%\"xG\"\"\",(\"\"'F**$ F)\"\"#!\"\"F)F/F*/F);\"\"!F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(``(6*x-x^3-x^2),x = 0 .. 2);" "6#/%!G-%$IntG6$-F$6#,(*&\"\"'\"\"\"%\"xGF-F-*$F.\"\"$!\"\"*$F.\"\"#F1 /F.;\"\"!F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=3*x^2-x^4/4-x^3/3" "6#/%!G,(*&\"\"$\"\"\"*$%\"xG\"\" #F(F(*&F*\"\"%F-!\"\"F.*&F*F'F'F.F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([2, ``],[0, ``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F(" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1 2-4-8/3" "6#/%!G,(\"#7\"\"\"\"\"%!\"\"*&\"\")F'\"\"$F)F)" }{XPPEDIT 18 0 "``=16/3" "6#/%!G*&\"#;\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "The moment of the region about the " } {TEXT 297 1 "x" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "M[x] = Int((f(x)^2-g(x)^2)/2,x = 0 .. 2);" "6 #/&%\"MG6#%\"xG-%$IntG6$*&,&*$-%\"fG6#F'\"\"#\"\"\"*$-%\"gG6#F'F1!\"\" F2F1F7/F';\"\"!F1" }{XPPEDIT 18 0 "`` = Int(((6-x^2)^2-x^2)/2,x = 0 .. 2);" "6#/%!G-%$IntG6$*&,&*$,&\"\"'\"\"\"*$%\"xG\"\"#!\"\"F0F-*$F/F0F1 F-F0F1/F/;\"\"!F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((36-13*x^2+x^4)/2,x = 0 .. 2);" "6#/%!G-%$Int G6$*&,(\"#O\"\"\"*&\"#8F+*$%\"xG\"\"#F+!\"\"*$F/\"\"%F+F+F0F1/F/;\"\"! F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Int(18-13*x^2/2+x^4/2,x = 0 .. 2);" "6#/%!G-%$IntG6$,(\"#=\"\" \"*(\"#8F**$%\"xG\"\"#F*F/!\"\"F0*&F.\"\"%F/F0F*/F.;\"\"!F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 18*x- 13*x^3/6+x^5/10;" "6#/%!G,(*&\"#=\"\"\"%\"xGF(F(*(\"#8F(*$F)\"\"$F(\" \"'!\"\"F/*&F)\"\"&\"#5F/F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE ([2, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=36-52/3+16/5" "6#/%!G,(\"#O\"\"\"*&\"#_F'\"\"$! \"\"F+*&\"#;F'\"\"&F+F'" }{XPPEDIT 18 0 "``=36-17" "6#/%!G,&\"#O\"\"\" \"# " 0 " " {MPLTEXT 1 0 341 "f := x -> 6-x^2:\ng := x -> x:\na := 0: b := 2:\nI nt(f(x)-g(x),x=a..b);\nA := value(%);\nInt(x*(f(x)-g(x)),x=a..b);\nMy \+ := value(%);\nInt((f(x)^2-g(x)^2)/2,x=a..b);\nexpand(%);\nMx := value( %);\nxG := My/A;\nyG := Mx/A;\nxGf := evalf(evalf(xG,13)):\nyGf := eva lf(evalf(yG,13)):\nprint(`The centroid is located at the point .. `,`` (xG,yG)*`~`*``(xGf,yGf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ ,(\"\"'\"\"\"*$)%\"xG\"\"#F(!\"\"F+F-/F+;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG#\"#A\"\"$" }}{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$,&*&\"\"#!\"\",&\"\"'\"\"\"*$)%\"xGF(F,F)F(F,* &F(F)F/F(F)/F/;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\" \"\"#F&-%$IntG6$,(\"#OF&*&\"#8F&)%\"xGF'F&!\"\"*$)F0\"\"%F&F&/F0;\"\"! F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MxG#\"$G$\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xGG#\"\")\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yGG#\"$k\"\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%I The~centroid~is~located~at~the~point~..~G*(-%!G6$#\"\")\"#6#\"$k\"\"#b \"\"\"%\"|irGF.-F&6$$\"+tssss!#5$\"+#===)H!\"*F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can draw a picture whi ch shows the centroid of the region. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 658 "f := x -> 6-x^2:\ng := \+ x -> x:\np1 := plot([f(x),g(x)],x=0..2.6,color=[red,blue],thickness=2) :\na := 0: b := 2:\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25 ):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,num points=25):\ngpts := op(1,op(1,pp)):\np2 := plots[polygonplot]([seq([f pts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n style=patch nogrid,color=COLOR(RGB,.95,.95,.95)):\np3 := plot([[[a,g(a)],[a,f(a)]] ,[[b,g(b)],[b,f(b)]],\n [[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],linest yle=[1$2,2$2],color=black):\np4 := plot([[[xG,yG]]$3],style=point,symb ol=[circle,diamond,cross],color=black):\nplots[display]([p1,p2,p3,p4], labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 290 300 300 {PLOTDATA 2 "6.-%'CURVESG6%7S7$$\"\"!F)$\"\"'F)7$$\"3'HLLL3gsm&!#>$\"3 3'y9j@)y'*f!#<7$$\"3smm\"zFJ)f5!#=$\"3!\\\\Bmdn())fF27$$\"3RLL$eszVh\" F6$\"3'G=35yPR(fF27$$\"3cLL$33/E<#F6$\"3?j%z]\"zz_fF27$$\"3Qmm\"HHv\"G FF6$\"30\">6dfqb#fF27$$\"3ELLeC4EVKF6$\"3K32v&e7[*eF27$$\"3)****\\7E-m x$F6$\"3S8Kg`FPdeF27$$\"3eLLeMY=GVF6$\"3Xn&*ox\"oE\"eF27$$\"3E++Dw!)*z ([F6$\"3O85oZ80idF27$$\"3Anmm\"y[NW&F6$\"3c`hhmxn.dF27$$\"32ML$3:'oTfF 6$\"3]`*\\oljpk&F27$$\"3<+++X3Z-lF6$\"3G]#*4H(yrd&F27$$\"3g*****\\3wF6$\"3w)\\10O`6U&F27$$\"3BLLeM`'45) F6$\"3a\"zykgVPM&F27$$\"3kmmmhV\"po)F6$\"3I%pI()=v`C&F27$$\"3qmmmY+H$= *F6$\"3;Hz=R=nc^F27$$\"37++DrHpg(*F6$\"3G3*4ss)GZ]F27$$\"3emmmR#zr-\"F 2$\"3E3yf4G!\\%\\F27$$\"39+]7Z\\D$3\"F2$\"3X8#H&>(el#[F27$$\"3>+]P\"G_ m8\"F2$\"3>2]C\"f@!3ZF27$$\"3wmm\"zzmB>\"F2$\"3wafA>9EyXF27$$\"3xm;H;, `V7F2$\"3oLc()\\Gj`WF27$$\"3WLL3nmr)H\"F2$\"3]d*R'=]L8VF27$$\"3UL$eM)* RgN\"F2$\"3%HKiKcb6;%F27$$\"3$***\\())4SfS\"F2$\"3[b`PQCLBSF27$$\"3]LL e!fL)f9F2$\"3?sX!y)e))oQF27$$\"3++++T3^::F2$\"3%HZ63*oA.PF27$$\"33++Do ,)*p:F2$\"3C,p)H!pO$F27$$\"33++v* [)>\"o\"F2$\"3!*=B1Q;dtJF27$$\"3immm>\"yPt\"F2$\"3q<^wJM,%*HF27$$\"3?+ +](4=**y\"F2$\"3#*z>C/K>'z#F27$$\"3@L$e9E*yS=F2$\"37%o**[*[\\6EF27$$\" 3A++]!)[S'*=F2$\"34!y:#H&[OS#F27$$\"3]m;zV[t[>F2$\"38pAf3DV-AF27$$\"3C +]iLZV.?F2$\"3)R#\\5o#\\i)>F27$$\"3#ommT(Q\"p0#F2$\"3?t1E9`5pA#F2$\"3q*)z&*oq&G1\"F27$$\"3wm;z,blwAF2$\"3qTthhsRo\")F67$ $\"3:+++,u!pK#F2$\"3uYUrr%>]&eF67$$\"3immT[<]%Q#F2$\"3e'fyz69:9$F67$$ \"3ILLL>[,OCF2$\"3#HxQ))**zJe'F/7$$\"39+]7(\\Q4\\#F2$!3kVNfkfuZ?F67$$ \"3B+]([Y2Na#F2$!3G-&[(QAI%p%F67$$\"33+++++++EF2$!3u1++++++wF6-%'COLOU RG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6%7S7$F(F(7$F-F-7$F 4F47$F:F:7$F?F?7$FDFD7$FIFI7$FNFN7$FSFS7$FXFX7$FgnFgn7$F\\oF\\o7$FaoFa o7$FfoFfo7$F[pF[p7$F`pF`p7$FepFep7$FjpFjp7$F_qF_q7$FdqFdq7$FiqFiq7$F^r F^r7$FcrFcr7$FhrFhr7$F]sF]s7$FbsFbs7$FgsFgs7$F\\tF\\t7$FatFat7$FftFft7 $F[uF[u7$F`uF`u7$FeuFeu7$FjuFju7$F_vF_v7$FdvFdv7$FivFiv7$F^wF^w7$FcwFc w7$FhwFhw7$F]xF]x7$FbxFbx7$FgxFgx7$F\\yF\\y7$FayFay7$FfyFfy7$F[zF[z7$F `zF`z7$FezFez-Fjz6&F\\[lF(F(F][lF`[l-%)POLYGONSG6<7&F'7$$\"+n;')=()!#6 $\"+X\")R#*f!\"*7$F__lF__lFg[l7&F^_l7$$\"+e'40j\"!#5$\"+$Q9M(fFd_l7$Fh _lFh_lFe_l7&Fg_l7$$\"+<6m$[#Fj_l$\"+vUJQfFd_l7$F``lF``lF]`l7&F_`l7$$\" +(>%Fj_l$\"+Hd$Q# eFd_l7$F^alF^alF[al7&F]al7$$\"+#>K'*)\\Fj_l$\"+1d.^dFd_l7$FealFealFbal 7&Fdal7$$\"+Dt:5eFj_l$\"+>2UicFd_l7$F\\blF\\blFial7&F[bl7$$\"+#fX(emFj _l$\"+r5hcbFd_l7$FcblFcblF`bl7&Fbbl7$$\"+DCh/vFj_l$\"+Cz!oV&Fd_l7$Fjbl FjblFgbl7&Fibl7$$\"+L/pu$)Fj_l$\"+,ck)H&Fd_l7$FaclFaclF^cl7&F`cl7$$\"+ Fd_l$\"+3%f*RBFd_l7$F\\i lF\\ilFihl7&F[il7$$Fc[lF)FcilFbilF`il-%&COLORG6&F\\[l$\"#&*!\"#FgilFgi l-%&STYLEG6#%,PATCHNOGRIDG-F$6%7$Fg[lF'-%*LINESTYLEG6#\"\"\"-Fjz6&F\\[ lF)F)F)-F$6%7$FbilFbilFajlFejl-F$6%7$Fg[lFg[l-FbjlFb[lFejl-F$6%7$7$Fci lF(FbilF][mFejl-F$6&7#7$$\"3#HFFFFFFF(F6$\"3#>======)HF2-%'SYMBOLG6#%' CIRCLEGFejl-F[jl6#%&POINTG-F$6&Fd[m-F[\\m6#%(DIAMONDGFejlF^\\m-F$6&Fd[ m-F[\\m6#%&CROSSGFejlF^\\m-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAUL TG-%%VIEWG6$;F($\"#E!\"\"Fc]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 16 "Pappus' theorem " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{TEXT 313 1 "R" }{TEXT -1 86 " is a plane region which lies entirely on one side o f a line in its plane. The volume " }{XPPEDIT 18 0 "V[L]" "6#&%\"VG6#% \"LG" }{TEXT -1 49 " of the solid of revolution obtained by rotating \+ " }{TEXT 314 1 "R" }{TEXT -1 16 " about the line " }{TEXT 315 1 "L" } {TEXT -1 28 " is the product of the area " }{TEXT 316 1 "A" }{TEXT -1 4 " of " }{TEXT 317 1 "R" }{TEXT -1 47 " and the distance travelled by the centroid of " }{TEXT 318 1 "R" }{TEXT -1 31 " during the rotation , that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[L] = 2*Pi*conjugate(z)*`.`*A;" "6#/&% \"VG6#%\"LG*,\"\"#\"\"\"%#PiGF*-%*conjugateG6#%\"zGF*%\".GF*%\"AGF*" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 319 8 "_____ ___" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{XPPEDIT 18 0 "conjugate(z);" "6#-%*conjugateG6 #%\"zG" }{TEXT -1 50 " is the perpendicular distance of the centroid o f " }{TEXT 320 1 "R" }{TEXT -1 6 " from " }{TEXT 321 1 "L" }{TEXT -1 2 ". 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>\"F9F_[oF/-Fejp6$Fgjp\"#:Fhjp-F$6&7$7$$\"#VFf^n$!\"$Ff^n7$F(Fd\\q7%7$ $\"*+++W$F`]m$!+++++bF[\\mFf\\q7$Fi\\q$!*++++&F[\\mFg^nF/-F$6&7$7$$\"# dFf^nFd\\q7$$\"$+\"Ff^nFd\\q7%7$$\"++++c'*F`]mF^]qFf]q7$F[^qF[]qFg^nF/ -F$6&7$7$$\"#aFf^n$F\\[mF+7$F(Fd^q7%7$$\"*+++K%F`]m$\"++++] " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 349 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 121 "Use Pappus' theorem to obtain a f ormula for the volume of a solid torus (doughnut shape) with its large r radius equal to " }{TEXT 351 1 "R" }{TEXT -1 34 ", and its smaller r adius equal to " }{TEXT 352 1 "r" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 99 "In detail, a solid torus is the solid of revolution obtai ned by rotating a circular disc of radius " }{TEXT 353 1 "r" }{TEXT -1 85 ", say, around an axis such that the centre of the disc traces o ut a circle of radius " }{TEXT 354 1 "R" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "R>r" "6#2%\"rG%\"RG" 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;T([SUA#4'*F?7$$\"3mt'*ygm3sHF<$\"37az2/\\&ez*F?7$F^[m$\"3+iI7++++5F3fo8[#*F?7$Fi[o$!3kcl\\\\@;\"3*F?7$F^\\o $!38Yh'[fo5%*)F?7$Fc\\o$!3@<=dU@I8))F?7$Fh\\o$!3;@Y=zg*=q)F?7$F]]o$!3z %*4Mt^w8')F?7$Fb]o$!3'=zF)z(>'[&)F?7$Fg]o$!3tg%QaRKO^)F?7$F\\^o$!3(4e- uE++])F?7$Fb^o$!3'*yq&o!z)R^)F?7$Fg^o$!3O7Fi<,Z`&)F?7$F\\_o$!30$eE'>+( 3h)F?7$Fa_o$!3[.#Q!Rsk/()F?7$Ff_o$!3ev-C%\\lW!))F?7$F[`o$!3hl[%4>%3U*) F?7$F``o$!3kVKN#*G:\"3*F?7$Fe`o$!3O1gdQRv\\#*F?7$Fj`o$!3S+x+iq8B%*F?7$ F_ao$!3M'z2gUBUh*F?7$Fdao$!33$>U#\\i$fz*F?7$Fiao$!3wh_%e.[`***F?7$F^bo $!3Q')H%y(>D?5F<7$Fcbo$!3.LTUXU)z.\"F<7$Fhbo$!3kxZ/`$3l0\"F<7$F]co$!3' G!oHjoju5F<7$Fbco$!35Ya.k'o54\"F<7$Fgco$!3MB5=<-Z06F<7$F\\do$!3Z9r\")G *\\%>6F<7$Fado$!3#=?u6N%)*H6F<7$Ffdo$!3aH?5**\\\"*Q6F<7$F[eo$!3>z&R/B1 [9\"F<7$F`eo$!3'e-'*)HOu[6F<7$Feeo$!37a)G*)H***\\6F<7$Fjeo$!3?rinF6v[6 F<7$F_fo$!3@F?W%>@]9\"F<7$Fdfo$!3^gD)y:E&Q6F<7$Fifo$!3OS$eT@v)H6F<7$F^ go$!3SMxP4@v=6F<7$Fcgo$!3fG)34$3k06F<7$Fhgo$!3oRfFyD'>4\"F<7$F]ho$!3K( p)e/Hju5F<7$Fbho$!3)yg]'G&**y0\"F<7$Fgho$!3*f7&fdx2R5F<7$F\\io$!3f/Af4 XT?5F<7$F^[m$!2+Qp()*********F " 0 "" {MPLTEXT 1 0 535 "d := evalf( 4/(3*Pi)):\nu := evalf(Pi/60,15):\nsector1 := [[0,0],seq([cos(i*u),sin (i*u)],i=0..30)]:\nsector2 := [[0,0],seq([cos(i*u),sin(i*u)],i=60..90) ]:\np1 := plots[polygonplot](sector1,color=COLOR(RGB,.8,.8,.93)):\np2 \+ := plots[polygonplot](sector2,color=COLOR(RGB,.9,.9,.96)):\np3 := plot ([[[-1,1],[1,-1]],[[0,0],[d,d]]],color=[red,black]):\np4 := plot([[[d, d]]$3],style=point,symbol=[circle,diamond,cross],color=black):\nt1 := \+ plots[textplot]([-.6,.9,`y = -x`],color=red):\nplots[display]([p1,p2,p 3,p4,t1],labels=[`x`,`y`],tickmarks=[3,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 315 280 280 {PLOTDATA 2 "6--%)POLYGONSG6$7B7$$\"\"!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#F 07$$\"+j^c5&*F0$\"+W*p,4$F07$$\"+lU!eL*F0$\"+&\\zOe$F07$$\"+wXXN\"*F0$ \"+JkOnSF07$$\"+U_15*)F0$\"+(*\\!*RXF07$$\"+QSDg')F0$\"+++++]F07$$\"+z cq'Q)F0$\"+^.RYaF07$$\"+W*p,4)F0$\"+BD&y(eF07$$\"+9'f9x(F0$\"+6R?$H'F0 7$$\"+b#[9V(F0$\"+jgI\"p'F07$$\"+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$FF FD7$$\"+/p6z?F0$\"+3gZ\"y*F07$$\"+_YMk:F0F:7$$\"+HYGX5F0F57$$\"+[cfL_F 3$\"+Z`H')**F07$$!+3Q.^?!#>F+-%&COLORG6&%$RGBG$\"\")!\"\"Fhs$\"#$*!\"# -F$6$7BF'7$$FjsF)$!+:w1-TFcs7$$!+[`H')**F0$!+4cfL_F37$$!+`*=_%**F0$!+O YGX5F07$$!+1M)o()*F0$!+[YMk:F07$$!+2gZ\"y*F0$!+5p6z?F07$$!+k#e#f'*F0$! +Z/>)e#F07$$!+j^c5&*F0$!+W*p,4$F07$$!+jU!eL*F0$!++&zOe$F07$$!+xXXN\"*F 0$!+IkOnSF07$$!+S_15*)F0$!+,]!*RXF07$$!+RSDg')F0$!+)*******\\F07$$!+yc q'Q)F0$!+_.RYaF07$$!+Y*p,4)F0$!+?D&y(eF07$$!+9'f9x(F0$!+7R?$H'F07$$!+e #[9V(F0$!+ggI\"p'F07$$!+7y1rqF0F\\y7$FixFgx7$$!+6R?$H'F0Fbx7$F_xF]x7$F jwFhw7$FewFcw7$$!++]!*RXF0F^w7$F[wFiv7$FfvFdv7$FavF_v7$F\\vFju7$FguFeu 7$$!+ZYMk:F0F`u7$$!+NYGX5F0F[u7$$!+2cfL_F3Fft7$$!+y&)*o%QFcsFbt-Fes6&F gs$\"\"*FjsF[[l$\"#'*F]t-%'CURVESG6$7$7$FbtF+7$F+Fbt-%'COLOURG6&Fgs$\" *++++\"!\")F(F(-F`[l6$7$F'7$$\"31+++9=8WU!#=F_\\l-Ff[l6&FgsF)F)F)-F`[l 6&7#F^\\l-%'SYMBOLG6#%'CIRCLEGFb\\l-%&STYLEG6#%&POINTG-F`[l6&Ff\\l-Fh \\l6#%(DIAMONDGFb\\lF[]l-F`[l6&Ff\\l-Fh\\l6#%&CROSSGFb\\lF[]l-%%TEXTG6 %7$$!\"'FjsF[[lQ'y~=~-x6\"Fe[l-%*AXESTICKSG6$\"\"$Fd^l-%+AXESLABELSG6% %\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$F]_lF]_l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 30 "The centroid lies on the line " }{XPPEDIT 18 0 "y= x" "6#/%\"yG%\"xG" }{TEXT -1 19 " at a distance of " }{XPPEDIT 18 0 " 4/(3*Pi)" "6#*&\"\"%\"\"\"*&\"\"$F%%#PiGF%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sec(Pi/4)= 4*sqrt(2)/(3*Pi)" "6#/-%$secG6#*&%#PiG\"\"\" \"\"%!\"\"*(F*F)-%%sqrtG6#\"\"#F)*&\"\"$F)F(F)F+" }{TEXT -1 41 " from \+ the origin and, since the the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\" xG" }{TEXT -1 30 " is perpendicular to the line " }{XPPEDIT 18 0 "y=-x " "6#/%\"yG,$%\"xG!\"\"" }{TEXT -1 153 ", this is the perpendicular di stance of the centroid frm the axis of rotation. The distance travelle d by the centroid when it is rotated around the line " }{XPPEDIT 18 0 "y=-x" "6#/%\"yG,$%\"xG!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "2*Pi* `.`*``(sqrt(2)/(3*Pi))=8*sqrt(2)/3" "6#/**\"\"#\"\"\"%#PiGF&%\".GF&-%! G6#*&-%%sqrtG6#F%F&*&\"\"$F&F'F&!\"\"F&*(\"\")F&-F.6#F%F&F1F2" }{TEXT -1 34 ". Since the area of the sector is " }{XPPEDIT 18 0 "A=Pi/4" "6# /%\"AG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 40 ", Pappus' theorem shows t hat the volume " }{TEXT 360 1 "V" }{TEXT -1 40 " of the solid of revol ution is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V= ``(8*sqrt(2)/3)*`.`*A" "6#/%\"VG*(-%!G6#*(\"\")\"\"\"-%%sqrtG6#\" \"#F+\"\"$!\"\"F+%\".GF+%\"AGF+" }{XPPEDIT 18 0 "``=``(8*sqrt(2)/3)*`. `*``(Pi/4)" "6#/%!G*(-F$6#*(\"\")\"\"\"-%%sqrtG6#\"\"#F*\"\"$!\"\"F*% \".GF*-F$6#*&%#PiGF*\"\"%F0F*" }{XPPEDIT 18 0 "``=2*sqrt(2)*Pi/3" "6#/ %!G**\"\"#\"\"\"-%%sqrtG6#F&F'%#PiGF'\"\"$!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 78 "Find the coordinates of the centroid of the plane region bounded by t he curve " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 14 " and the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/2,2/ 5);" "6#-%!G6$*&\"\"\"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 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 11 "A triangle " }{TEXT 362 1 "T" }{TEXT -1 27 " has vertices at the points" }{XPPEDIT 18 0 "` `(1,0),``(2,1);" "6$-%!G6$\"\"\"\"\"!-F$6$\"\"#F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(2,-1)" "6#-%!G6$\"\"#,$\"\"\"!\"\"" }{TEXT -1 102 ". Use Pappus' theorem to find the volume of the solid of revolution o btained by rotating the triangle " }{TEXT 363 1 "T" }{TEXT -1 12 " aro und the " }{TEXT 361 1 "y" }{TEXT -1 7 " 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 "10*Pi/3" "6#*(\"#5\"\"\"%#PiGF% \"\"$!\"\"" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 46 "__________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_______________________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 28 "Find the area of the region " }{TEXT 364 1 "R" }{TEXT -1 23 " boun ded by the curves " }{XPPEDIT 18 0 "y=x^2+1" "6#/%\"yG,&*$%\"xG\"\"#\" \"\"F)F)" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=3-x^2" "6#/%\"yG,&\" \"$\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Locate the centroid of the region " }{TEXT 365 1 "R" }{TEXT -1 77 " without performing any integration, by exploiting symmetries of t he region. " }}{PARA 0 "" 0 "" {TEXT -1 98 "Use Pappus' theorem to fin d the volume of the solid of revolution obtained by rotating the regio n " }{TEXT 367 1 "R" }{TEXT -1 12 " around the " }{TEXT 366 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 1 " " } {XPPEDIT 18 0 "32*Pi/3;" "6#*(\"#K\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 46 "________________________________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "______________________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 30 "The sector of t he unit circle " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$ %\"yGF'F(F(" }{TEXT -1 56 " which lies in the first quadrant and is bo unded by the " }{TEXT 368 1 "x" }{TEXT -1 5 " and " }{TEXT 369 1 "y" } {TEXT -1 43 " axes has its centroid 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 -1 115 " Use Pappus' theorem to find the volume of the solid of revolution obta ined by rotating this sector around the line " }{XPPEDIT 18 0 "x+y = - 2;" "6#/,&%\"xG\"\"\"%\"yGF&,$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {XPPEDIT 18 0 " 2*sqrt(2)*Pi/3+sqrt(2)/2*Pi^2" "6#,&** \"\"#\"\"\"-%%sqrtG6#F%F&%#PiGF&\"\"$!\"\"F&*(-F(6#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 22 "moment of a rea picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1926 "f := x -> 2+sin(x-2)/4:\ng := x -> x^2/8-3*x/4+13 /8:\na := 1: b := 5: c := 3.34: m := 3.4: d := 3.46:\nfgm2 := (f(m)+g( m))/2: fgm4 := fgm2/2:\np1 := plot([f(x),g(x)],x=0.5..5.5,color=[red,b lue],thickness=2):\np2 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)] ],\n [[c,g(c)],[c,f(c)]],[[d,g(d)],[d,f(d)]]],color=black):\np3 := \+ plots[polygonplot]([[c,g(c)],[c,f(c)],[m,f(m)],[d,f(d)],\n [d ,g(d)],[m,g(m)]],color=COLOR(RGB,.8,.8,.83),style=patchnogrid):\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp) ):\npp := plot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1 ,op(1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i], gpts[i-1]],i=2..25)],\n style=patchnogrid,color=COLOR(RGB,.9 5,.95,.95)):\np5 := plot([[[m,(f(m)+g(m))/2]]$3],style=point,\n sy mbol=[circle,diamond,cross],color=black):\np6 := plot([[[m,g(m)],[4.8, g(m)]],[[m,f(m)],[4.8,f(m)]],\n [[m,fgm2],[4.1,fgm2]],[[a,0],[a,g(a) ]],[[b,0],[b,g(b)]]],\n color=black,linestyle=2):\np7 := plott ools[arrow]([m/2+.2,fgm2],[m,fgm2],0,.04,.06,arrow,color=black):\np8 : = plottools[arrow]([m/2-.2,fgm2],[0,fgm2],0,.04,.06,arrow,color=black) :\np9 := plottools[arrow]([4.6,fgm2+.1],[4.6,f(m)],0,.08,.08,arrow,col or=black):\np10 := plottools[arrow]([4.6,fgm2-.1],[4.6,g(m)],0,.08,.08 ,arrow,color=black):\np11 := plottools[arrow]([4,1],[4,fgm2],0,.08,.15 ,arrow,color=black):\np12 := plottools[arrow]([4,.73],[4,0],0,.08,.08, arrow,color=black):\nt1 := plots[textplot]([1.5,2.05,`y = f(x)`],color =red):\nt2 := plots[textplot]([1.56,.6,`y = g(x)`],color=blue):\nt3 := plots[textplot]([[-.1,2.54,`y`],[5.93,-.05,`x`],[a,-.05,`x = a`],\n \+ [b,-.05,`x = b`],[4,.92,`f(x) + g(x)`],\n [4,.89,`_______`],[m/2,1 .4,`x`],[4.6,fgm2,`f(x) - g(x)`]],color=black):\nt4 := plots[textplot] ([4,.79,`2`],font=[HELVETICA,9],color=black):\nplots[display]([p1,p2,p 3,p4,p5,p6,p7,p8,p9,p10,p11,p12,t1,t2,t3,t4],\n tickmarks=[0,0],view =[-.1..6,-.1..2.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Pappus' theorem pictures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1595 "f := t -> 1+(.3+.1*sin(5*t +3))*cos(t):\ng := t -> (.3+.1*sin(5*t+2))*sin(t):\np1 := plot([[0,-1] ,[0,1]],color=black):\np2 := plot([f(t),g(t),t=0..2*Pi],\n color=bl ack,linestyle=1,thickness=1,adaptive=false,numpoints=60):\np3 := plot( [-f(t),g(t),t=0..2*Pi],\n color=black,linestyle=2,thickness=1,adapt ive=false,numpoints=60):\np4 := plots[polygonplot]([[1,0],op(op(op(1,p 2)))],\n style=patchnogrid,color=COLOR(RGB,.8,.8,.93)):\np5 : = plots[polygonplot]([[-1,0],op(op(op(1,p3)))],\n style=patch nogrid,color=COLOR(RGB,.9,.9,.96)):\nr := .15:\np6 := plot([1.2*cos(t) ,.2+r*sin(t),t=0..2*Pi],\n color=COLOR(RGB,.7,.7,.8),thickn ess=3):\nt1 := evalf(5): a := f(t1): b := g(t1):\np7 := plot([a*cos(t) ,b+r*sin(t),t=0..2*Pi],color=black,linestyle=2):\np8 := plot([[[1,-.03 ]]$3],style=point,symbol=[circle,diamond,cross],color=black):\np9 := p lot([[1.2,.2],[-1.2,.2]],style=point,symbol=circle,symbolsize=15,color =black):\np10 := plottools[arrow]([.43,-.03],[0,-.03],0,.05,.08,arrow, color=black):\np11 := plottools[arrow]([.57,-.03],[1,-.03],0,.05,.08,a rrow,color=black):\np12 := plottools[arrow]([.54,.2],[0,.2],0,.05,.08, arrow,color=black):\np13 := plottools[arrow]([.66,.2],[1.2,.2],0,.05,. 08,arrow,color=black):\nt1 := plots[textplot]([[.5,-.03,`z`],[.5,.06,` _`],[.6,.21,`z`],[1.33,.2,'A'],\n [-.34,0,`2 z A`],[.05,.93,` L`]],font=[HELVETICA,10]):\nt2 := plots[textplot]([[1.29,.2,'d'],[-.39 ,0,`p`],\n [-.305,0,'d']],font=[SYMBOL,10]):\nt3 := plots[tex tplot]([1.12,0,`R`],font=[TIMES,ITALIC,14]):\nplots[display]([p1,p2,p3 ,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,t1,t2,t3],axes=none);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "f := t -> 1+(.3+.1*sin(5*t+3))*cos(t):\ng := t -> (.3+.1*sin(5*t+2))*sin(t ):\np1 := plot3d([f(t)*cos(s),f(t)*sin(s),g(t)],\nt=0..2*Pi,s=0..2*Pi, scaling=unconstrained,style=patchnogrid,axes=none,\n color=COLOR(R GB,.7,.7,.8),lightmodel=light4,grid=[50,50]):\np2 := plots[pointplot3d ]([[0,0,-1],[0,0,1]],style=line,color=black):\nplots[display]([p1,p2]) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 9 "Examples \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "p1 := plot3d([(3+cos(t))*cos(s),(3+cos(t))*sin(s),sin(t)],\n \+ t=0..2*Pi,s=0..2*Pi,color=COLOR(RGB,.7,.7,.87),\n lightmodel=light4, grid=[30,40],scaling=constrained):\np2 := plots[pointplot3d]([[0,0,-4] ,[0,0,4]],style=line,color=black):\nplots[display]([p1,p2]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1037 "f := t -> 3+cos(t):\ng := t -> sin(t):\np1 := plot([[0,-2],[0,2] ],color=black):\np2 := plot([f(t),g(t),t=0..2*Pi],\n color=black,li nestyle=1,thickness=1,adaptive=false,numpoints=60):\np3 := plot([-f(t) ,g(t),t=0..2*Pi],\n color=black,linestyle=2,thickness=1,adaptive=fa lse,numpoints=60):\np4 := plots[polygonplot]([[3,0],op(op(op(1,p2)))], \n style=patchnogrid,color=COLOR(RGB,.8,.8,.93)):\np5 := plot s[polygonplot]([[-3,0],op(op(op(1,p3)))],\n style=patchnogrid ,color=COLOR(RGB,.9,.9,.96)):\nr := 0.15:\np6 := plot([[3*cos(t),1+r*s in(t),t=0..2*Pi],\n [3*cos(t),-1+r*sin(t),t=0..2*Pi]],color=black,l inestyle=2):\np7 := plottools[arrow]([1.25,0],[0,0],0,.1,.09,arrow,col or=black):\np8 := plottools[arrow]([1.65,0],[3,0],0,.1,.09,arrow,color =black):\np9 := plottools[arrow]([3.2,.2],[3,0],0,.11,.25,arrow,color= black):\np10 := plottools[arrow]([3.5,.5],[3.707,.707],0,.11,.25,arrow ,color=black):\nt1 := plots[textplot]([[1.5,0,`R`],[3.35,.35,'r']],fon t=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,t1] ,axes=none);" }}}{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 }