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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 23 "The arc length formula " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Suppose that we wish to calcula te the arc length of the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG- %\"fG6#%\"xG" }{TEXT -1 15 " from the point" }{XPPEDIT 18 0 "``(a,f(a) );" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 13 " to the point" }{XPPEDIT 18 0 "``(b,f(b));" "6#-%!G6$%\"bG-%\"fG6#F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Suppose that there is a function " }{XPPEDIT 18 0 "s=s(x)" "6#/%\"sG-F$6#%\"xG" }{TEXT -1 49 " which gives the arc \+ length starting at the point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$% \"aG-%\"fG6#F&" }{TEXT -1 19 " as far as a value " }{TEXT 263 1 "x" } {TEXT -1 9 " between " }{TEXT 264 1 "a" }{TEXT -1 5 " and " }{TEXT 265 1 "b" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Consider a ( small) change " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 266 1 "x" } {TEXT -1 17 " in the variable " }{TEXT 267 1 "x" }{TEXT -1 10 ", and l et " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 268 1 "y" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 269 1 "s" }{TEXT -1 33 " be the corresponding changes in " }{TEXT 270 1 "y" }{TEXT -1 5 " and " }{TEXT 271 1 "s" }{TEXT -1 15 " respectively. 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[ZF*7$$\"3-++v)[Dxy)F*$\"3wTPaKy;(*[F*7$$\"33mmm\"4!pv!*F*$\"392o2_?'> /&F*7$$\"3Y)**\\PMirP*F*$\"3gT7'z)>[#>&F*7$$\"3OMLL`f^n'*F*$\"3_1cGV&> kL&F*7$$\"3GKL$eXWW'**F*$\"3G2!y\")>qD[&F*7$$\"3cm;/C9*e-\"!#<$\"3jA3I edYEcF*7$$\"3++++R,&H0\"Fix$\"3-Hzw]MxddF*7$$\"3smm\"*zC'R3\"Fix$\"3e8 9xKj=2fF*7$$\"3ELLL(G+<6\"Fix$\"3VB\">aBX)RgF*7$$\"3****\\PvXFT6Fix$\" 3%z1\\fJq-='F*7$$\"3!***\\iU4ep6Fix$\"3h\\:?,?p8jF*7$$\"3%************ **>\"Fix$\"3_***********fX'F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[ l-F$6$7&7$Fa[lFa[l7$$\"\"\"Fb[lFa[l7$Fh[l$\"3U+++++++bF*Ff[l-F[[l6&F][ lFb[lFb[lFb[l-%)POLYGONSG6$7%Ff[lFg[l7$Fh[l$\"#b!\"#-%&COLORG6&F][l$\" \"'F`[lFj\\lFh[l-%%TEXTG6&7$$\"#_Ff\\l$!#[!\"$Q\"d6\"F]\\l-%%FONTG6$%' SYMBOLG\"#6-F]]l6&7$$\"$0\"Ff\\l$\"\"$F`[lFe]lF]\\lFg]l-F]]l6&7$$\"#XF f\\l$\"$U$Fd]lFe]l-F[[l6&F][lFa[lFa[l$\"*++++\"!\")Fg]l-F]]l6&7$$\"#cF f\\l$!\"&Ff\\lQ\"xFf]lF]\\l-Fh]l6$%*HELVETICAGF_[l-F]]l6&7$$\"$4\"Ff\\ lFa^lQ\"yFf]lF]\\lFg_l-F]]l6&7$$\"#\\Ff\\l$\"#MFf\\lQ\"sFf]lFj^lFg_l-% *AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Ff_lQ!Ff]l-Fh]l6#%(DEFAULTG-%%VIEW G6$;$Ff\\lF`[l$\"#7F`[lFbal" 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" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "This approximation progressively improves as " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 275 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 276 1 "y" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 277 1 "s" }{TEXT -1 12 " tend to 0. " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{XPPEDIT 18 0 "s^2" "6#*$%\"sG\"\"#" }{TEXT -1 1 " " }{TEXT 278 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{XPPEDIT 18 0 "x^2+delta" "6# ,&*$%\"xG\"\"#\"\"\"%&deltaGF'" }{XPPEDIT 18 0 "y^2" "6#*$%\"yG\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 280 1 "s" }{TEXT -1 1 " " }{TEXT 279 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(delta*x^2+delta*y^2)" "6#-%%sqrtG6#,&*&%&deltaG\"\"\"*$%\"xG\"\" #F)F)*&F(F)*$%\"yGF,F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=sqrt(delta *x^2*(1+(delta*y^2)/(delta*x^2))" "6#/%!G-%%sqrtG6#*(%&deltaG\"\"\"*$% \"xG\"\"#F*,&F*F**(F)F**$%\"yGF-F**&F)F**$F,F-F*!\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = delta*x*sqrt(1+(delta*y/(delta*x))^2);" "6#/%! G*(%&deltaG\"\"\"%\"xGF'-%%sqrtG6#,&F'F'*$*(F&F'%\"yGF'*&F&F'F(F'!\"\" \"\"#F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(delta*s)/(delta*x)" "6#*(%&deltaG\"\"\"%\"sGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " }{TEXT 281 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+(delta*y/(delta*x))^2 );" "6#-%%sqrtG6#,&\"\"\"F'*$*(%&deltaGF'%\"yGF'*&F*F'%\"xGF'!\"\"\"\" #F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 3 "As " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 282 1 "x " }{TEXT -1 13 " tends to 0, " }{XPPEDIT 18 0 "delta*y/(delta*x)" "6#* (%&deltaG\"\"\"%\"yGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 25 " tends to the \+ derivative " }{XPPEDIT 18 0 "dy/dx = `f '`(x);" "6#/*&%#dyG\"\"\"%#dxG !\"\"-%$f~'G6#%\"xG" }{TEXT -1 8 ". Hence " }{XPPEDIT 18 0 "delta*s/(d elta*x)" "6#*(%&deltaG\"\"\"%\"sGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 56 " \+ also tends to a limit which can only be the derivative " }{XPPEDIT 18 0 "ds/dx = `s '`(x);" "6#/*&%#dsG\"\"\"%#dxG!\"\"-%$s~'G6#%\"xG" } {TEXT -1 28 " of the arc length function " }{XPPEDIT 18 0 "s=s(x)" "6# /%\"sG-F$6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dx=sqrt(1+(dy/ dx)^2)" "6#/*&%#dsG\"\"\"%#dxG!\"\"-%%sqrtG6#,&F&F&*$*&%#dyGF&F'F(\"\" #F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 20 "It follows that the " }{TEXT 259 10 "arc length" } {TEXT -1 15 " along a curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\" fG6#%\"xG" }{TEXT -1 13 " from a point" }{XPPEDIT 18 0 "``(a,f(a));" " 6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 24 " on the curve to a point" } {XPPEDIT 18 0 "``(b,f(b));" "6#-%!G6$%\"bG-%\"fG6#F&" }{TEXT -1 26 " i s given by the integral:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = a .. b) = Int(sqrt(1+`f '`(x) ^2),x = a .. b);" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*$*&%#dyGF+%#dxG!\" \"\"\"#F+/%\"xG;%\"aG%\"bG-F%6$-F(6#,&F+F+*$-%$f~'G6#F3F1F+/F3;F5F6" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 283 24 "___ _____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " provided that the derivative " }{XPPEDIT 18 0 "dy/dx = `f '`(x);" "6# /*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6#%\"xG" }{TEXT -1 37 " exists througho ut the interval from " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "A more ri gorous derivation of this arc length formula can be made along the fol lowing lines. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Subdivide the interval \+ from " }{TEXT 307 1 "a" }{TEXT -1 4 " to " }{TEXT 308 1 "b" }{TEXT -1 6 " into " }{TEXT 309 1 "n" }{TEXT -1 29 " subintervals of equal width " }{XPPEDIT 18 0 "h=(b-a)/n" "6#/%\"hG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"n GF*" }{TEXT -1 11 " by points " }{XPPEDIT 18 0 "x[0] = a,x[1],` . . . \+ `,x[n] = b;" "6&/&%\"xG6#\"\"!%\"aG&F%6#\"\"\"%(~.~.~.~G/&F%6#%\"nG%\" bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The subinterval ar e then " }{XPPEDIT 18 0 "[x[i-1],x[i]]" "6#7$&%\"xG6#,&%\"iG\"\"\"F)! \"\"&F%6#F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "i=1,2,` . . . `,n " "6&/%\"iG\"\"\"\"\"#%(~.~.~.~G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "P[0],P[1],` . . . `,P[n]" "6&&% \"PG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 28 " be the corr esponding points" }{XPPEDIT 18 0 "``(x[0],f(x[0])),``(x[1],f(x[1])),` \+ . . . `,``(x[n],f(x[n]));" "6&-%!G6$&%\"xG6#\"\"!-%\"fG6#&F'6#F)-F$6$& F'6#\"\"\"-F+6#&F'6#F3%(~.~.~.~G-F$6$&F'6#%\"nG-F+6#&F'6#F=" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6# %\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "For each interv al " }{XPPEDIT 18 0 "[x[i-1],x[i]]" "6#7$&%\"xG6#,&%\"iG\"\"\"F)!\"\"& F%6#F(" }{TEXT -1 17 " choose a number " }{XPPEDIT 18 0 "x[i]^`*`" "6# )&%\"xG6#%\"iG%\"*G" }{TEXT -1 71 " in the interval such that the tang ent line at the corresponding point " }{XPPEDIT 18 0 "P[i]^`*`" "6#)&% \"PG6#%\"iG%\"*G" }{TEXT -1 17 " with coordinates" }{XPPEDIT 18 0 "``( x[i]^`*`,f(x[i]^`*`));" "6#-%!G6$)&%\"xG6#%\"iG%\"*G-%\"fG6#)&F(6#F*F+ " }{TEXT -1 52 " is parallel to the line segment joining the points " }{XPPEDIT 18 0 "P[i-1]" "6#&%\"PG6#,&%\"iG\"\"\"F(!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P[i]" "6#&%\"PG6#%\"iG" }{TEXT -1 93 ". It is i ntuitively clear that this can be done. More formally, this is a conse quence of the " }{TEXT 259 18 "mean value theorem" }{TEXT -1 54 " of d ifferential calculus. Thus the slope of the line " }{XPPEDIT 18 0 "P[i -1]*P[i]" "6#*&&%\"PG6#,&%\"iG\"\"\"F)!\"\"F)&F%6#F(F)" }{TEXT -1 4 " \+ is " }{XPPEDIT 18 0 "`f '`(x[i]^`*`);" "6#-%$f~'G6#)&%\"xG6#%\"iG%\"*G " }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 390 295 295 {PLOTDATA 2 "6>-%'CURVESG6$7S7$$!35+++++++?!#=$!3B++++++?=F*7$ $!37++](oUIn\"F*$!3)o\"4SR%eg]\"F*7$$!3C+]7y)e&)Q\"F*$!3\"H%o9>z6Q7F*7 $$!35++D\"3F'o5F*$!3\")o#fAM>fU*!#>7$$!3+++](oXdY(F<$!3RviQ5f48lF<7$$! 3!>+]i:F0E%F<$!3yK*fxN/fn$F<7$$!3=-+D\"Gz))G\"F<$!3T3uY=5`+6F<7$$\"3%z **\\(o**3)y\"F<$\"3dAB61AnF<7$$\"34++]7*309\"F*$\"3!3U-Ce(4/ $*F<7$$\"3$****\\i&e*yU\"F*$\"3GAO_!)[a_6F*7$$\"3;++]([D9v\"F*$\"3chqA Fpo'R\"F*7$$\"3!)****\\iNGw?F*$\"38px;DC^N;F*7$$\"39++]7XM*Q#F*$\"35%e U'>Qnf=F*7$$\"3#)**\\(o%QjtEF*$\"3S\"3H9BR\"e?F*7$$\"3'*****\\i8o6IF*$ \"3?/\"[#>C#yG#F*7$$\"3'*******\\>0)H$F*$\"3!f.`vvHqZ#F*7$$\"3z**\\(=- p6j$F*$\"3g55j9?$4p#F*7$$\"3Y*****\\2Mg#RF*$\"3qm\"3ql;Z(GF*7$$\"3A+]( =xZ&\\UF*$\"3Q;.8=fNqIF*7$$\"3G+]i:$4wb%F*$\"3%p4l!QQ\"3D$F*7$$\"3O++v =#R!z[F*$\"3.#[%*\\iKIV$F*7$$\"3#)**\\P4A@u^F*$\"3Q!RR$=i!\\f$F*7$$\"3 I++Dchf#\\&F*$\"3'=$Huc$[Ow$F*7$$\"3L+](of2L#eF*$\"3M(34j6%[KRF*7$$\"3 B+]7yG>6hF*$\"3ixof%f8T2%F*7$$\"3w++voo6AkF*$\"3u+5;'y\"\\@UF*7$$\"3\" ))****\\xJLu'F*$\"3Qf#=fIcwO%F*7$$\"3+++v$*yddqF*$\"3fQq*y.fY]%F*7$$\" 3Q+](=')F*$\"3ka[)=QZy4&F*7$$\"3I,+](Q(zS*)F*$\"3H'>X;/ U:?&F*7$$\"3U**\\(=-,FC*F*$\"3=u1?.-Z$H&F*7$$\"3k+]P4tFe&*F*$\"3-g8Gfd r$Q&F*7$$\"3!)****\\73\"o')*F*$\"37#*\\LA03\"Fiw $\"3a7[9Pnj%o&F*7$$\"38+v=(4bM6\"Fiw$\"3Xv_tl:-XdF*7$$\"37++]xlWU6Fiw$ \"3O4cUhLC&z&F*7$$\"33+]i&3uc<\"Fiw$\"3kf>)41,m%eF*7$$\"3/+++lJR07Fiw$ \"39[Q\"e&Q#p)eF*7$$\"33+v=-*zqB\"Fiw$\"3hZ]S:\"zS#fF*7$$\"33+D\"G:3uE \"Fiw$\"3Yo%f9-**R&fF*7$$\"3/+++++++8Fiw$\"3w************zfF*-%'COLOUR G6&%$RGBG$\"#5!\"\"$\"\"!Fa[lF`[l-F$6$7&7$F`[lF`[l7$$\"\"\"Fa[lF`[l7$F g[l$\"3U+++++++bF*Fe[l-Fjz6&F\\[lFa[lFa[lFa[l-F$6&7%Fe[l7$$\"3++++++++ ]F*$\"3w*************\\$F*Fi[l-%'SYMBOLG6#%'CIRCLEGF\\\\l-%&STYLEG6#%& POINTG-F$6&F`\\l-Fg\\l6#%(DIAMONDGF\\\\lFj\\l-F$6&F`\\l-Fg\\l6#%&CROSS GF\\\\lFj\\l-F$6$7$7$$\"3/+++++++5F*$\"3-+++++++8F*7$$\"3A+++++++!*F*$ \"3]*************p&F*-Fjz6&F\\[lF`[lF`[l$\"*++++\"!\")-F$6%7$7$F`[l$!3 %**************\\\"F*Fe[lF\\\\l-%*LINESTYLEG6#\"\"#-F$6%7$7$Fg[lF^_lFf [lF\\\\lF`_l-F$6%7$7$Fb\\lF^_lFa\\lF\\\\lF`_l-%%TEXTG6&7$$Fi^l!\"#$\" \"&Fa`lQ\"P6\"F\\\\l-%%FONTG6$%*HELVETICAGF^[l-F]`l6&7$$\"#'*Fa`l$\"\" 'F_[lFd`lF\\\\lFf`l-F]`l6&7$$\"#WFa`l$\"#RFa`lFd`lF\\\\lFf`l-F]`l6&7$$ \"%X5!\"$F`[lQ\"QFe`lF\\\\lFf`l-F]`l6&7$$\"#ZFa`l$\"\"%F_[lQ\"*Fe`lF\\ \\lFf`l-F]`l6&7$$F_[lFa`l$F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve \+ " }{XPPEDIT 18 0 "y=sqrt(1-x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG \"\"#!\"\"" }{TEXT -1 104 " is a semi-circle with its centre at the or igin and radius 1. Hence the arc length along this curve from" } {XPPEDIT 18 0 "``(0,1);" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 3 " to" } {XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 8 " units. " }}{PARA 0 "" 0 "" {TEXT -1 55 "We can check that the value of the a rc length integral " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 0 .. 1);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F **$*&%#dyGF*%#dxG!\"\"\"\"#F*/%\"xG;\"\"!F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "agrees with this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "f := x -> sqrt(1-x^2 ):\n'f(x)'=f(x);\np1 := plot([[[0,1],[1,0]]$3],style=point,\n s ymbol=[circle,diamond,cross],color=black):\np2 := plot([f(x),f(x)],x=0 ..1,y,color=[red,blue],thickness=[1,2]):\nplots[display]([p1,p2],scali ng=constrained,tickmarks=[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"xG*$,&\"\"\"F**$)F'\"\"#F*!\"\"#F*F-" }}{PARA 13 "" 1 "" {GLPLOT2D 402 258 258 {PLOTDATA 2 "6+-%'CURVESG6&7$7$$\"\"!F)$\"\"\"F) 7$F*F(-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBGF)F)F)-%&STYLEG6#%&POINTG -F$6&F&-F.6#%(DIAMONDGF1F5-F$6&F&-F.6#%&CROSSGF1F5-F$6%7YF'7$$\"3emmm; arz@!#>$\"3e(=:\"QTi(***!#=7$$\"3[LL$e9ui2%FI$\"3!e2E0a)o\"***FL7$$\"3 nmmm\"z_\"4iFI$\"3)f\\s^f/2)**FL7$$\"3[mmmT&phN)FI$\"3)37fk0E]'**FL7$$ \"3CLLe*=)H\\5FL$\"3R\\N+$H'zW**FL7$$\"3gmm\"z/3uC\"FL$\"3^l4$yi$*=#** FL7$$\"3%)***\\7LRDX\"FL$\"3%p#zm\"3WR*)*FL7$$\"3]mm\"zR'ok;FL$\"3iaa \"Q\\n/')*FL7$$\"3w***\\i5`h(=FL$\"3\"Rj1r%eUA)*FL7$$\"3WLLL3En$4#FL$ \"3_.Ayu2Py(*FL7$$\"3qmm;/RE&G#FL$\"3wIBk#=x`t*FL7$$\"3\")*****\\K]4]# FL$\"31q9\"H%H@#o*FL7$$\"3$******\\PAvr#FL$\"3B_^A([sOi*FL7$$\"3)***** *\\nHi#HFL$\"3m:#e7,zAc*FL7$$\"3jmm\"z*ev:JFL$\"3)>#\\;*p8A]*FL7$$\"3? LLL347TLFL$\"3pW!GHZL`U*FL7$$\"3,LLLLY.KNFL$\"3#=1W(>`Yb$*FL7$$\"3w*** \\7o7Tv$FL$\"3!QZNcQ$eo#*FL7$$\"3'GLLLQ*o]RFL$\"3=`?'yv9l=*FL7$$\"3A++ D\"=lj;%FL$\"3]kxU3;t!4*FL7$$\"31++vV&R5j')FL7$$\"3&em;zRQb@&FL$\"3e0&=5t$=K&)FL7$$\"3 \\***\\(=>Y2aFL$\"3ld&yZ.e=T)FL7$$\"39mm;zXu9cFL$\"3]/X@$oS\\F)FL7$$\" 3l******\\y))GeFL$\"3k!yShX>b7)FL7$$\"3'*)***\\i_QQgFL$\"35nMlUm1rzFL7 $$\"3@***\\7y%3TiFL$\"3bX$z*QqP8yFL7$$\"35****\\P![hY'FL$\"3QXhIao;GwF L7$$\"3kKLL$Qx$omFL$\"3q\"p\\+YH?X(FL7$$\"3!)*****\\P+V)oFL$\"378IrHv- `sFL7$$\"3?mm\"zpe*zqFL$\"3QynnKd;iqFL7$$\"3%)*****\\#\\'QH(FL$\"3G\"Q TZ_=5%oFL7$$\"3GKLe9S8&\\(FL$\"3a%Rp()p\"*)>mFL7$$\"3R***\\i?=bq(FL$\" 3=&FL7$$\"3=LLe9tOc()FL$\"3PnBuRIqH[FL7$$\"3u******\\Qk\\*)FL$ \"3+@I:/fPhWFL7$$\"3CLL$3dg6<*FL$\"3au)*4GH?')RFL7$$\"3ImmmmxGp$*FL$\" 3s!RL6R._\\$FL7$$\"3A++D\"oK0e*FL$\"3!yTAE(=!f'GFL7$$\"3C+++]oi\"o*FL$ \"3?UJ1O#=K]#FL7$$\"3A++v=5s#y*FL$\"37vO>U4Dt?FL7$$\"3;+D1k2/P)*FL$\"3 mLW(eL^zz\"FL7$$\"35+]P40O\"*)*FL$\"31H.f%oH+Z\"FL7$$\"3k]7.#Q?&=**FL$ \"3G#>z^*=&RF\"FL7$$\"31+voa-oX**FL$\"3(pV\\8H')3/\"FL7$$\"3[\\PMF,%G( **FL$\"3?&=hJ*p=ltFIF,-F26&F4$\"*++++\"!\")F(F(-%*THICKNESSG6#F+-F$6%F E-F26&F4F(F(F]^l-Fa^l6#\"\"#-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6$ \"\"$Fa_l-%+AXESLABELSG6%Q!6\"Fe_l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F*F j_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" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y=sqrt(1-x^2)" "6#/%\"yG-%%sqrtG 6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{XPPEDIT 18 0 "``=(1-x^2)^(1/2)" "6#/% !G),&\"\"\"F'*$%\"xG\"\"#!\"\"*&F'F'F*F+" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=-1/2" "6#/*&%#dyG\"\" \"%#dxG!\"\",$*&F&F&\"\"#F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^2) ^(-1/2)*`.`*2*x=-x/sqrt(1-x^2)" "6#/**),&\"\"\"F'*$%\"xG\"\"#!\"\",$*& F'F'F*F+F+F'%\".GF'F*F'F)F',$*&F)F'-%%sqrtG6#,&F'F'*$F)F*F+F+F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+(dy/dx)^2=1+x^2/(1-x^2)" "6#/,& \"\"\"F%*$*&%#dyGF%%#dxG!\"\"\"\"#F%,&F%F%*&%\"xGF+,&F%F%*$F.F+F*F*F% " }{XPPEDIT 18 0 "``=(1-x^2+x^2)/(1-x^2)" "6#/%!G*&,(\"\"\"F'*$%\"xG\" \"#!\"\"*$F)F*F'F',&F'F'*$F)F*F+F+" }{XPPEDIT 18 0 "``=1/(1-x^2)" "6#/ %!G*&\"\"\"F&,&F&F&*$%\"xG\"\"#!\"\"F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(1+(dy/dx)^2)=1/sqrt(1-x^2)" "6#/-%%sqrtG6#,&\"\"\" F(*$*&%#dyGF(%#dxG!\"\"\"\"#F(*&F(F(-F%6#,&F(F(*$%\"xGF.F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 0 .. 1) = Int (1/sqrt(1-x^2),x = 0 .. 1);" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*$*&%#dy GF+%#dxG!\"\"\"\"#F+/%\"xG;\"\"!F+-F%6$*&F+F+-F(6#,&F+F+*$F3F1F0F0/F3; F5F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=arcsin*x" "6#/%!G*&%'arcsinG\" \"\"%\"xGF'" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ` `],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = arcsin*1;" "6#/%!G*&%'arcsinG\"\"\"F'F'" } {XPPEDIT 18 0 "`` = Pi/2;" "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 " Note" }{TEXT -1 22 ": Since the integrand " }{XPPEDIT 18 0 "1/sqrt(1-x ^2)" "6#*&\"\"\"F$-%%sqrtG6#,&F$F$*$%\"xG\"\"#!\"\"F," }{TEXT -1 21 " \+ is not defined when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 38 ", we should (strictly speaking) treat " }{XPPEDIT 18 0 "Int(1/sqrt (1-x^2),x = 0 .. 1);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&F'F'*$%\"xG\" \"#!\"\"F//F-;\"\"!F'" }{TEXT -1 7 " as an " }{TEXT 259 17 "improper i ntegral" }{TEXT -1 21 " and evaluate it as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int(1/sqrt(1-x^2),x = 0 .. t),t = 1,left)=Limit(``,t = 1,left)" "6#/-%&LimitG6%-%$IntG6$*&\"\"\"F+-%%sq rtG6#,&F+F+*$%\"xG\"\"#!\"\"F3/F1;\"\"!%\"tG/F7F+%%leftG-F%6%%!G/F7F+F 9" }{TEXT -1 1 " " }{TEXT 301 1 "[" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ar csin*x" "6#*&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 2 " " }{XPPEDIT 18 0 " PIECEWISE([t, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%\"tG%!G7$F( F(7$\"\"!F(" }{TEXT -1 3 " " }{TEXT 302 1 "]" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Limit(``(arcsin*t),t = 1,left);" "6#/%!G-%&LimitG6 %-F$6#*&%'arcsinG\"\"\"%\"tGF,/F-F,%%leftG" }{XPPEDIT 18 0 "`` = Pi/2; " "6#/%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "f := x -> s qrt(1-x^2):\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqrt(1+D( f)(x)^2),x=0..1);\n``=simplify(%,symbolic);\n``=value(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,&\"\"\"F**$)F'\"\"#F* !\"\"#F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xG F*,$*&,&\"\"\"F.*$)F*\"\"#F.!\"\"#F2F1F*F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*&,&F(F(*$)%\"xG\"\"#F(!\"\"F/F-F. F(#F(F./F-;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*& \"\"\"F)*$,&F)F)*$)%\"xG\"\"#F)!\"\"#F)F/F0/F.;\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 303 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "Find the arc length along the curve " }{XPPEDIT 18 0 "y = x^3/6+1/(2* x);" "6#/%\"yG,&*&%\"xG\"\"$\"\"'!\"\"\"\"\"*&F+F+*&\"\"#F+F'F+F*F+" } {TEXT -1 26 " from the the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"x G\"\"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 3;" " 6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 304 8 "Solutio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 292 "f := x -> x^3/6+1/(2*x):\n'f(x)'=f(x);\np1 \+ := plot(f(x),x=.2..3.5,y=0..7,color=red):\np2 := plot(f(x),x=1..3,colo r=blue,thickness=3):\np3 := plot([[[1,2/3],[3,14/3]]$3],style=point,\n symbol=[circle,diamond,cross],color=black):\nplots[display]([p 1,p2,p3],tickmarks=[3,3],view=[0..3.5,0..7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"'!\"\"F'\"\"$\"\"\"*&F-F-*&\"\"#F -F'F-F+F-" }}{PARA 13 "" 1 "" {GLPLOT2D 375 345 345 {PLOTDATA 2 "6*-%' CURVESG6$7U7$$\"35+++++++?!#=$\"3>LLLLLL,D!#<7$$\"3!****\\P/`'fBF*$\"3 8(*p3I`9@@F-7$$\"3p****\\(31$>FF*$\"3)**[[+vb?%=F-7$$\"35+D\"yFQA.$F*$ \"3QD!\\qc$f`;F-7$$\"3'***\\7o/ku%=pPP(F*7$$\"3O)* \\i]_I\">)F*$\"3CI3!z))f+-(F*7$$\"34++]2'>\"4*)F*$\"3agNmv1z!z'F*7$$\" 3)****\\P)3PT&*F*$\"35p'*Rza/)o'F*7$$\"3)****\\sg8`-\"F-$\"3?D#>!)QAIn 'F*7$$\"3#****\\PQ#y'4\"F-$\"3ksIyFJrdnF*7$$\"3'****\\Fzbc;\"F-$\"3%4r l#fA;HpF*7$$\"3*)*\\7jW*>G7F-$\"3I)ezToW)erF*7$$\"3$****\\(**)pDI\"F-$ \"3kg_gp3*>_(F*7$$\"3%)******G9dl8F-$\"3I=D)=ZLc!zF*7$$\"3!)*\\7[=d)Q9 F-$\"3TSRoM2yR%)F*7$$\"3f****\\'\\FP]\"F-$\"3g[pYp\"4@**)F*7$$\"3'**\\ 7)40!\\d\"F-$\"3wQX]N/@&o*F*7$$\"3%)*\\P%\\SnU;F-$\"3k4A_559V5F-7$$\"3 #***\\7G')Q8F-$\"3/9X,^:*>W\"F -7$$\"3g*\\(=LCY%)>F-$\"3a>)3IRcWb\"F-7$$\"3%***\\76d'G0#F-$\"3%fGSb7Y ao\"F-7$$\"3y****\\!*H`B@F-$\"3QL,yu0VJ=F-7$$\"3))**\\iOrm#>#F-$\"3s25 T2I,&)>F-7$$\"3!**\\7y(zbfAF-$\"3<-vroi,W@F-7$$\"3$***\\P_)GQL#F-$\"3% >gVrOtGL#F-7$$\"3o****\\OXc+CF-$\"3y_\\a(e5R^#F-7$$\"3-++vB\">=Z#F-$\" 3'3Q)ft0P>FF-7$$\"3e*\\7.P'QODF-$\"3.R;X7zm;HF-7$$\"3#****\\_Uvpg#F-$ \"3Mm&H(>nwWJF-7$$\"3w*\\7[A%RtEF-$\"3/.BJ!f-:P$F-7$$\"3t*\\i!35#Gu#F- $\"3dbnGT^O@OF-7$$\"3t***\\(y$)p5GF-$\"3Or!4R[c'yQF-7$$\"3w*\\7`pf<)GF -$\"3Qu7)oO>@;%F-7$$\"3E+++*=+-&HF-$\"3g*=gDK!3\\WF-7$$\"3.+]()y/>?IF- $\"3'R[*)>[Aqv%F-7$$\"3=+D\"Q@,'*3$F-$\"3g&zNR5,s2&F-7$$\"3)*****\\qCQ `JF-$\"3W;&[]qrYQ&F-7$$\"3#***\\P))H[EKF-$\"3Q5-b')*=Iv&F-7$$\"3k***** H'\\'=H$F-$\"3u%4GHT.s4'F-7$$\"33+D\"[yv:O$F-$\"3#zon'H\\\")zkF-7$$\"3 n*\\(=OzHGMF-$\"3+Hj9'zQ9'oF-7$$\"3++++++++NF-$\"3Gw/>w/p)G(F--%'COLOU RG6&%$RGBG$\"*++++\"!\")$\"\"!Fj[lFi[l-F$6%7S7$$\"\"\"Fj[l$\"3Immmmmmm mF*7$$\"3ALLL3VfV5F-$\"3'y\"HiVCT&o'F*7$$\"3smm\"H[D:3\"F-$\"3h0ZAj(G: t'F*7$$\"3XLL$e0$=C6F-$\"3jWa'p7bb\"oF*7$$\"3QLL$3RBr;\"F-$\"3Ni\\&pf^ P$pF*7$$\"3imm\"zjf)47F-$\"3iiHvr]G%3(F*7$$\"3WLLe4;[\\7F-$\"3ox))3D'> GD(F*7$$\"3-++Dmy]!H\"F-$\"3/>l/4g[cuF*7$$\"3>LLezs$HL\"F-$\"3=co$R37# )p(F*7$$\"31++D@1Bv8F-$\"3m'y]+e;1(zF*7$$\"3pmmm@Xt=9F-$\"3G!3.]+nOG)F *7$$\"3MLL$3y_qX\"F-$\"3/FA(o\"*Hre)F*7$$\"3'******\\1!>+:F-$\"3?\"puB _\\+'*)F*7$$\"3*******\\Z/Na\"F-$\"3gig+s*Q\"o$*F*7$$\"35+++NfC&e\"F-$ \"3'3ftKZNOz*F*7$$\"3LLLez6:B;F-$\"31F3D0Qx?5F-7$$\"3_mmm\"=C#o;F-$\"3 eh2x%3!\\t5F-7$$\"3gmmmEpS15k3T'*H\"F-7$$\"3!****\\(3zMu=F-$\"3oi%[xRYUO\"F-7$$\"3emm;H_? <>F-$\"3%)*GstX+`V\"F-7$$\"3mmm\"zihl&>F-$\"3??aCm,)Q]\"F-7$$\"3KLL$3# G,**>F-$\"3k\\fp2M[\"e\"F-7$$\"3u4X'R*R#>F-7$$\"3z****\\_qn2AF-$\"3[9,@TCz>?F-7$ $\"3%)***\\i&p@[AF-$\"3?B$)=6iK;@F-7$$\"3#)****\\2'HKH#F-$\"3\"oS!pC9, GAF-7$$\"3_mmmwanLBF-$\"3xau%\\xpCL#F-7$$\"3u*****\\2goP#F-$\"3A%\\861 f$[CF-7$$\"3CLLeR<*fT#F-$\"3!*RCYC\"=tb#F-7$$\"3'******\\)HxeCF-$\"3-i 007')z!o#F-7$$\"3Cmm\"H!o-*\\#F-$\"3m-Wg&\\/7!GF-7$$\"3))***\\7k.6a#F- $\"3r;I([!R]JHF-7$$\"3emmmT9C#e#F-$\"3!yP=un[L1$F-7$$\"3\"****\\i!*3`i #F-$\"3I,)f%>m;1KF-7$$\"3;LLL$*zymEF-$\"35O#)4+nT[LF-7$$\"30LL$3N1#4FF -$\"3?xog%fF()\\$F-7$$\"3kmm\"HYt7v#F-$\"3)zVt+P(p_OF-7$$\"3%*******p( G**y#F-$\"3+a%f8KX&)z$F-7$$\"3Umm;9@BMGF-$\"3G08?R%=4(RF-7$$\"3/LLL`v& Q(GF-$\"3>L?^;X()HTF-7$$\"30++DOl5;HF-$\"3#Rk,Tn*R/VF-7$$\"3/++v.UacHF -$\"3)*R7C6UQwWF-7$$\"\"$Fj[l$\"3'pmmmmmmm%F--Fc[l6&Fe[lFi[lFi[lFf[l-% *THICKNESSG6#F`[m-F$6&7$F^\\lF^[m-%'SYMBOLG6#%'CIRCLEG-Fc[l6&Fe[lFj[lF j[lFj[l-%&STYLEG6#%&POINTG-F$6&Fj[m-F\\\\m6#%(DIAMONDGF_\\mFa\\m-F$6&F j[m-F\\\\m6#%&CROSSGF_\\mFa\\m-%*AXESTICKSG6$F`[mF`[m-%+AXESLABELSG6%Q \"x6\"Q\"yFf]m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fi[l$\"#N!\"\";Fi[l$\"\"( Fj[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" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y=x^3/6+1/(2*x)" "6#/%\"yG,&*&% \"xG\"\"$\"\"'!\"\"\"\"\"*&F+F+*&\"\"#F+F'F+F*F+" }{TEXT -1 10 ", we h ave " }{XPPEDIT 18 0 "dy/dx = 1/2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F& \"\"#F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-1/2" "6#,&*$%\"xG\"\"#\" \"\"*&F'F'F&!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-2)=1/2" "6#/) %\"xG,$\"\"#!\"\"*&\"\"\"F*F'F(" }{XPPEDIT 18 0 "``(x^2-1/x^2)" "6#-%! G6#,&*$%\"xG\"\"#\"\"\"*&F*F**$F(F)!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1+(dy/dx)^2 = 1+1/4;" "6#/,&\"\"\"F%*$*&%#dyGF%%#dxG!\" \"\"\"#F%,&F%F%*&F%F%\"\"%F*F%" }{XPPEDIT 18 0 " ``(x^4-2+1/x^4)" "6#- %!G6#,(*$%\"xG\"\"%\"\"\"\"\"#!\"\"*&F*F**$F(F)F,F*" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*& \"\"\"F&\"\"%!\"\"" }{XPPEDIT 18 0 " ``(x^4+2+1/x^4)" "6#-%!G6#,(*$%\" xG\"\"%\"\"\"\"\"#F**&F*F**$F(F)!\"\"F*" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/4;" "6#/%!G*&\"\"\"F&\"\" %!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x^2+1/x^2)^2" "6#*$,&*$%\"xG \"\"#\"\"\"*&F(F(*$F&F'!\"\"F(F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+(dy/dx)^2) = 1/2;" "6#/-%%sqrtG6#,&\"\"\"F(*$*&%#dyGF(%#dxG! \"\"\"\"#F(*&F(F(F.F-" }{XPPEDIT 18 0 "``(x^2+1/x^2)" "6#-%!G6#,&*$%\" xG\"\"#\"\"\"*&F*F**$F(F)!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The required arc length is given by " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 1 .. 3) = 1 /2;" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*$*&%#dyGF+%#dxG!\"\"\"\"#F+/%\" xG;F+\"\"$*&F+F+F1F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x^2+1/(x^ 2)),x = 1 .. 3)" "6#-%$IntG6$-%!G6#,&*$%\"xG\"\"#\"\"\"*&F-F-*$F+F,!\" \"F-/F+;F-\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(x^3/3-1/x);" "6#-%!G6#,&*&%\"xG\"\"$F)!\"\"\"\"\" *&F+F+F(F*F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[``, \+ ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(``(27/3-1/3)-(1/3-1)) = 1/2;" "6#/-%!G6#,&-F%6#,&*&\"#F\"\"\" \"\"$!\"\"F-*&F-F-F.F/F/F-,&*&F-F-F.F/F-F-F/F/*&F-F-\"\"#F/" } {XPPEDIT 18 0 "``(26/3-(-2/3))=14/3" "6#/-%!G6#,&*&\"#E\"\"\"\"\"$!\" \"F*,$*&\"\"#F*F+F,F,F,*&\"#9F*F+F," }{TEXT -1 1 " " }{TEXT 305 1 "~" }{TEXT -1 14 " 4.666666667. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "f := x -> x^3/6+1/(2*x):\n' f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqrt(1+rhs(%)^2),x=1..3 );\n``=simplify(%,symbolic);\n``=expand(rhs(%));\n``=value(rhs(%));\n` `=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%\"xG,&*&\"\"'!\"\"F'\"\"$\"\"\"*&F-F-*&\"\"#F-F'F-F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&*&\"\"#!\"\"F*F-\" \"\"*&F/F/*&F-F/)F*F-F/F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*$,&\"\"\"F(*$),&*&\"\"#!\"\"%\"xGF-F(*&F(F(*&F-F()F/F-F(F.F.F-F(F(# F(F-/F/;F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\" #F(-%$IntG6$*&%\"xG!\"#,&*$)F.\"\"%F(F(F(F(F(/F.;F(\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"#F(-%$IntG6$,&*$)%\"xGF)F (F(*&F(F(*$F/F(!\"\"F(/F0;F(\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#9\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+nmmmY!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "Find the arc length along the curve " }{XPPEDIT 18 0 "y = sqrt(x)*(3-x)/3;" "6#/%\"yG*(-%%sqrtG6#%\"xG\"\"\",&\"\"$F*F )!\"\"F*F,F-" }{TEXT -1 36 " from the origin to the point where " } {XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "f := x -> sqrt(x) *(3-x)/3:\n'f(x)'=f(x);\np1 := plot(f(x),x=0..3.5,y,color=red):\np2 := plot(f(x),x=0..3,color=blue,thickness=3):\np3 := plot([[[0,0],[3,0]]$ 3],style=point,\n symbol=[circle,diamond,cross],color=black):\n plots[display]([p1,p2,p3],tickmarks=[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*(\"\"$!\"\"F'#\"\"\"\"\"#,&F*F-F'F+F-F -" }}{PARA 13 "" 1 "" {GLPLOT2D 431 194 194 {PLOTDATA 2 "6*-%'CURVESG6 $7Y7$$\"\"!F)F(7$$\"3Emm\"z%\\DO&*!#?$\"3=z4:GLLM(*!#>7$$\"3DLLe*)4D2> F0$\"3!fyQ#HGGv;F67$$\"3]mm;z>] 9QF0$\"3)*>.WW;CG>F67$$\"3v***\\(oHv@dF0$\"3%fs5()z'RYBF67$$\"3+LLLeR+ HwF0$\"3;b^M3e#=p#F67$$\"31+vVt\")z%4\"F6$\"3ciIqxa-)=$F67$$\"3'omT5&f pE9F6$\"3E%=(*[7Mvf$F67$$\"3MLL3xM?t@F6$\"3dfn(zRkSK%F67$$\"3oLLeR$fY# HF6$\"3)G1&y1Az!)[F67$$\"3'pmTNmVDn$F6$\"3wt?$\\;z#=`F67$$\"3OL$3x;GfO %F6$\"37?'Rp*)=fk&F67$$\"3U+]Pfw)Q3&F6$\"3:2jz'[W=#fF67$$\"3;M$3FR-k#e F6$\"3m-!\\!z%R1:'F67$$\"3)***\\(=(e`mlF6$\"3UQ;P$)RqHjF67$$\"3nnm;HT& yK(F6$\"3!*yy]F67$$\"3[L$3Fgg^'>Fiq$\"3;1%\\W&RgN [F67$$\"3******\\Z26S?Fiq$\"3KzMsw46qXF67$$\"3>+](=%[V8@Fiq$\"3Y@s&e_$ >'H%F67$$\"3G+vVt'zV=#Fiq$\"3lBOn@N>=SF67$$\"3#***\\78=:jAFiq$\"3u.wX+ H*\\p$F67$$\"3Umm;%3KRL#Fiq$\"3k$**RG$p)=R$F67$$\"3V++DJ^]4CFiq$\"3c)o !ygZLbIF67$$\"3=L3FWb)zZ#Fiq$\"3e463zu6RFF67$$\"3]++vBF&Gb#Fiq$\"3O#>, &*4a9Q#F67$$\"3emT50pHBEFiq$\"3Peq(z:pP.#F67$$\"35+v=s8$pp#Fiq$\"3G[_A m;.f;F67$$\"3umm\"H_A*oFFiq$\"3w&R6#yer\"G\"F67$$\"3$)*\\Pfe!HWGFiq$\" 3sg**RaqZ`()F07$$\"3#RLL$))*yo\"HFiq$\"3/N]*G(R/KZF07$$\"3_L$eR666*HFi q$\"3z0V8Z@RC^F-7$$\"3;nT5g&GZ1$Fiq$!3MUeeYi?xPF07$$\"3Y++]Z`PKJFiq$!3 z!Rj.^\"\\4yF07$$\"3\"pm\"z*>1*4KFiq$!3%*GP;&fuND\"F67$$\"3[LLL=2DzKFi q$!3EJolQEi&o\"F67$$\"33+vVQk=`LFiq$!3KgR,Cc\"e:#F67$$\"3I+DccB&RU$Fiq $!3(prK$=v#\\h#F67$$\"3++++++++NFiq$!3f<;JAy/=JF6-%'COLOURG6&%$RGBG$\" *++++\"!\")F(F(-F$6%7gnF'7$$\"3%)***\\iSmp3%F-$\"3S![RvOHUQ'F07$$\"3m* ***\\7G$R<)F-$\"3hVZ.HqM;!*F07$$\"3&***\\(=#**3E7F0$\"3\\[UK$*Rw-6F67$ $\"3$*****\\ilyM;F0$\"3W'>T%G.ir7F67$$\"3!****\\P%)z@X#F0$\"3c*>Nf$Q9` :F67$$\"3')*****\\7t&pKF0$\"3m6()RK$*[)y\"F67$$\"3z****\\(ofV!\\F0$\"3 +.-*zDv$y@F67$$\"3s******\\i9RlF0$\"3s&z[jHO9]#F67$$\"33++vVV)RQ*F0$\" 3/_1?%>3v'HF67$$\"3/++vVA)GA\"F6$\"35oZq0rUaLF67$$\"3;+]iSS\"Ga\"F6$\" 3$p/qS?oes$F67$$\"3+++]Peui=F6$\"3au_Sh(pz/%F67$$\"37+++]$)z%=#F6$\"3c ^')oqxxLVF67$$\"3A++]i3&o]#F6$\"3GyQP?cY)e%F67$$\"3%)***\\(oX*y9$F6$\" 3^;]U(z*)=-&F67$$\"3z***\\P9CAu$F6$\"37eL,$='Ga`F67$$\"3!)***\\P*zhdVF 6$\"359E_^;PUcF67$$\"31++v$>fS*\\F6$\"3mRk7XUX!*eF67$$\"3$)***\\(=$f%G cF6$\"3)\\mvSTeZ4'F67$$\"3Q+++Dy,\"G'F6$\"3AI!p)z:*fE'F67$$\"33++]7Rg'>_'F 67$$\"33+]7j=_68Fiq$\"3#zl#)[FzbW'F67$$\"33++vVy!eP\"Fiq$\"3E67%os6.N' F67$$\"34+](=WU[V\"Fiq$\"3+'3#=@&3%\\iF67$$\"3)****\\7B>&)\\\"Fiq$\"3^ !y)eKKuEhF67$$\"3)***\\P>:mk:Fiq$\"3!)y/`bAr%)fF67$$\"3'***\\iv&QAi\"F iq$\"3w+xbRRQ\\eF67$$\"31++vtLU%o\"Fiq$\"3km-aK&=9p&F67$$\"3!******\\N m'[Fiq$\"37#G7U%)))=#\\F6 7$$\"3z*****\\@80+#Fiq$\"3?^Ipp.B7ZF67$$\"31++]7,Hl?Fiq$\"3/L?Bl%3wZ%F 67$$\"3()**\\P4w)R7#Fiq$\"3iDH7%*4kbUF67$$\"3;++]x%f\")=#Fiq$\"3zKDBB5 ..SF67$$\"3!)**\\P/-a[AFiq$\"3g'4Ei\"*zgv$F67$$\"3/+](=Yb;J#Fiq$\"3/hx \"\\xf&)[$F67$$\"3')****\\i@OtBFiq$\"3i+BqRd$z@$F67$$\"3')**\\PfL'zV#F iq$\"3&)eJ+4%4_#HF67$$\"3>+++!*>=+DFiq$\"3xk(\\C4oVj#F67$$\"3-++DE&4Qc #Fiq$\"3p\"*pbl%z!GBF67$$\"3=+]P%>5pi#Fiq$\"3U%4$>E#[c,#F67$$\"39+++bJ *[o#Fiq$\"3q)R(\\]h2@isAs$F07$$\"\"$F)F(-Fg\\l6&Fi\\lF(F(Fj\\l-%*THICKN ESSG6#F__m-F$6&7$F'F]_m-%'SYMBOLG6#%'CIRCLEG-Fg\\l6&Fi\\lF)F)F)-%&STYL EG6#%&POINTG-F$6&Fg_m-Fi_m6#%(DIAMONDGF\\`mF^`m-F$6&Fg_m-Fi_m6#%&CROSS GF\\`mF^`m-%*AXESTICKSG6$F__mF__m-%+AXESLABELSG6%Q\"x6\"Q\"yFcam-%%FON TG6#%(DEFAULTG-%%VIEWG6$;F($\"#N!\"\"Fham" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " } {XPPEDIT 18 0 "y = 1/3;" "6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(x)*(3-x)" "6#*&-%%sqrtG6#%\"xG\"\"\",&\"\"$F(F' !\"\"F(" }{XPPEDIT 18 0 "`` = x^(1/2)-1/3;" "6#/%!G,&)%\"xG*&\"\"\"F) \"\"#!\"\"F)*&F)F)\"\"$F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2)" "6#)%\"xG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "dy/dx = 1/2;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&\"\"#F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-1/2)-1/2;" "6#,&)%\"xG,$*&\"\"\"F(\"\"#! \"\"F*F(*&F(F(F)F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(1/2) = 1/(2*s qrt(x))-sqrt(x)/2;" "6#/)%\"xG*&\"\"\"F'\"\"#!\"\",&*&F'F'*&F(F'-%%sqr tG6#F%F'F)F'*&-F.6#F%F'F(F)F)" }{XPPEDIT 18 0 "``=(1-x)/(2*sqrt(x))" " 6#/%!G*&,&\"\"\"F'%\"xG!\"\"F'*&\"\"#F'-%%sqrtG6#F(F'F)" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+(dy/dx)^2=1+(1-x)^2/(4*x)" "6#/,&\"\"\"F%*$*&% #dyGF%%#dxG!\"\"\"\"#F%,&F%F%*&,&F%F%%\"xGF*F+*&\"\"%F%F/F%F*F%" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=( 4*x+(1-2*x+x^2))/(4*x)" "6#/%!G*&,&*&\"\"%\"\"\"%\"xGF)F),(F)F)*&\"\"# F)F*F)!\"\"*$F*F-F)F)F)*&F(F)F*F)F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(x^2+2*x+1)/(4*x)" "6#/%!G*&,(*$% \"xG\"\"#\"\"\"*&F)F*F(F*F*F*F*F**&\"\"%F*F(F*!\"\"" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(x+1)^2/(4*x)" "6 #/%!G*&,&%\"xG\"\"\"F(F(\"\"#*&\"\"%F(F'F(!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+(dy/dx)^2)=(x+1)/(2*sqrt(x))" "6#/-%%sqrtG6# ,&\"\"\"F(*$*&%#dyGF(%#dxG!\"\"\"\"#F(*&,&%\"xGF(F(F(F(*&F.F(-F%6#F1F( F-" }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x)+1/(2*sqrt(x))" "6#,&-%%sqrtG6#%\"xG\"\"\"*& F(F(*&\"\"#F(-F%6#F'F(!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The required arc length is given by " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 0 .. 3) = I nt(``(sqrt(x)/2+1/(2*sqrt(x))),x = 0 .. 3);" "6#/-%$IntG6$-%%sqrtG6#,& \"\"\"F+*$*&%#dyGF+%#dxG!\"\"\"\"#F+/%\"xG;\"\"!\"\"$-F%6$-%!G6#,&*&-F (6#F3F+F1F0F+*&F+F+*&F1F+-F(6#F3F+F0F+/F3;F5F6" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\" \"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(3/2) +sqrt(x)" "6#, &)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F(-%%sqrtG6#F%F(" }{TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([3, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6 %7$\"\"$%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*sqrt(3 )" "6#/%!G*&\"\"#\"\"\"-%%sqrtG6#\"\"$F'" }{TEXT -1 1 " " }{TEXT 298 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3.464101615" "6#-%&FloatG6$\"+:; 5kM!\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 22 ": Since the integrand " }{XPPEDIT 18 0 "sqrt(x)/2+1/(2*sqrt(x)) " "6#,&*&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F)*&F)F)*&F*F)-F&6#F(F)F+F)" } {TEXT -1 21 " is not defined when " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG \"\"!" }{TEXT -1 38 ", we should (strictly speaking) treat " } {XPPEDIT 18 0 "Int(``(sqrt(x)/2+1/(2*sqrt(x))),x = 0 .. 3)" "6#-%$IntG 6$-%!G6#,&*&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F/*&F/F/*&F0F/-F,6#F.F/F1F/ /F.;\"\"!\"\"$" }{TEXT -1 7 " as an " }{TEXT 259 17 "improper integral " }{TEXT -1 21 " and evaluate it as: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int(``(sqrt(x)/2+1/(2*sqrt(x))),x = t .. 3), t = 0,right) = Limit(``,t = 0,right);" "6#/-%&LimitG6%-%$IntG6$-%!G6#, &*&-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"F3*&F3F3*&F4F3-F06#F2F3F5F3/F2;%\"tG \"\"$/F<\"\"!%&rightG-F%6%F+/F " 0 "" {MPLTEXT 1 0 212 "f := x -> sqrt(x)*(x-3)/3: \n'f'(x)=f(x);\nDiff(rhs(%),x)=diff(rhs(%),x);\n``=simplify(rhs(%));\n Int(sqrt(1+rhs(%)^2),x=0..3);\n``=simplify(%,symbolic);\n``=expand(rhs (%));\n``=value(rhs(%));\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*(\"\"$!\"\"F'#\"\"\"\"\"#,&F'F-F* F+F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,$*(\"\"$!\"\"%\" xG#\"\"\"\"\"#,&F+F-F)F*F-F-F+,&*(\"\"'F*F+#F*F.F/F-F-*&F)F*F+F,F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#!\"\",&%\"xG\"\"\"F+F(F+F *#F(F'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#/%!G,$*&#\"\"\"\"\"#F(-%$IntG6$*&,&%\"xGF(F(F(F( F/#!\"\"F)/F/;\"\"!\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$ *&#\"\"\"\"\"#F(-%$IntG6$,&*$%\"xGF'F(*&F(F(*$F/#F(F)!\"\"F(/F/;\"\"! \"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"\"\"$# F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+:;5kM!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 " " {TEXT 286 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "Find the arc length along the curve " }{XPPEDIT 18 0 "y = x^(2/3); " "6#/%\"yG)%\"xG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 36 " from the orig in to the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "f := x -> x^(2/3):\n'f(x)'= f(x);\np1 := plot(f(x),x=0..1.3,y,color=red):\np2 := plot(f(x),x=0..1, color=blue,thickness=3):\np3 := plot([[[0,0],[1,1]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nplots[display]([p1 ,p2,p3],tickmarks=[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6# %\"xG*$)F'#\"\"#\"\"$\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 402 258 258 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3[mmmT+jLG!#>$\"3'=5?_]e XH*F-7$$\"3mLLe*Qc\"*H&F-$\"36uoaQ&=3T\"!#=7$$\"3'pmm\"H')*=2)F-$\"3/Q v4mIun=F57$$\"3ymmTS?I'3\"F5$\"3#e!HsN;nwAF57$$\"3?L$ekk(3k8F5$\"332CE g?*)\\EF57$$\"3jm;Hi/j@;F5$\"31PfuS\"GP(HF57$$\"3)***\\iI6I))=F5$\"3U8 6F$H,9H$F57$$\"3ym;H&)=o.RF57$$\"3hLL$3Ru%F57$$\"3/nmTvI%3(HF5$\"3CU2 6o5K_WF57$$\"33++]AaB^KF5$\"31KfcAQBGZF57$$\"3!)****\\(3zF`$F5$\"3Keo' [R-u*\\F57$$\"3C++]x&)4/QF5$\"3o3SR;K5]_F57$$\"3im;HnE[]SF5$\"33i\"GHq ;WZ&F57$$\"3KLL$3=dMM%F5$\"3yiu*=v6`t&F57$$\"3MLLLB]k\"f%F5$\"3Uj02Q'F57$$\"3)GLL$)>'*e8&F5$\"3o!zan)* RKT'F57$$\"3t+]iNZF;aF5$\"3W#*GS6PdWmF57$$\"3#4+voShKo&F5$\"3%eEp_\"Q< hoF57$$\"3)QL$e*)R$='fF5$\"3of2(*Hte$3(F57$$\"3$QLe9e]w@'F5$\"3s!4zFY) z%G(F57$$\"3AnmTNLe$\\'F5$\"3mR#[7ti()\\(F57$$\"3;n;H<**>!y'F5$\"3#)f: z$HEyr(F57$$\"3n**\\P%\\+(HqF5$\"3'yR!R-&>g!zF57$$\"3anm\"H&z;*H(F5$\" 3x6Ch`'*y1\")F57$$\"3)******\\?avd(F5$\"3h8j\"[zE;J)F57$$\"3W++DT3!*\\ yF5$\"3J-$pw*eg4&)F57$$\"3()**\\i:-T8\")F5$\"3oAG8*\\#**)p)F57$$\"3S++ v[C*fS)F5$\"3lD3V<\"))o!*)F57$$\"33LLL)f!*)o')F5$\"3_9qudCk\"4*F57$$\" 3'4++v[!f\\*)F5$\"31J*)4rt&oG*F57$$\"32m;H2j%R?*F5$\"3;D7zu]*>Y*F57$$ \"35,+]-W-#[*F5$\"3'39g=1J;l*F57$$\"3_K$e*=UnV(*F5$\"3Dc\"[/nx$G)*F57$ $\"37+D\"oO<<+\"!#<$\"3Z$z_PeW6+\"F]w7$$\"3TLL3PpXG5F]w$\"3xkyOLC))=5F ]w7$$\"3!**\\i!*y]k0\"F]w$\"3yF')4%Q)GP5F]w7$$\"3ommm&>7M3\"F]w$\"3gm; D-C'[0\"F]w7$$\"3!omT!GT)46\"F]w$\"3SgyYZUos5F]w7$$\"3QLe*3vF$Q6F]w$\" 3yV/b$[8-4\"F]w7$$\"33++]+PXj6F]w$\"3:HScjw>16F]w7$$\"3IL$3U(3D#>\"F]w $\"3tq*)H;lPC6F]w7$$\"3lmmm4u+=7F]w$\"3E,5*zC709\"F]w7$$\"33+Dc[#paC\" F]w$\"3o\\p2F:fd6F]w7$$\"37+vVKPvr7F]w$\"3[2iyv8#Q<\"F]w7$$\"3/+++++++ 8F]w$\"3uKk>D%Q6>\"F]w-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6%7SF'7$$ \"3emmm;arz@F-$\"3+2cXk%)3.yF-7$$\"3[LL$e9ui2%F-$\"3=A!R.o:F57$$\"3[mmmT&phN)F-$\"3=7&)oQ4M6>F57$$\"3 CLLe*=)H\\5F5$\"3w,/]'zrYA#F57$$\"3gmm\"z/3uC\"F5$\"3GBuVoGa'\\#F57$$ \"3%)***\\7LRDX\"F5$\"3k'3d!f*RKw#F57$$\"3]mm\"zR'ok;F5$\"3s9VL@\\8EIF 57$$\"3w***\\i5`h(=F5$\"3M6uD8(psF$F57$$\"3WLLL3En$4#F5$\"39i0]&[?f_$F 57$$\"3qmm;/RE&G#F5$\"3pM3E0/(yt$F57$$\"3\")*****\\K]4]#F5$\"3iU<:k#3& pRF57$$\"3$******\\PAvr#F5$\"357SlfT[&>%F57$$\"3)******\\nHi#HF5$\"3\" Q\\g^$[j2WF57$$\"3jmm\"z*ev:JF5$\"3%[=GlS`ff%F57$$\"3?LLL347TLF5$\"3#z 3viS$)\\\"[F57$$\"3,LLLLY.KNF5$\"3QBmp4.q'*\\F57$$\"3w***\\7o7Tv$F5$\" 3[P&[:'4,/_F57$$\"3'GLLLQ*o]RF5$\"3Y9iBAl7%Q&F57$$\"3A++D\"=lj;%F5$\"3 u\\p2:)Q$ybF57$$\"31++vV&R$HhGe6'F57$$\"3cmm;/T1&*\\F5$\"3M fZMu'eaH'F57$$\"3&em;zRQb@&F5$\"35gIMh(p$zkF57$$\"3\\***\\(=>Y2aF5$\"3 jEb)y;ktj'F57$$\"39mm;zXu9cF5$\"3Ny`jKt\"f!oF57$$\"3l******\\y))GeF5$ \"3Q.Fjp[)y(pF57$$\"3'*)***\\i_QQgF5$\"3WH&3s&[4WrF57$$\"3@***\\7y%3Ti F5$\"3b6tUc64.tF57$$\"35****\\P![hY'F5$\"3gD]$eQEwZ(F57$$\"3kKLL$Qx$om F5$\"3\"QO;!\\PtKwF57$$\"3!)*****\\P+V)oF5$\"3W-QK3Ji'z(F57$$\"3?mm\"z pe*zqF5$\"3kwc$y)plVzF57$$\"3%)*****\\#\\'QH(F5$\"3*>m'\\aE'G5)F57$$\" 3GKLe9S8&\\(F5$\"3K\"RTFEZ7D)F57$$\"3R***\\i?=bq(F5$\"3#R,py-R\\S)F57$ $\"3\"HLL$3s?6zF5$\"3a%oml7aQb)F57$$\"3a***\\7`Wl7)F5$\"3X,p^a\"y$3()F 57$$\"3#pmmm'*RRL)F5$\"3yrx/My\"f&))F57$$\"3Qmm;a<.Y&)F5$\"3\\nW_.\"Qb +*F57$$\"3=LLe9tOc()F5$\"3q<;!G#=q_\"*F57$$\"3u******\\Qk\\*)F5$\"3'\\ M7nG%*oG*F57$$\"3CLL$3dg6<*F5$\"3\"R*[2_<^R%*F57$$\"3ImmmmxGp$*F5$\"3= v%=Baw\\d*F57$$\"3A++D\"oK0e*F5$\"3+&p\"=CFO=(*F57$$\"3A++v=5s#y*F5$\" 3a]7$yl " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = x^(2 /3);" "6#/%\"yG)%\"xG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 10 ", we have \+ " }{XPPEDIT 18 0 "dy/dx = 2/3;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"#F&\" \"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-1/3) = 2/(3*x^(1/3));" "6#/ )%\"xG,$*&\"\"\"F(\"\"$!\"\"F**&\"\"#F(*&F)F()F%*&F(F(F)F*F(F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+(dy/dx)^2 =1+4/(9*x^(2/3))" "6#/,& \"\"\"F%*$*&%#dyGF%%#dxG!\"\"\"\"#F%,&F%F%*&\"\"%F%*&\"\"*F%)%\"xG*&F+ F%\"\"$F*F%F*F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=(9*x^(2/3)+4)/(9*x ^(2/3))" "6#/%!G*&,&*&\"\"*\"\"\")%\"xG*&\"\"#F)\"\"$!\"\"F)F)\"\"%F)F )*&F(F))F+*&F-F)F.F/F)F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1 +(dy/dx)^2)=sqrt(9*x^(2/3)+4)/(3*x^(1/3))" "6#/-%%sqrtG6#,&\"\"\"F(*$* &%#dyGF(%#dxG!\"\"\"\"#F(*&-F%6#,&*&\"\"*F()%\"xG*&F.F(\"\"$F-F(F(\"\" %F(F(*&F8F()F6*&F(F(F8F-F(F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The required arc length is given by " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 0 .. 1) = 1 /3;" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*$*&%#dyGF+%#dxG!\"\"\"\"#F+/%\" xG;\"\"!F+*&F+F+\"\"$F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^(-1/3)* sqrt(9*x^(2/3)+4),x = 0 .. 1);" "6#-%$IntG6$*&)%\"xG,$*&\"\"\"F+\"\"$! \"\"F-F+-%%sqrtG6#,&*&\"\"*F+)F(*&\"\"#F+F,F-F+F+\"\"%F+F+/F(;\"\"!F+ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(x^(-1/3)*sqrt(9*x^(2/3 )+4),x = 0 .. 1)" "6#-%$IntG6$*&)%\"xG,$*&\"\"\"F+\"\"$!\"\"F-F+-%%sqr tG6#,&*&\"\"*F+)F(*&\"\"#F+F,F-F+F+\"\"%F+F+/F(;\"\"!F+" }{TEXT -1 47 " can be found with the aid of the substitution " }{XPPEDIT 18 0 "u = \+ 9*x^(2/3)+4;" "6#/%\"uG,&*&\"\"*\"\"\")%\"xG*&\"\"#F(\"\"$!\"\"F(F(\" \"%F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "In t(x^(-1/3)*sqrt(9*x^(2/3)+4),x = 0 .. 1)" "6#-%$IntG6$*&)%\"xG,$*&\"\" \"F+\"\"$!\"\"F-F+-%%sqrtG6#,&*&\"\"*F+)F(*&\"\"#F+F,F-F+F+\"\"%F+F+/F (;\"\"!F+" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = 9*x^( 2/3)+4,`x =`*0*` implies `*u= 4],[du=6*x^(-1/3)*dx,`x =`*1*` implies `*u= 13],[``(1/6)*du=x^(-1/3)*dx,``])" "6#-%*PIECEWISEG6%7$/%\"uG,&*& \"\"*\"\"\")%\"xG*&\"\"#F,\"\"$!\"\"F,F,\"\"%F,/**%%x~~=GF,\"\"!F,%*~i mplies~GF,F(F,F37$/%#duG*(\"\"'F,)F.,$*&F,F,F1F2F2F,%#dxGF,/**F6F,F,F, F8F,F(F,\"#87$/*&-%!G6#*&F,F,F=F2F,F;F,*&)F.,$*&F,F,F1F2F2F,FAF,FI" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/18;" "6#/%!G*&\"\"\"F&\"#=!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(u),u=4..13)" "6#-%$IntG6$-% %sqrtG6#%\"uG/F);\"\"%\"#8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = 1/27;" "6#/%!G*&\"\"\" F&\"#F!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(3/2)" "6#)%\"uG*&\"\"$ \"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([13,``],[4 ,``])" "6#-%*PIECEWISEG6$7$\"#8%!G7$\"\"%F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/27" "6#/%!G*&\"\"\"F&\"#F!\"\"" }{XPPEDIT 18 0 "`` (13^(3/2)-8)" "6#-%!G6#,&)\"#8*&\"\"$\"\"\"\"\"#!\"\"F+\"\")F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The required arc length is " }{XPPEDIT 18 0 "1/27" "6#*&\"\"\"F$\"#F!\"\"" }{XPPEDIT 18 0 "`` (13^(3/2)-8)" "6#-%!G6#,&)\"#8*&\"\"$\"\"\"\"\"#!\"\"F+\"\")F-" } {TEXT -1 1 " " }{TEXT 288 1 "~" }{TEXT -1 14 " 1.439709873. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "f := x -> x^(2/3):\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqr t(1+D(f)(x)^2),x=0..1);\n``=simplify(%,symbolic);\n``=student[changeva r](9*x^(2/3)+4=u,rhs(%),u);\n``=simplify(value(rhs(%)));\n``=evalf(eva lf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$)F '#\"\"#\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"f G6#%\"xGF*,$*(\"\"#\"\"\"\"\"$!\"\"F*#F0F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"$!\"\",&\"\"*\"\"\"*&\"\"%F,%\"xG#!\"# F(F,#F,\"\"#F,/F/;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&# \"\"\"\"\"$F(-%$IntG6$*&,&*&\"\"*F()%\"xG#\"\"#F)F(F(\"\"%F(#F(F4F2#! \"\"F)/F2;\"\"!F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\" \"\"\"$F(-%$IntG6$,$*&\"\"'!\"\"%\"uG#F(\"\"#F(/F1;\"\"%\"#8F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"#8\"\"\"\"#F!\"\"F'#F(\"\"#F (#\"\")F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+t)4(R9!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 " " 0 "" {TEXT 284 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "Find the arc length along the curve " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 36 " from the origin to t he point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 8 "Solut ion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "f := x -> exp(x):\n'f(x)'=f(x);\np1 := p lot(f(x),x=-.2..1.1,y=0..3,color=red):\np2 := plot(f(x),x=0..1,color=b lue,thickness=3):\np3 := plot([[[0,1],[1,exp(1)]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nplots[display]([p 1,p2,p3],tickmarks=[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"xG-%$expGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 287 376 376 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!35+++++++?!#=$\"3A=)z2`2t=)F*7$$!3dLL$e*pj;\"F*$\"3 eJ!3l\\$ev))F*7$$!3ILL$efzp8*!#>$\"3QmMfI9!o7*F*7$$!3?pmTNN7fjF?$\"3s@ x(=,&)QQ*F*7$$!3#[L$3x`p$y$F?$\"3FG4./#*pG'*F*7$$!3@,+v$p))p6\"F?$\"3O LrDYE#*)))*F*7$$\"3zmm\"H5\"FV7$$\"3)*****\\AaB^7F*$\"3n.TU`%)GL6FV7$$\"3p****\\(3zF `\"F*$\"33!3cz()[c;\"FV7$$\"38++]x&)4/=F*$\"3Cml]_\"3x>\"FV7$$\"3]m;Hn E[]?F*$\"34DH,:VeF7FV7$$\"3@LL$3=dMM#F*$\"3!RO%[K93k7FV7$$\"3CLLLB]k\" f#F*$\"3'RjVr&p%eH\"FV7$$\"3%***\\i&[Y.)GF*$\"3U3tp'*e8$F*$\"3?C?g'3G$o8FV7$$\"3h+]iNZF;MF*$\"3N@^DnfB29FV7$$\"3!3+voSh Ko$F*$\"3M]/gPLJX9FV7$$\"3wLLe*)R$='RF*$\"37:nU\\=9'[\"FV7$$\"3rL$e9e] w@%F*$\"3pO!Q%y-lC:FV7$$\"35nmTNLe$\\%F*$\"3qo1'\\<1tc\"FV7$$\"3/n;H<* *>!y%F*$\"3')=gFFx(Gh\"FV7$$\"36+]P%\\+(H]F*$\"3Mr@VL`i`;FV7$$\"3'zm;H &z;*H&F*$\"3/X?Lb4z)p\"FV7$$\"3K*****\\?avd&F*$\"3;eqX$QZnu\"FV7$$\"3w ***\\7%3!*\\eF*$\"3)e!>#o=t\\z\"FV7$$\"3?**\\i:-T8hF*$\"3188d76!H%=FV7 $$\"3u***\\([C*fS'F*$\"3-(GGjnDn\"[> FV7$$\"3U,+]([!f\\pF*$\"35`+B@qi.?FV7$$\"3_m;H2j%R?(F*$\"3'4#G68TCb?FV 7$$\"3W++]-W-#[(F*$\"3'*H6z+!)>8@FV7$$\"3&=Le*=UnVxF*$\"3kfeK#\\>#p@FV 7$$\"3l,]7oO<7M))F*$\"3b$ft!>-9>CFV7 $$\"3TomT!GT)4\"*F*$\"39RBHG'on[#FV7$$\"3CM$e*3vF$Q*F*$\"31w#*3pSqbDFV 7$$\"3@,++0q`M'*F*$\"3\"o\"f+&3K2i#FV7$$\"3cLL3U(3D#**F*$\"3;*[6_#*)H( p#FV7$$\"3qmmm4u+=5FV$\"3S.\"*fCWnnFFV7$$\"37+Dc[#pa/\"FV$\"3[I0U(4LZ% GFV7$$\"3;+vVKPvr5FV$\"3i?MV\"y'\\?HFV7$$\"33+++++++6FV$\"3RLk%R-mT+$F V-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fa[lF`[l-F$6%7S7$F`[l$\"\"\"Fa[ l7$$\"3emmm;arz@F?$\"3)y_2wWO?-\"FV7$$\"3[LL$e9ui2%F?$\"3aNdbY\\gT5FV7 $$\"3nmmm\"z_\"4iF?$\"3#=fT9tfS1\"FV7$$\"3[mmmT&phN)F?$\"3;Hr`&G_r3\"F V7$$\"3CLLe*=)H\\5F*$\"3L^LEiEj56FV7$$\"3gmm\"z/3uC\"F*$\"3r]yZ%yaG8\" FV7$$\"3%)***\\7LRDX\"F*$\"3')oPGkJLc6FV7$$\"3]mm\"zR'ok;F*$\"3CAm\"\\ \\E6=\"FV7$$\"3w***\\i5`h(=F*$\"36)e/'[$pj?\"FV7$$\"3WLLL3En$4#F*$\"3^ RSV5x*GB\"FV7$$\"3qmm;/RE&G#F*$\"3A#3j2pYnD\"FV7$$\"3\")*****\\K]4]#F* $\"3=m9jYu9%G\"FV7$$\"3$******\\PAvr#F*$\"3%*p#e<$=E78FV7$$\"3)****** \\nHi#HF*$\"3PYw>&\\P*R8FV7$$\"3jmm\"z*ev:JF*$\"3AKG01]dl8FV7$$\"3?LLL 347TLF*$\"3?9H.#p*p'R\"FV7$$\"3,LLLLY.KNF*$\"35Lbfo2iB9FV7$$\"3w***\\7 o7Tv$F*$\"3gYdFH**eb9FV7$$\"3'GLLLQ*o]RF*$\"3%pS?a_'[%[\"FV7$$\"3A++D \"=lj;%F*$\"3\\;8&[1^o^\"FV7$$\"31++vV&R'=e\"FV7$$\"3GLLeR\"3Gy%F*$\"3+.c]c%)H8;FV7$$\"3 cmm;/T1&*\\F*$\"3q*eh)zw!zk\"FV7$$\"3&em;zRQb@&F*$\"3G,G!GGVYo\"FV7$$ \"3\\***\\(=>Y2aF*$\"3L6+U5yG<jrpbKv\"FV7$$ \"3l******\\y))GeF*$\"3I!=v7P07z\"FV7$$\"3'*)***\\i_QQgF*$\"3nOlB#\\E \"H=FV7$$\"3@***\\7y%3TiF*$\"3!*eT_<6em=FV7$$\"35****\\P![hY'F*$\"3DV) H1Jn!4>FV7$$\"3kKLL$Qx$omF*$\"3uZcmrs1[>FV7$$\"3!)*****\\P+V)oF*$\"3OR APIze!*>FV7$$\"3?mm\"zpe*zqF*$\"3hLeRd*=*H?FV7$$\"3%)*****\\#\\'QH(F*$ \"3Mh'*f?z!Q2#FV7$$\"3GKLe9S8&\\(F*$\"3^**33Q,(f6#FV7$$\"3R***\\i?=bq( F*$\"35?.C'Qe4;#FV7$$\"3\"HLL$3s?6zF*$\"3o(p,G?ne?#FV7$$\"3a***\\7`Wl7 )F*$\"3EvYezG)QD#FV7$$\"3#pmmm'*RRL)F*$\"3^8U**za6,BFV7$$\"3Qmm;a<.Y&) F*$\"3+^n_#[T/N#FV7$$\"3=LLe9tOc()F*$\"3wf*=(>KS+CFV7$$\"3u******\\Qk \\*)F*$\"3='G]\"H'[sW#FV7$$\"3CLL$3dg6<*F*$\"3*>O4Z;k?]#FV7$$\"3Immmmx Gp$*F*$\"3e[Xr/78_DFV7$$\"3A++D\"oK0e*F*$\"335')HYrh1EFV7$$\"3A++v=5s# y*F*$\"3!*=xB3j&)fEFV7$Ff[l$\"34X!f%G=G=FFV-Fjz6&F\\[lF`[lF`[lF][l-%*T HICKNESSG6#\"\"$-F$6&7$Fe[lFcjl-%'SYMBOLG6#%'CIRCLEG-Fjz6&F\\[lFa[lFa[ lFa[l-%&STYLEG6#%&POINTG-F$6&F^[m-F`[m6#%(DIAMONDGFc[mFe[m-F$6&F^[m-F` [m6#%&CROSSGFc[mFe[m-%*AXESTICKSG6$F[[mF[[m-%+AXESLABELSG6%Q\"x6\"Q\"y Fj\\m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!\"#!\"\"$\"#6Ff]m;F`[l$F[[mFa[l " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\" xG" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "dy/dx = exp(x);" "6#/*&% #dyG\"\"\"%#dxG!\"\"-%$expG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+(dy/dx)^2 = 1+exp(x)^2;" "6#/,&\"\"\"F%*$*&%#dyGF%%#dxG!\"\"\"\"# F%,&F%F%*$-%$expG6#%\"xGF+F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+e xp(2*x);" "6#/%!G,&\"\"\"F&-%$expG6#*&\"\"#F&%\"xGF&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1+(dy/dx)^2) = sqrt(1+exp(2*x));" "6#/-% %sqrtG6#,&\"\"\"F(*$*&%#dyGF(%#dxG!\"\"\"\"#F(-F%6#,&F(F(-%$expG6#*&F. F(%\"xGF(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "The requi red arc length is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 0 .. 1)=Int(sqrt(1+exp(2*x)), x = 0 .. 1)" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*$*&%#dyGF+%#dxG!\"\"\" \"#F+/%\"xG;\"\"!F+-F%6$-F(6#,&F+F+-%$expG6#*&F1F+F3F+F+/F3;F5F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(sqrt(1+exp(2*x)),x = 0 .. 1);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F*-%$expG6#*&\"\"#F*%\"xGF*F*/F 0;\"\"!F*" }{TEXT -1 48 " can be found with the aid of the substitutio n " }{XPPEDIT 18 0 "u = sqrt(1+exp(2*x));" "6#/%\"uG-%%sqrtG6#,&\"\" \"F)-%$expG6#*&\"\"#F)%\"xGF)F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "u^ 2=1+exp(2*x)" "6#/*$%\"uG\"\"#,&\"\"\"F(-%$expG6#*&F&F(%\"xGF(F(" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (sqrt(1+exp(2*x)),x = 0 .. 1);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F*-%$exp G6#*&\"\"#F*%\"xGF*F*/F0;\"\"!F*" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u^2 = 1+exp(2*x), exp(2*x) = u^2-1],[2*u*du = 2*exp(2*x) *dx, `x =`*0*` implies `*u = sqrt(2)],[``(u/(u^2-1))*du = dx, `x =`* 1*` implies `*u = sqrt(1+exp(2))]);" "6#-%*PIECEWISEG6%7$/*$%\"uG\"\"# ,&\"\"\"F,-%$expG6#*&F*F,%\"xGF,F,/-F.6#*&F*F,F1F,,&*$F)F*F,F,!\"\"7$/ *(F*F,F)F,%#duGF,*(F*F,-F.6#*&F*F,F1F,F,%#dxGF,/**%%x~~=GF,\"\"!F,%*~i mplies~GF,F)F,-%%sqrtG6#F*7$/*&-%!G6#*&F)F,,&*$F)F*F,F,F8F8F,F " 0 "" {MPLTEXT 1 0 128 "sqrt(1+exp(2))+1/2*ln((sqr t(1+exp(2))-1)/(sqrt(1+exp(2))+1))\n -sqrt(2)-1/2*ln((sqrt(2)-1)/(sqr t(2)+1));\n``=evalf(evalf[15](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,**$,&\"\"\"F&-%$expG6#\"\"#F&#F&F*F&*&F+F&-%#lnG6#*&,&F$F&F&!\"\"F&,& F&F&F$F&F2F&F&*$F*F+F2*&#F&F*F&-F.6#*&,&F4F&F&F2F&,&F&F&F4F&F2F&F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+7r\\.?!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "f := x -> exp(x):\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqrt(1+D(f)( x)^2),x=0..1);\n``=simplify(%);\n``=student[changevar](1+exp(2*x)=u^2, rhs(%),u);\n``=map(convert,rhs(%),parfrac,u);\n``=value(rhs(%));\n``=e valf(evalf[14](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"xG-%$expGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#% \"xGF*-%$expGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\" F(*$)-%$expG6#%\"xG\"\"#F(F(#F(F//F.;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*$,&\"\"\"F*-%$expG6#,$*&\"\"#F*%\"xGF*F*F *#F*F0/F1;\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&% \"uG\"\"#,&\"\"\"!\"\"*$)F)F*F,F,F-/F);*$F*#F,F**$,&F,F,-%$expG6#F*F,F 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,(\"\"\"F)*&F)F)*&\" \"#F),&%\"uGF)F)!\"\"F)F/F)*&F)F)*&F,F),&F.F)F)F)F)F/F//F.;*$F,#F)F,*$ ,&F)F)-%$expG6#F,F)F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,.*&#\"\" \"\"\"#F(-%#lnG6#,&F(F(*$F)F'F(F(F(F.!\"\"*&#F(F)F(-F+6#,&F.F(F(F/F(F/ *&#F(F)F(-F+6#,&*$,&F(F(-%$expG6#F)F(F'F(F(F(F(F/F:F(*&F'F(-F+6#,&F:F( F(F/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+7r\\.?!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 0 " " 0 "" {TEXT 293 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "Find the arc length along the curve " }{XPPEDIT 18 0 "y = ln(cos*x);" "6#/%\"yG-%#lnG6#*&%$cosG\"\"\"%\"xGF*" }{TEXT -1 36 " fr om the origin to the point where " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"x G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 294 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "f := x -> ln(cos(x)):\n'f(x )'=f(x);\np1 := plot(f(x),x=-.4..0.9,y,color=red):\np2 := plot(f(x),x= 0..Pi/4,color=blue,thickness=3):\np3 := plot([[[0,0],[Pi/4,-ln(2)/2]]$ 3],style=point,\n symbol=[circle,diamond,cross],color=black):\n plots[display]([p1,p2,p3],tickmarks=[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%#lnG6#-%$cosGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 402 258 258 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!3A+++++++S!#=$! 3oSb]2>!HA)!#>7$$!3SLL$e*pj;PF*$!3M@*z%Qx\"=2(F-7$$!31n;/hV3qMF*$!331. lX9gXhF-7$$!3QLL3P,\"G>$F*$!3=!Q:tQYg=&F-7$$!3WLLefzp8HF*$!36()e&*o_F1 VF-7$$!3Jn;a`B\"fj#F*$!3--KJPx+:NF-7$$!3gL$3x`p$yBF*$!3#*3^=ZZRbGF-7$$ !3B+]Pp))p6@F*$!3=Z8iQnSYAF-7$$!3VL$3Fo2f$=F*$!3?pp$\\pJ[p\"F-7$$!35+] (='4+h:F*$!3v\"p%*HTULA\"F-7$$!3gmm;4cAy7F*$!3'))>JtyY;>)!#?7$$!3>LLeC p:H5F*$!3xn$=h$\\>0`Fjn7$$!3S,++vdk([(F-$!3$4e?RFle!GFjn7$$!3?/++D\"4A n%F-$!3i#y#)RCu=4\"Fjn7$$!3y*****\\A9!f>F-$!3$fwJY6\"**=>!#@7$$\"3WRm; HnE[]Fjn$!3-IX%3EbUF\"!#A7$$\"3)4LL$3=dMMF-$!3168u? '*e8\"F*$!3#=>^?)=AlkFjn7$$\"3]+]iNZF;9F*$!3FYn*e\"zG15F-7$$\"3q+](oSh Ko\"F*$!3Gik)4XDMU\"F-7$$\"3mLLe*)R$='>F*$!3KCZ(G$)oo$>F-7$$\"3gL$e9e] w@#F*$!3oCj#>y5%zCF-7$$\"3)pm;aL$e$\\#F*$!3>_(onhU<9$F-7$$\"3#pm\"H<** >!y#F*$!3#4H!=3Xf:RF-7$$\"3W**\\P%\\+(HIF*$!3e&pR\"\\N_hYF-7$$\"3Inm\" H&z;*H$F*$!3&er+8\"F*7$$\"3u++ ]([!f\\\\F*$!3M.N-O#o%y7F*7$$\"3&em\"H2j%R?&F*$!3!*>Xo#ez*>9F*7$$\"3)3 ++DSC?[&F*$!33.m*QLMXe\"F*7$$\"3IK$e*=UnVdF*$!3y<[(4Rd!\\K&)\\A9$>F*7$$\"3)QLL3PpXG'F*$!3aA/Y#Gf.7#F*7$$\"3u)*\\ i!*y]klF*$!38#ov%3o()HBF*7$$\"3nmmmc>7MoF*$!3,EY2u1ZVDF*7$$\"3unmT!GT) 4rF*$!3;L#y0F8Wx#F*7$$\"3eL$e*3vF$Q(F*$!3'pl0g+Cl,$F*7$$\"3a+++0q`MwF* $!3MalwIz.^KF*7$$\"3*GL$3U(3D#zF*$!3-h^]nZtMNF*7$$\"3Immm'4u+=)F*$!3w \\#Hw&4q-QF*7$$\"3a+]i&[#pa%)F*$!3p#e9RhpS5%F*7$$\"3&4+vVKPvr)F*$!3F]E DNOc3WF*7$$\"3A+++++++!*F*$!3AU\"eeVCWv%F*-%'COLOURG6&%$RGBG$\"*++++\" !\")$\"\"!Fc[lFb[l-F$6%7S7$Fb[lFb[l7$$\"3>tfb\"[W>r\"F-$!3KQK@P&[aY\"F _p7$$\"3WC/C@#)\\,KF-$!36Jqf?6nD^F_p7$$\"3W&)z2*=dm([F-$!3'\\x'>x3c*= \"Fjn7$$\"3%ewRz>?Hc'F-$!3)\\yti%Q9b@Fjn7$$\"3>*[rIpo6C)F-$!3w-/FkRp*R $Fjn7$$\"3ah&>)y)>rz*F-$!3D:pit\\(o![Fjn7$$\"3sp^u?<#39\"F*$!3C#\\/zYN :_'Fjn7$$\"3()f@%pjTuI\"F*$!3%>N%Qb-[r&)Fjn7$$\"3=mZ&4?FNZ\"F*$!3%=R$Q FGf*3\"F-7$$\"3USD0=mOW;F*$!3q1`&H$y5e8F-7$$\"3Y$4A'p?%[z\"F*$!3VLO99A X>;F-7$$\"3EQc2)yTU'>F*$!3Fj,p'4d;%>F-7$$\"31V([!yqLM@F*$!3CT#z&=M?&H# F-7$$\"3k>'GySb#)H#F*$!3IrP_&=rXm#F-7$$\"3c$QD\\&*3rW#F*$!3j>*H)32aCIF -7$$\"3Tcn%)>-6CEF*$!3Y5vJ[>B$[$F-7$$\"3m,$H&3N0uFF*$!3-&e8qNs!)*QF-7$ $\"39!=X\"*>t%[HF*$!3!o/S^^F7T%F-7$$\"3%RW>'zT'G5$F*$!3A5Vu!)4<$*[F-7$ $\"3ulP&[bbAF$F*$!3iIxfpJ>_aF-7$$\"3%ylZ;?cNV$F*$!38'4t@-aU,'F-7$$\"3a h!G]]c=g$F*$!3fCezuQ*>j'F-7$$\"3gj**>@(3kv$F*$!3m`)*yC8vFsF-7$$\"3RI0i lT6BRF*$!3oWyzx.J,zF-7$$\"3)f)=mqUF'4%F*$!3?S!p#H2UN')F-7$$\"3%>vN6l5q C%F*$!320!=(fEY.$*F-7$$\"3n7cgr+\")4WF*$!3sQo/`?d05F*7$$\"3yLd)G!y*zd% F*$!39b/RI>p'3\"F*7$$\"3DY2d&oODu%F*$!33deB;OXp6F*7$$\"3,9-+:lt,\\F*$! 3E?$[ZY&y_7F*7$$\"3;jPv#y+&y]F*$!3k-\"3Z!)G\"\\8F*7$$\"3%)Q(Q#R8LP_F*$ !3^Ysw+i=R9F*7$$\"3WE**)*fo\"pS&F*$!3E'fVp6d!R:F*7$$\"3#*H[:ZlegbF*$!3 _HykP@$Hj\"F*7$$\"3#*G%4Z5)eGdF*$!3)y]nFe5$RF*7$$\"366vD*fZM@'F*$!3oij*QU c!p?F*7$$\"334!\\o8tDQ'F*$!3Ytfb78K#>#F*7$$\"3KplfI6YXlF*$!3wqAUC9B:BF *7$$\"3nrwjIw.7nF*$!356yRyhGXCF*7$$\"3Cud:$zMs(oF*$!3Msp^pKtyDF*7$$\"3 +:jzGQ.HqF*$!3[k(e-g%Q0FF*7$$\"3\\fF:aE,.sF*$!3m\\QWyjTbGF*7$$\"3M(>o& *Q@'etF*$!3;k\\7R<9%*HF*7$$\"37\\s " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " } {XPPEDIT 18 0 "y = ln(cos*x);" "6#/%\"yG-%#lnG6#*&%$cosG\"\"\"%\"xGF* " }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "dy/dx = -tan*x;" "6#/*&%#d yG\"\"\"%#dxG!\"\",$*&%$tanGF&%\"xGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "The required arc length is given by " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2),x = 0 .. P i/4) = Int(sqrt(1+tan^2*x),x = 0 .. Pi/4);" "6#/-%$IntG6$-%%sqrtG6#,& \"\"\"F+*$*&%#dyGF+%#dxG!\"\"\"\"#F+/%\"xG;\"\"!*&%#PiGF+\"\"%F0-F%6$- F(6#,&F+F+*&%$tanGF1F3F+F+/F3;F5*&F7F+F8F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(sec*x,x = 0 .. Pi/4) " "6#/%!G-%$IntG6$*&%$secG\"\"\"%\"xGF*/F+;\"\"!*&%#PiGF*\"\"%!\"\"" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+ ln(abs(sec*x+tan*x))" "6#/%!G-%#lnG6#-%$absG6#,&*&%$secG\"\"\"%\"xGF.F .*&%$tanGF.F/F.F." }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi/4,`` ],[0,``])" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"\"\"\"%!\"\"%!G7$\"\"!F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ln(sqrt(2)+1)" "6#/%!G-%#lnG6#,&-%%s qrtG6#\"\"#\"\"\"F-F-" }{TEXT -1 1 " " }{TEXT 295 1 "~" }{TEXT -1 15 " 0.8813735870. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "ln(sqrt(2)+1);\nevalf(evalf[15](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"\"F'*$\"\"##F'F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qet8))!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "f := x -> ln(cos(x)) :\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqrt(1+rhs(%)^2),x= 0..Pi/4);\n``=simplify(%,symbolic);\n``=value(rhs(%));\n``=evalf(evalf [15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%#lnG 6#-%$cosGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"x GF*,$*&-%$sinGF)\"\"\"-%$cosGF)!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*&-%$sinG6#%\"xG\"\"#-%$cosGF,!\"#F(#F(F./F -;\"\"!,$*&\"\"%!\"\"%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G -%$IntG6$*&\"\"\"F)-%$cosG6#%\"xG!\"\"/F-;\"\"!,$*&\"\"%F.%#PiGF)F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(-%#lnG6#F)F(!\" \"-F+6#,&F)F(*$F)#F(F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+ qet8))!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example \+ 7 " }}{PARA 0 "" 0 "" {TEXT 290 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 39 "Find the arc length along the parabola " } {XPPEDIT 18 0 "y=x^2/2" "6#/%\"yG*&%\"xG\"\"#F'!\"\"" }{TEXT -1 36 " f rom the origin to the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 292 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "f := x -> x^2/2:\n'f(x)'=f(x);\np1 := plot(f(x),x=-. 4..1.3,y,color=red):\np2 := plot(f(x),x=0..1,color=blue,thickness=3): \np3 := plot([[[0,0],[1,.5]]$3],style=point,\n symbol=[circle,d iamond,cross],color=black):\nplots[display]([p1,p2,p3],tickmarks=[3,3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&\"\"#!\"\"F'F* \"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 402 258 258 {PLOTDATA 2 "6*-%'CUR VESG6$7S7$$!3A+++++++S!#=$\"3c,++++++!)!#>7$$!32nm;z$[%HOF*$\"3'Qx<&ox W'e'F-7$$!3hL$3_RLqI$F*$\"3;z:^Q\\BoaF-7$$!30nmTDSWWHF*$\"3[`fS4`([L%F -7$$!3Amm\"z<^%zDF*$\"3VQyi*=%yELF-7$$!3HL$3x2$>;AF*$\"3In6%)yevbCF-7$ $!3ym;a=jSz=F*$\"3kqr3bS3m(R0\")F-$\"37;DU$=t[G$FV7$$! 3#*GLLecc2WF-$\"3Vb'fi^b;$F[o7$$\"3M.++v.)y>'F-$\"38409dgo?>FV7$$\"3A6 ++vW!fu*F-$\"3#>E\"z,F8\\ZFV7$$\"3eL$ek-&y'H\"F*$\"3'\\qKU-d#3%)FV7$$ \"3qnm;Wb!*z;F*$\"3\"R&4m=8/69F-7$$\"36nmmw)eW+#F*$\"3d)*Q7%pF*3?F-7$$ \"3k+]7e:*>Q#F*$\"3<%Q*[\"*=%p$GF-7$$\"39nmm^><;FF*$\"3/c.^N]z)o$F-7$$ \"3#3+D\"33#G3$F*$\"3qTU]n?*=v%F-7$$\"3$=+vVAd>V$F*$\"3o$))p\\>l\"*)eF -7$$\"3#zm;zWWiz$F*$\"3#Q1pT&ft0sF-7$$\"3.o;HPQxITF*$\"3LmmUZikJ&)F-7$ $\"3]ML3x*3;\\%F*$\"3))z2:gvs35F*7$$\"3gM$ekF:k'[F*$\"3]_t9#))*4%=\"F* 7$$\"3y+](=E&o#>&F*$\"3M#pW9,*>[8F*7$$\"35NLe%yl]a&F*$\"3%*RywFxQP:F*7 $$\"3'4++]M4\"4fF*$\"3)pJeDmyeu\"F*7$$\"3y,+DY\\DliF*$\"3w0b2x4ni>F*7$ $\"3)3+D\"GT%)4mF*$\"3sFa!*p>]%=#F*7$$\"3E-+vj;X#*pF*$\"3h3S\\8!>ZW#F* 7$$\"3\"zmm;bTiL(F*$\"3)y-?_+A5p#F*7$$\"3c,+]P1J.xF*$\"3'Q.\"*)Q(\\q'H F*7$$\"3bM$ekyHf.)F*$\"3y9Wkw$3)GKF*7$$\"3q.+]s.d*R)F*$\"3n#**G@7Rw_$F *7$$\"3ao;zCysT()F*$\"3'R%o8o-*3#QF*7$$\"3=,]i]4Q*4*F*$\"3Rd)HUoO*RTF* 7$$\"3%)om;aA0\\%*F*$\"3K5')4D%HUY%F*7$$\"3l,]7.d7:)*F*$\"3u_sSGY$o\"[ F*7$$\"3\"QLLVzpn,\"!#<$\"3=\\KMtS5p^F*7$$\"3wL$3#)RDG0\"Fax$\"38$oibf 1Aa&F*7$$\"3)o;zMW#e)3\"Fax$\"3Rn`D\"oe]#fF*7$$\"3Q++]a%R97\"Fax$\"3?# HY0DK\")G'F*7$$\"3'omTqH(4f6Fax$\"3r!\\Y+sKvr'F*7$$\"3kLLL?*yF>\"Fax$ \"3^lx&Rw2O6(F*7$$\"3J+D\"eb!pG7Fax$\"3WaTr4CS[vF*7$$\"3E+v=tD1j7Fax$ \"3)o9N*=`jwzF*7$$\"3/+++++++8Fax$\"3%3++++++X)F*-%'COLOURG6&%$RGBG$\" *++++\"!\")$\"\"!Fd[lFc[l-F$6%7S7$Fc[lFc[l7$$\"3emmm;arz@F-$\"3'*pr#)[ 'zbP#F[o7$$\"3[LL$e9ui2%F-$\"3SOY*fX0!3$)F[o7$$\"3nmmm\"z_\"4iF-$\"3*z 48&>*yw#>FV7$$\"3[mmmT&phN)F-$\"3]&)QXq%y7\\$FV7$$\"3CLLe*=)H\\5F*$\"3 O.VJ`M80bFV7$$\"3gmm\"z/3uC\"F*$\"3iVj.!>M,y(FV7$$\"3%)***\\7LRDX\"F*$ \"3Q)49WDN\\0\"F-7$$\"3]mm\"zR'ok;F*$\"3/$Q/lUa\\K<>#F-7$$\"3qmm;/RE&G #F*$\"3$z`Veb:7h#F-7$$\"3\")*****\\K]4]#F*$\"3P-)eSEwt7$F-7$$\"3$***** *\\PAvr#F*$\"3W?GJHRY#p$F-7$$\"3)******\\nHi#HF*$\"3i-`Ub+T\"G%F-7$$\" 3jmm\"z*ev:JF*$\"33Z7q2u'R&[F-7$$\"3?LLL347TLF*$\"3uy50iWa\"e&F-7$$\"3 ,LLLLY.KNF*$\"3-mI`DVjPiF-7$$\"3w***\\7o7Tv$F*$\"3_J5w65oYqF-7$$\"3'GL LLQ*o]RF*$\"3da8z,L(R!yF-7$$\"3A++D\"=lj;%F*$\"3)R>L0c&*F-7$$\"3WLL$e9Ege%F*$\"3!pM80z\"e^5F*7$$\"3GLL eR\"3Gy%F*$\"3DHK+&oiP9\"F*7$$\"3cmm;/T1&*\\F*$\"39Y2aF*$\"3Crl8?A.i9F*7$$\"39mm; zXu9cF*$\"3?tSYMyEw:F*7$$\"3l******\\y))GeF*$\"3)4)QRymz)p\"F*7$$\"3'* )***\\i_QQgF*$\"3&*e)=*G[5B=F*7$$\"3@***\\7y%3TiF*$\"3YF*7$$\" 35****\\P![hY'F*$\"3WaK9ANb!4#F*7$$\"3kKLL$Qx$omF*$\"33vvKYGOBAF*7$$\" 3!)*****\\P+V)oF*$\"3%oDhEez'pBF*7$$\"3?mm\"zpe*zqF*$\"3z#H5#e2H1DF*7$ $\"3%)*****\\#\\'QH(F*$\"3Sis?xK-gEF*7$$\"3GKLe9S8&\\(F*$\"3u-#G[p^)3G F*7$$\"3R***\\i?=bq(F*$\"3E5DMT0voHF*7$$\"3\"HLL$3s?6zF*$\"3Ik#fY(*f$H JF*7$$\"3a***\\7`Wl7)F*$\"3lj%>4IO?I$F*7$$\"3#pmmm'*RRL)F*$\"3/+-Soxss MF*7$$\"3Qmm;a<.Y&)F*$\"3[\\7;PHt^OF*7$$\"3=LLe9tOc()F*$\"3'o;&RF%)pLQ F*7$$\"3u******\\Qk\\*)F*$\"3qS@4_i![+%F*7$$\"3CLL$3dg6<*F*$\"3O*R+3J4 b?%F*7$$\"3ImmmmxGp$*F*$\"3Q#[IFmx\"*Q%F*7$$\"3A++D\"oK0e*F*$\"3+m\\#G KI$*e%F*7$$\"3A++v=5s#y*F*$\"3#>lMl_\"3&y%F*7$$\"\"\"Fd[l$\"3++++++++] F*-F][l6&F_[lFc[lFc[lF`[l-%*THICKNESSG6#\"\"$-F$6&7$Fh[lFdjl-%'SYMBOLG 6#%'CIRCLEG-F][l6&F_[lFd[lFd[lFd[l-%&STYLEG6#%&POINTG-F$6&Fa[m-Fc[m6#% (DIAMONDGFf[mFh[m-F$6&Fa[m-Fc[m6#%&CROSSGFf[mFh[m-%*AXESTICKSG6$F^[mF^ [m-%+AXESLABELSG6%Q\"x6\"Q\"yF]]m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!\"%! \"\"$\"#8Fi]mFb]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y=x^2/2 " "6#/%\"yG*&%\"xG\"\"#F'!\"\"" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "dy/dx=x" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 36 "The required arc length is given by " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(dy/dx)^2) ,x = 0 .. 1) = Int(sqrt(1+x^2),x = 0 .. 1);" "6#/-%$IntG6$-%%sqrtG6#,& \"\"\"F+*$*&%#dyGF+%#dxG!\"\"\"\"#F+/%\"xG;\"\"!F+-F%6$-F(6#,&F+F+*$F3 F1F+/F3;F5F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "This int egral can be found with the aid of the substitution " }{XPPEDIT 18 0 " x=tan*theta" "6#/%\"xG*&%$tanG\"\"\"%&thetaGF'" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+x^2),x = 0 .. 1)" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#F*/F,;\"\"!F*" } {TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([x=tan*theta,`x =`*0 *` implies `*theta=0],[dx=sec^2*theta*d*theta,`x =`*1*` implies `*the ta=Pi/4])" "6#-%*PIECEWISEG6$7$/%\"xG*&%$tanG\"\"\"%&thetaGF+/**%%x~~= GF+\"\"!F+%*~implies~GF+F,F+F07$/%#dxG**%$secG\"\"#F,F+%\"dGF+F,F+/**F /F+F+F+F1F+F,F+*&%#PiGF+\"\"%!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(sqrt(1+tan^2*theta)*sec^2*theta,theta=0..Pi/4)" "6#/%!G-%$IntG6$*( -%%sqrtG6#,&\"\"\"F-*&%$tanG\"\"#%&thetaGF-F-F-*$%$secGF0F-F1F-/F1;\" \"!*&%#PiGF-\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(sec^3*t heta,theta = 0 .. Pi/4)" "6#/%!G-%$IntG6$*&%$secG\"\"$%&thetaG\"\"\"/F +;\"\"!*&%#PiGF,\"\"%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 18 "using the formula " }{XPPEDIT 18 0 "1+tan^2*theta=sec^2*theta" "6#/,&\"\"\"F%*&%$tanG\"\"#%&thetaGF%F%*&%$secGF(F)F%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "sec*theta " "6#*&%$secG\"\"\"%&thetaGF%" }{TEXT -1 17 " is positive for " } {XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "sec*theta=sqrt(1+tan^2*theta)" "6#/*&%$secG\"\"\"%&thetaGF&-%%sqrtG 6#,&F&F&*&%$tanG\"\"#F'F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The indefinite integral " } {XPPEDIT 18 0 "Int(sec^3*theta,theta)" "6#-%$IntG6$*&%$secG\"\"$%&thet aG\"\"\"F)" }{TEXT -1 53 " can be found using the integration by parts formula " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v - Int(v*``(du/dx),x) " "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)% \"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sec^3*theta,theta)" "6#-%$IntG6$*&%$secG\"\"$%&thetaG\"\"\"F )" }{TEXT -1 8 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=sec*theta,v=tan *theta],[du/dx=sec*theta*tan*theta,dv/dx=sec^2*theta])" "6#-%*PIECEWIS EG6$7$/%\"uG*&%$secG\"\"\"%&thetaGF+/%\"vG*&%$tanGF+F,F+7$/*&%#duGF+%# dxG!\"\"**F*F+F,F+F0F+F,F+/*&%#dvGF+F5F6*&F*\"\"#F,F+" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=Int(u*``(dv/dx),x)" "6#/%!G-%$IntG6$*&%\"uG\"\"\"-F$ 6#*&%#dvGF*%#dxG!\"\"F*%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=u*v-Int( v*``(du/dx),x)" "6#/%!G,&*&%\"uG\"\"\"%\"vGF(F(-%$IntG6$*&F)F(-F$6#*&% #duGF(%#dxG!\"\"F(%\"xGF3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sec*thet a*tan*theta-Int(sec*theta*tan^2*theta,theta);" "6#/%!G,&**%$secG\"\"\" %&thetaGF(%$tanGF(F)F(F(-%$IntG6$**F'F(F)F(F*\"\"#F)F(F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sec*theta*tan*theta-Int(sec*theta*(sec^2*th eta-1),theta);" "6#/%!G,&**%$secG\"\"\"%&thetaGF(%$tanGF(F)F(F(-%$IntG 6$*(F'F(F)F(,&*&F'\"\"#F)F(F(F(!\"\"F(F)F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sec*theta*tan*theta-Int(sec^3*theta,theta)+Int(sec*theta,th eta)" "6#/%!G,(**%$secG\"\"\"%&thetaGF(%$tanGF(F)F(F(-%$IntG6$*&F'\"\" $F)F(F)!\"\"-F,6$*&F'F(F)F(F)F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Int(sec^3*theta,theta)=sec*theta*tan*theta+Int(sec*theta,theta)" "6 #/*&\"\"#\"\"\"-%$IntG6$*&%$secG\"\"$%&thetaGF&F-F&,&**F+F&F-F&%$tanGF &F-F&F&-F(6$*&F+F&F-F&F-F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sec*thet a*tan*theta+ln(abs(sec*theta+tan*theta))+c" "6#/%!G,(**%$secG\"\"\"%&t hetaGF(%$tanGF(F)F(F(-%#lnG6#-%$absG6#,&*&F'F(F)F(F(*&F*F(F)F(F(F(%\"c GF(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sec^3*theta,theta)=1/2 " "6#/-%$IntG6$*&%$secG\"\"$%&thetaG\"\"\"F**&F+F+\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sec*theta*tan*theta+ 1/2" "6#,&**%$secG\"\"\" %&thetaGF&%$tanGF&F'F&F&*&F&F&\"\"#!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(sec*theta+tan*theta)) + c[1]" "6#,&-%#lnG6#-%$absG6#,&*&% $secG\"\"\"%&thetaGF-F-*&%$tanGF-F.F-F-F-&%\"cG6#F-F-" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[1]=c/2" "6#/ &%\"cG6#\"\"\"*&F%F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sec^3*theta,theta = 0 .. Pi/4)=1/2" "6#/-%$IntG6$*& %$secG\"\"$%&thetaG\"\"\"/F*;\"\"!*&%#PiGF+\"\"%!\"\"*&F+F+\"\"#F2" } {XPPEDIT 18 0 " ``(sec*theta*tan*theta+ln(abs(sec*theta+tan*theta)))" "6#-%!G6#,&**%$secG\"\"\"%&thetaGF)%$tanGF)F*F)F)-%#lnG6#-%$absG6#,&*& F(F)F*F)F)*&F+F)F*F)F)F)" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([ Pi/4,``],[0,``])" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"\"\"\"%!\"\"%!G7$\"\" !F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(sqrt(2)+ln( sqrt(2)+1));" "6#-%!G6#,&-%%sqrtG6#\"\"#\"\"\"-%#lnG6#,&-F(6#F*F+F+F+F +" }{TEXT -1 1 " " }{TEXT 291 1 "~" }{TEXT -1 14 " 1.147793575. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f := x -> x^2/2:\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt (sqrt(1+D(f)(x)^2),x=0..1);\n``=simplify(%,symbolic);\n``=value(rhs(%) );\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"fG6#%\"xG,$*&\"\"#!\"\"F'F*\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%DiffG6$-%\"fG6#%\"xGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6$*$,&\"\"\"F(*$)%\"xG\"\"#F(F(#F(F,/F+;\"\"!F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G-%$IntG6$*$,&\"\"\"F**$)%\"xG\"\"#F*F*#F*F./F-;\" \"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\"F'#\"\"\"F' F**&#F*F'F*-%#lnG6#,&*$F'F)F*F*F(F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+vNzZ6!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Calculate the arc length along the curve " }{XPPEDIT 18 0 "y=sqrt( 1-x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 15 " from the point" }{XPPEDIT 18 0 "``(-1/sqrt(2),1/sqrt(2))" "6#-%!G6$,$ *&\"\"\"F(-%%sqrtG6#\"\"#!\"\"F-*&F(F(-F*6#F,F-" }{TEXT -1 13 " to the point" }{XPPEDIT 18 0 "``(1/sqrt(2),1/sqrt(2))" "6#-%!G6$*&\"\"\"F'-% %sqrtG6#\"\"#!\"\"*&F'F'-F)6#F+F," }{TEXT -1 1 "." }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "f := x -> sqrt(1-x^2):\n'f'(x)=f(x );\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqrt(1+D(f)(x)^2),x=-1/sqrt(2).. 1/sqrt(2));\n``=simplify(%,symbolic);\n``=value(rhs(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,&\"\"\"F**$)F'\"\"#F*!\"\"#F*F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,$*&,&\" \"\"F.*$)F*\"\"#F.!\"\"#F2F1F*F.F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$*$,&\"\"\"F(*&,&F(F(*$)%\"xG\"\"#F(!\"\"F/F-F.F(#F(F./F-;,$*& F.F/F.F0F/,$*&F.F/F.F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$Int G6$*&\"\"\"F)*$,&F)F)*$)%\"xG\"\"#F)!\"\"#F)F/F0/F.;,$*&F/F0F/#F)F/F0, $*&F/F0F/F6F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#!\"\"%#P iG\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(1-x^2),x = -1/sqrt(2 ) .. 1/sqrt(2)) = Pi/2" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&F(F(*$%\"xG \"\"#!\"\"F0/F.;,$*&F(F(-F*6#F/F0F0*&F(F(-F*6#F/F0*&%#PiGF(F/F0" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{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 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Calculate the arc length along the curve " }{XPPEDIT 18 0 "y=x^2/2 -ln*x/4" "6#/%\"yG,&*&%\"xG\"\"#F(!\"\"\"\"\"*(%#lnGF*F'F*\"\"%F)F)" } {TEXT -1 22 " from the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\" \"\"" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x=2" "6#/%\" xG\"\"#" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "f := x -> x^2/2-ln(x)/4:\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x ),x);\nInt(sqrt(1+D(f)(x)^2),x=1..2);\n``=map(expand@simplify,%,symbol ic);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#% \"xG,&*&\"\"#!\"\"F'F*\"\"\"*&#F,\"\"%F,-%#lnGF&F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&F*\"\"\"*&F,F,*&\"\"%F,F *F,!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*$) ,&%\"xGF(*&F(F(*&\"\"%F(F,F(!\"\"F0\"\"#F(F(#F(F1/F,;F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,&*&\"\"\"F**&\"\"%F*%\"xGF*!\"\"F *F-F*/F-;F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\" \"%F(-%#lnG6#\"\"#F(F(#\"\"$F-F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(``(1/(4*x)+x),x = \+ 1 .. 2) = 1/4;" "6#/-%$IntG6$-%!G6#,&*&\"\"\"F,*&\"\"%F,%\"xGF,!\"\"F, F/F,/F/;F,\"\"#*&F,F,F.F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln*2+3/2" " 6#,&*&%#lnG\"\"\"\"\"#F&F&*&\"\"$F&F'!\"\"F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 41 "Calculate the arc length along the curve \+ " }{XPPEDIT 18 0 "y = x^4+1/(32*x^2);" "6#/%\"yG,&*$%\"xG\"\"%\"\"\"*& F)F)*&\"#KF)*$F'\"\"#F)!\"\"F)" }{TEXT -1 22 " from the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 20 " to the point wher e " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "f := x -> x^4+1/(32*x^2):\n 'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\nInt(sqrt(1+D(f)(x)^2),x=1. .2);\n``=map(expand@simplify,%,symbolic);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"%\"\"\"F,*&F,F,*&\" #KF,)F'\"\"#F,!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-% \"fG6#%\"xGF*,&*&\"\"%\"\"\")F*\"\"$F.F.*&F.F.*&\"#;F.F/F.!\"\"F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*$),&*&\"\"%F()% \"xG\"\"$F(F(*&F(F(*&\"#;F(F.F(!\"\"F4\"\"#F(F(#F(F5/F/;F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,&*&\"\"\"F**&\"#;F*)%\"xG\"\"$ F*!\"\"F**&\"\"%F*F-F*F*/F.;F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G#\"%B>\"$G\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(``(1/(16*x^3)+4*x^3),x = 1 .. 2) \+ = 1923/128;" "6#/-%$IntG6$-%!G6#,&*&\"\"\"F,*&\"#;F,*$%\"xG\"\"$F,!\" \"F,*&\"\"%F,*$F0F1F,F,/F0;F,\"\"#*&\"%B>F,\"$G\"F2" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 41 "Calculate the arc length along the curve \+ " }{XPPEDIT 18 0 "y = 2*ln(1-x^2/4);" "6#/%\"yG*&\"\"#\"\"\"-%#lnG6#,& F'F'*&%\"xGF&\"\"%!\"\"F/F'" }{TEXT -1 36 " from the origin to the poi nt where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "A ns " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "f := x -> 2*ln(1-x^2/4):\n'f'(x)=f(x);\nDiff('f'(x), x)=diff(f(x),x);\nInt(sqrt(1+D(f)(x)^2),x=0..1);\n``=simplify(%,symbol ic);\n``=-map(convert,-rhs(%),parfrac,x);\n``=value(rhs(%));\n``=evalf (evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG ,$*&\"\"#\"\"\"-%#lnG6#,&F+F+*&\"\"%!\"\"F'F*F2F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,$*&F*\"\"\",&F-F-*&\"\"%! \"\"F*\"\"#F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\" \"F(*&%\"xG\"\"#,&F(F(*&\"\"%!\"\"F*F+F/!\"#F(#F(F+/F*;\"\"!F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$-%$IntG6$*&,&\"\"%\"\"\"*$)%\"xG \"\"#F,F,F,,&F+!\"\"F-F,F2/F/;\"\"!F,F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$-%$IntG6$,(\"\"\"F**&\"\"#F*,&%\"xGF*F,F*!\"\"F/*&F,F*,&F. F*F,F/F/F*/F.;\"\"!F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&\"\" \"!\"\"*&\"\"#F&-%#lnG6#\"\"$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G$\"+xXA(>\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "-Int((x^2+4)/(x^2-4),x = 0 .. 1) = \+ -Int(``(1-2/(x+2)+2/(x-2)),x = 0 .. 1);" "6#/,$-%$IntG6$*&,&*$%\"xG\" \"#\"\"\"\"\"%F-F-,&*$F+F,F-F.!\"\"F1/F+;\"\"!F-F1,$-F&6$-%!G6#,(F-F-* &F,F-,&F+F-F,F-F1F1*&F,F-,&F+F-F,F1F1F-/F+;F4F-F1" }{XPPEDIT 18 0 "``= 2*ln*3-1" "6#/%!G,&*(\"\"#\"\"\"%#lnGF(\"\"$F(F(F(!\"\"" }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 15 " 1.197224577. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{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 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Calculate the arc length along the curve " }{XPPEDIT 18 0 "y = cosh*x;" "6#/%\"yG*&%%coshG\"\"\"%\"xGF'" }{TEXT -1 36 " from t he origin to the point where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "f := x -> cosh(x):\n'f(x)'=f(x);\n p1 := plot(f(x),x=-.5..1.2,y,color=red):\np2 := plot(f(x),x=0..1,color =blue,thickness=3):\np3 := plot([[[0,1],[1,cosh(1)]]$3],style=point,\n symbol=[circle,diamond,cross],color=black):\nplots[display]([p 1,p2,p3],tickmarks=[3,3],view=[-.5..1.2,0..1.8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%%coshGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 298 246 246 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!3++++++++]!#=$\"3q!Q1_'fi F6!#<7$$!3&omm\"z$[%HYF*$\"3pY`xEl346F-7$$!3RL$3_RLqI%F*$\"3GyOLAa>%4 \"F-7$$!3$om;a-WW%RF*$\"3WlE.pq!)y5F-7$$!3cmm\"z<^%zNF*$\"3_Zx7z#\\Z1 \"F-7$$!3jL$3x2$>;KF*$\"3Yh@K5F-7$$!3Wm;aBJ.q@F*$\"30hA;gxjB5F-7$$!3- +]P>(R0\"=F*$\"3&>(H#G4Nk,\"F-7$$!3ALL$ecc2W\"F*$\"3XnBUjoR55F-7$$!3'p m;HO^]6\"F*$\"3)=e,;9Bi+\"F-7$$!3]&****\\ZWQ[(!#>$\"3r5qo.@!QFbo$\"3775w#*Gs+5F-7$$!3ew****\\_&4a#!#?$\"30G&G#GK++5F-7$$ \"39FLek-&y'HFbo$\"3z]p3R/W+5F-7$$\"3?ommTa0*z'Fbo$\"3`EzF[AJ-5F-7$$\" 3Ammmw)eW+\"F*$\"3h)3\"3J*[]+\"F-7$$\"3u**\\7e:*>Q\"F*$\"3d'o[=rk&45F- 7$$\"3Dmmm^><;V#F*$\"38=1n=\"=(H5F-7$$\"3/nm\"zWWiz#F*$\"35P:J: .NR5F-7$$\"39n;HPQxIJF*$\"3]bbZV.T\\5F-7$$\"3]KL3x*3;\\$F*$\"3!>\\IUZy :1\"F-7$$\"3gK$ekF:k'QF*$\"3#o`:Xk\"ov5F-7$$\"3!***\\(=E&o#>%F*$\"3of2 #z9)=*3\"F-7$$\"35LLe%yl]a%F*$\"3%ouIiZy]5\"F-7$$\"3'*)****\\M4\"4\\F* $\"3!pBp\"3i$H7\"F-7$$\"3y***\\i%\\Dl_F*$\"3qjb,?m%=9\"F-7$$\"3*))*\\7 GT%)4cF*$\"3oW79%)=_h6F-7$$\"3E++vj;X#*fF*$\"3mx))Q))\\)\\=\"F-7$$\"3# fmm;bTiL'F*$\"3II\"e!)QYv?\"F-7$$\"3c****\\P1J.nF*$\"3AV?F,=@L7F-7$$\" 3cK$ekyHf.(F*$\"3+&4X$)o-zD\"F-7$$\"3s,+]s.d*R(F*$\"3Y&>\"oT**['G\"F-7 $$\"3Kk;zCysTxF*$\"3:`GNz6%\\J\"F-7$$\"3=**\\i]4Q*4)F*$\"3urJ=?sKY8F-7 $$\"3ikm;aA0\\%)F*$\"3iy'y:jx'y8F-7$$\"3m**\\7.d7:))F*$\"3_D_KfDN99F-7 $$\"3)\\LLL%zpn\"*F*$\"3!)*[*)>=.0X\"F-7$$\"3IKL3#)RDG&*F*$\"3e*y>(exL *[\"F-7$$\"3'zm\"zMW#e))*F*$\"3;lBy'*HwH:F-7$$\"32++]a%R9-\"F-$\"3![8J )*)Gjo:F-7$$\"3ym;/(H(4f5F-$\"3\"p!fvjwE:;F-7$$\"3MLLL?*yF4\"F-$\"3/K: ,br\"*e;F-7$$\"3++D\"eb!pG6F-$\"3'ym=b/Jvq\"F-7$$\"3%**\\(=tD1j6F-$\"3 Ea\">_hAhv\"F-7$$\"3%**************>\"F-$\"3suVKnbl5=F--%'COLOURG6&%$R GBG$\"*++++\"!\")$\"\"!Fb[lFa[l-F$6%7S7$Fa[l$\"\"\"Fb[l7$$\"3emmm;arz@ Fbo$\"3si1PnvB+5F-7$$\"3[LL$e9ui2%Fbo$\"33H-f:4$3+\"F-7$$\"3nmmm\"z_\" 4iFbo$\"3@XMK)HG>+\"F-7$$\"3[mmmT&phN)Fbo$\"3hvvW5L\\.5F-7$$\"3CLLe*=) H\\5F*$\"3yg)*Q'=5b+\"F-7$$\"3gmm\"z/3uC\"F*$\"3[u(QyA!z25F-7$$\"3%)** *\\7LRDX\"F*$\"3UJOt8zc55F-7$$\"3]mm\"zR'ok;F*$\"3\"z9o7$z)Q,\"F-7$$\" 3w***\\i5`h(=F*$\"3z7`@Q9l<5F-7$$\"3WLLL3En$4#F*$\"3VE0b.v*>-\"F-7$$\" 3qmm;/RE&G#F*$\"3%41[V*fAE5F-7$$\"3\")*****\\K]4]#F*$\"3/Vq\"46P9.\"F- 7$$\"3$******\\PAvr#F*$\"3FY]dPC:P5F-7$$\"3)******\\nHi#HF*$\"3=j&f<[? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "f := x -> sqrt(x-x^2)+arcsin(sqrt(x)):\n'f(x)'=f(x);\np1 := plot(f(x),x=0. .1,y,color=red):\np2 := plot(f(x),x=0..1,color=blue,thickness=3):\np3 \+ := plot([[[0,0],[1,Pi/2]]$3],style=point,\n symbol=[circle,diam ond,cross],color=black):\nplots[display]([p1,p2,p3],tickmarks=[3,3]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"xG,&*$,&F'\"\"\"*$)F' \"\"#F+!\"\"#F+F.F+-%'arcsinG6#*$F'F0F+" }}{PARA 13 "" 1 "" {GLPLOT2D 238 281 281 {PLOTDATA 2 "6*-%'CURVESG6$7W7$$\"\"!F)F(7$$\"3WmmmT&)G\\a !#?$\"3u2-:dR/v9!#=7$$\"3ILLL3x&)*3\"!#>$\"31n*GP@ET3#F07$$\"3$*****\\ ilyM;F4$\"3s'y\\W'3>]DF07$$\"3emmm;arz@F4$\"3&[T.!4&4?%HF07$$\"3.++D\" y%*z7$F4$\"3BA0H$Q*p=NF07$$\"3[LL$e9ui2%F4$\"3$4!oH?PN5SF07$$\"3nmmm\" z_\"4iF4$\"3y!elrUp:$\\F07$$\"3[mmmT&phN)F4$\"3%*4Etp*\\)*p&F07$$\"3CL Le*=)H\\5F0$\"3GVVP+:UjjF07$$\"3gmm\"z/3uC\"F0$\"3d'Qj/l)*R\"pF07$$\"3 %)***\\7LRDX\"F0$\"3t6:4A?mLuF07$$\"3]mm\"zR'ok;F0$\"3WIec?PpFzF07$$\" 3w***\\i5`h(=F0$\"3IPGFYw%QQ)F07$$\"3WLLL3En$4#F0$\"3c^nxek8@))F07$$\" 3qmm;/RE&G#F0$\"3Gk99)=-J=*F07$$\"3\")*****\\K]4]#F0$\"3WN!evdgxc*F07$ $\"3$******\\PAvr#F0$\"3^Paz/$[B$**F07$$\"3)******\\nHi#HF0$\"3[gyE%oG l-\"!#<7$$\"3jmm\"z*ev:JF0$\"3u\\i$HeR`0\"Faq7$$\"3?LLL347TLF0$\"31m/% \\f%)z3\"Faq7$$\"3,LLLLY.KNF0$\"3!e]&*4rrV6\"Faq7$$\"3w***\\7o7Tv$F0$ \"3]yE:m;rV6Faq7$$\"3'GLLLQ*o]RF0$\"3'oPr\")oY&o6Faq7$$\"3A++D\"=lj;%F 0$\"3_Xu.*4XY>\"Faq7$$\"31++vV&R+KE\"Faq7$$\"3cmm;/T1& *\\F0$\"3q4K5IV!\\G\"Faq7$$\"3&em;zRQb@&F0$\"3f#\\E-!R\\18Faq7$$\"3\\* **\\(=>Y2aF0$\"315i5sl_C8Faq7$$\"39mm;zXu9cF0$\"3X=K%R1NKM\"Faq7$$\"3l ******\\y))GeF0$\"3?uW&y'Hvh8Faq7$$\"3'*)***\\i_QQgF0$\"3eYD0/n4z8Faq7 $$\"3@***\\7y%3TiF0$\"3)4#*eC)*p^R\"Faq7$$\"35****\\P![hY'F0$\"3G@)piy ?AT\"Faq7$$\"3kKLL$Qx$omF0$\"3OI:d\"3UoU\"Faq7$$\"3!)*****\\P+V)oF0$\" 3YJix3`tT9Faq7$$\"3?mm\"zpe*zqF0$\"3i<]V:*)fa9Faq7$$\"3%)*****\\#\\'QH (F0$\"3'z/ap\"=)zY\"Faq7$$\"3GKLe9S8&\\(F0$\"3ciy,L\"H*z9Faq7$$\"3R*** \\i?=bq(F0$\"3KkV4\"e]<\\\"Faq7$$\"3\"HLL$3s?6zF0$\"3]h54\"fZE]\"Faq7$ $\"3a***\\7`Wl7)F0$\"3=!GSG[]L^\"Faq7$$\"3#pmmm'*RRL)F0$\"3tkeE9r'H_\" Faq7$$\"3Qmm;a<.Y&)F0$\"3`#e-tb%3K:Faq7$$\"3=LLe9tOc()F0$\"3w8]@G%)QS: Faq7$$\"3u******\\Qk\\*)F0$\"3uQPH$)QMZ:Faq7$$\"3CLL$3dg6<*F0$\"3a2?;^ UZa:Faq7$$\"3ImmmmxGp$*F0$\"3ulI`#oH+c\"Faq7$$\"3A++D\"oK0e*F0$\"3MDV< 4_*\\c\"Faq7$$\"3A++v=5s#y*F0$\"3k31^gqko:Faq7$$\"\"\"F)$\"3c'*[zEjzq: Faq-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6%F&-F]\\l6&F_\\lF(F(F`\\l-% *THICKNESSG6#\"\"$-F$6&7$F'Fg[l-%'SYMBOLG6#%'CIRCLEG-F]\\l6&F_\\lF)F)F )-%&STYLEG6#%&POINTG-F$6&F]]l-F_]l6#%(DIAMONDGFb]lFd]l-F$6&F]]l-F_]l6# %&CROSSGFb]lFd]l-%*AXESTICKSG6$Fj\\lFj\\l-%+AXESLABELSG6%Q\"x6\"Q\"yFi ^l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fh[lF^_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" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f := x -> sqrt(x-x^2)+arcsin(sqrt(x)):\n'f'(x)= f(x);\nDiff('f'(x),x)=diff(f(x),x);\n``=simplify(rhs(%),assume=positiv e);\nInt(sqrt(1+rhs(%)^2),x=0..1);\n``=simplify(%);\n``=value(rhs(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$,&F'\"\"\"*$)F' \"\"#F+!\"\"#F+F.F+-%'arcsinG6#*$F'F0F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&*(\"\"#!\"\",&F*\"\"\"*$)F*F-F0F.#F. F-,&F0F0*&F-F0F*F0F.F0F0*&F0F0*(F-F0F*#F0F-,&F0F0F*F.#F0F-F.F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%\"xG#!\"\"\"\"#,&\"\"\"F+F&F(#F +F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*&%\"xG!\" \",&F(F(F*F+F(F(#F(\"\"#/F*;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G-%$IntG6$*&\"\"\"F)*$%\"xG#F)\"\"#!\"\"/F+;\"\"!F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 47 ": The answer is given by an improper integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 37 " _____________________________________" }}{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 37 "_________________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Calculate th e arc length along the curve " }{XPPEDIT 18 0 "y=sqrt(exp(2*x)-1)-arct an(sqrt(exp(2*x)-1))" "6#/%\"yG,&-%%sqrtG6#,&-%$expG6#*&\"\"#\"\"\"%\" xGF/F/F/!\"\"F/-%'arctanG6#-F'6#,&-F+6#*&F.F/F0F/F/F/F1F1" }{TEXT -1 36 " from the origin to the point where " }{XPPEDIT 18 0 "x = 1;" "6#/ %\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "f := x -> sqrt(exp(2*x)-1)- arctan(sqrt(exp(2*x)-1)):\n'f(x)'=f(x);\np1 := plot(f(x),x=0..1.2,y,co lor=red):\np2 := plot(f(x),x=0..1,color=blue,thickness=3):\np3 := plot ([[[0,f(0)],[1,f(1)]]$3],style=point,\n symbol=[circle,diamond, cross],color=black):\nplots[display]([p1,p2,p3],tickmarks=[3,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"xG,&*$,&-%$expG6#,$*&\"\" #\"\"\"F'F1F1F1F1!\"\"#F1F0F1-%'arctanG6#F)F2" }}{PARA 13 "" 1 "" {GLPLOT2D 238 281 281 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3h *******\\ech#!#>$\"3FS*z=[5*>S!#?7$$\"3-+++v*G:*[F-$\"3&Qz^4)e;N5F-7$$ \"3u******\\L)4X(F-$\"3!)))e'[bo8'>F-7$$\"3)******\\MSF+\"!#=$\"3o&)*z D$f\\'3$F-7$$\"3#)****\\Fy:f7F>$\"3KQ(zhm4yP%F-7$$\"3')****\\d'*)o\\\" F>$\"3l!)))[=Nr;dF-7$$\"3w****\\(>ZIu\"F>$\"3UZ#)G,^;RsF-7$$\"3u****\\ 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\"3iy!*pC#)pd**F>7$$\"3=LLe9tOc()F>$\"3zCsb%3)3T5Ffv7$$\"3u******\\Qk \\*)F>$\"3_e\"oeKgP3\"Ffv7$$\"3CLL$3dg6<*F>$\"3E2**e`.!R8\"Ffv7$$\"3Im mmmxGp$*F>$\"3iEK5M-))z6Ffv7$$\"3A++D\"oK0e*F>$\"3@zjwtU5I7Ffv7$$\"3A+ +v=5s#y*F>$\"3-6#*fhXNz7Ffv7$$\"\"\"F)$\"3=$RvbS*eL8Ffv-Fiz6&F[[lF(F(F \\[l-%*THICKNESSG6#\"\"$-F$6&7$F'F]jl-%'SYMBOLG6#%'CIRCLEG-Fiz6&F[[lF) F)F)-%&STYLEG6#%&POINTG-F$6&Fjjl-F\\[m6#%(DIAMONDGF_[mFa[m-F$6&Fjjl-F \\[m6#%&CROSSGF_[mFa[m-%*AXESTICKSG6$FgjlFgjl-%+AXESLABELSG6%Q\"x6\"Q \"yFf\\m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#7!\"\"F[]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "f := x -> sqrt(exp(2*x)-1)-arctan( sqrt(exp(2*x)-1)):\n'f'(x)=f(x);\nDiff('f'(x),x)=diff(f(x),x);\n``=sim plify(rhs(%),assume=positive);\nInt(sqrt(1+rhs(%)^2),x=0..1);\n``=simp lify(%);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%\"xG,&*$,&-%$expG6#,$*&\"\"#\"\"\"F'F1F1F1F1!\"\"#F1F0F1-%'arctanG6 #F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&* &,&-%$expG6#,$*&\"\"#\"\"\"F*F4F4F4F4!\"\"#F5F3F.F4F4*&F4F4*$F-#F4F3F5 F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*$,&-%$expG6#,$*&\"\"#\"\"\" %\"xGF-F-F-F-!\"\"#F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$ -%$expG6#,$*&\"\"#\"\"\"%\"xGF-F-#F-F,/F.;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$-%$expG6#%\"xG/F+;\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$expG6#\"\"\"F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {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 37 "__ ___________________________________" }}{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 21 "arc length pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1252 "f := x -> 1.1-0.6*cos(x+ 0.1):\na := .5: b := 2.5: c := 1.6: d := 1.66:\np1 := plot(f(x),x=0..2 .8,color=red):\np2 := plot(f(x),x=a..d,color=blue,thickness=2):\np3 := plot([[[0,0],[3,0]],[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black): \np4 := plot([[[c,0],[c,f(c)]],[[d,0],[d,f(d)]],\n [[(c+d)/2,0] ,[(c+d)/2,-.1]]],color=black,linestyle=2):\np5 := plottools[arrow]([c- .17,.2],[c,.2],0,.04,.3,arrow,color=black):\np6 := plottools[arrow]([d +.17,.2],[d,.2],0,.04,.3,arrow,color=black):\np7 := plots[polygonplot] ([[c,f(c)],[d,f(d)],[d,f(c)]],color=blue):\np8 := plot([[[a,f(a)],[b,f (b)]]$3],style=point,\n symbol=[circle,diamond,cross],c olor=black):\nt1 := plots[textplot]([[1.44,.275,'d']],font=[SYMBOL,11] ,color=black):\nt2 := plots[textplot]([[1.49,.27,`x`],[.5,-.05,`x = a` ],\n [2.5,-.05,`x = b`],[3,-.035,`x`],[.43,.71,`(a,f(a))`],\n \+ [2.47,1.72,`(b,f(b))`],[(c+d)/2,-.15,`x`]],font=[HELVETICA,10],color =black):\nt3 := plots[textplot]([1.97,1.54,`y = f(x)`],font=[HELVETICA ,10],color=red):\nt4 := plots[textplot]([[1.57,1.275,'d']],font=[SYMBO L,11],color=blue):\nt5 := plots[textplot]([[.97,.92,`s`],[1.62,1.27,`s `]],font=[HELVETICA,10],color=blue):\nplots[display]([p1,p2,p3,p4,p5,p 6,p7,p8,t1,t2,t3,t4,t5],axes=none,view=[0..3,-.15..1.75]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 564 "i nterp([0,.5,1],[0,.29,.55],x):\np := unapply(%,x):\np1 := plot(p(x),x= -.2..1.2):\np2 := plot([[0,0],[1,0],[1,.55],[0,0]],color=black):\np3 : = plots[polygonplot]([[0,0],[1,0],[1,.55]],color=COLOR(RGB,.6,.6,1)): \nt1 := plots[textplot]([[.52,-.048,'d'],[1.05,.3,'d']],font=[SYMBOL,1 1],color=black):\nt2 := plots[textplot]([.45,.342,'d'],font=[SYMBOL,11 ],color=blue):\nt3 := plots[textplot]([[.56,-.05,`x`],[1.09,.3,`y`]],f ont=[HELVETICA,10],color=black):\nt4 := plots[textplot]([.49,.34,`s`], font=[HELVETICA,10],color=blue):\nplots[display]([p1,p2,p3,t1,t2,t3,t4 ],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 908 "interp([0,.5,1],[0,.35,.55],x):\np := unapply(% ,x):\np1 := plot(p(x),x=-.2..1.3):\np2 := plot([[0,0],[1,0],[1,.55],[0 ,0]],color=black):\np3 := plot([[[0,0],[.5,.35],[1,.55]]$3],style=poin t,\n symbol=[circle,diamond,cross],color=black):\np4 := plot([[.1,.13],[.9,.57]],color=blue):\np5 := plot([[[0,-.15],[0,0]],[ [1,-.15],[1,0]],[[.5,-.15],[.5,.35]]],\n color=bl ack,linestyle=2):\nt1 := plots[textplot]([[-.08,.05,`P`],[.96,.6,`P`], [.44,.39,`P`],[1.045,0,`Q`],\n [.47,.4,`*`],[-.01,-.19,`x`],[.49,-.19 ,`x`],[.99,-.19,`x`],[.51,-.18,`*`]],\n font=[HELVETICA,10],color=bl ack):\nt2 := plots[textplot]([[-.04,.03,`i-1`],[.98,.58,`i`],[.46,.37, `i`],\n [.04,-.21,`i-1`],[.52,-.21,`i`],[1.02,-.21,`i`]],\n \+ font=[HELVETICA,9],color=black):\nt3 := plots[textplot]( [1.25,.66,`y = f(x)`],font=[HELVETICA,10],color=red):\nplots[display]( [p1,p2,p3,p4,p5,t1,t2,t3],axes=none);" }}}{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 "" }}{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 }