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1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Hyperbolic sectors and arcs " }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Area of a hyperbolic sector " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The identity " } {XPPEDIT 18 0 "cosh^2*u-sinh^2*u = 1;" "6#/,&*&%%coshG\"\"#%\"uG\"\"\" F)*&%%sinhGF'F(F)!\"\"F)" }{TEXT -1 21 " means that, for any " }{TEXT 292 1 "t" }{TEXT -1 11 ", the point" }{XPPEDIT 18 0 "``(cosh*t,sinh*t) ;" "6#-%!G6$*&%%coshG\"\"\"%\"tGF(*&%%sinhGF(F)F(" }{TEXT -1 14 " lie s on the " }{TEXT 260 21 "rectangular hyperbola" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^2-y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Consider the area " }{XPPEDIT 18 0 "A(t)" "6#-%\"AG6#%\"t G" }{TEXT -1 8 " of the " }{TEXT 260 17 "hyperbolic sector" }{TEXT -1 52 " bounded by the section of the curve from the point " }{XPPEDIT 18 0 "S(1,0)" "6#-%\"SG6$\"\"\"\"\"!" }{TEXT -1 14 " to the point " } {XPPEDIT 18 0 "P(cosh*t,sinh*t);" "6#-%\"PG6$*&%%coshG\"\"\"%\"tGF(*&% %sinhGF(F)F(" }{TEXT -1 35 ", and the line segments OS and OP. 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This will be inves tigated in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "B(t)" "6#-%\"BG6#%\"tG" } {TEXT -1 74 " be the area of region bounded by the section of the curv e from the point " }{XPPEDIT 18 0 "S(1,0)" "6#-%\"SG6$\"\"\"\"\"!" } {TEXT -1 14 " to the point " }{XPPEDIT 18 0 "P(cosh*t,sinh*t);" "6#-% \"PG6$*&%%coshG\"\"\"%\"tGF(*&%%sinhGF(F)F(" }{TEXT -1 5 " the " } {TEXT 294 1 "x" }{TEXT -1 19 " axis and the line " }{XPPEDIT 18 0 "x = cosh*t;" "6#/%\"xG*&%%coshG\"\"\"%\"tGF'" }{TEXT -1 2 ". 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\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*t*cosh*t-Int(sqrt(x^2-1),x \+ = 1 .. cosh*t);" "6#,&**%%sinhG\"\"\"%\"tGF&%%coshGF&F'F&F&-%$IntG6$-% %sqrtG6#,&*$%\"xG\"\"#F&F&!\"\"/F1;F&*&F(F&F'F&F3" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We can fi nd " }{XPPEDIT 18 0 "Int(sqrt(x^2-1),x = 1 .. cosh*t);" "6#-%$IntG6$-% %sqrtG6#,&*$%\"xG\"\"#\"\"\"F-!\"\"/F+;F-*&%%coshGF-%\"tGF-" }{TEXT -1 17 " by substituting " }{XPPEDIT 18 0 "x = cosh*u;" "6#/%\"xG*&%%co shG\"\"\"%\"uGF'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "0 <= u;" "6#1 \"\"!%\"uG" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "dx/du = sinh*u; " "6#/*&%#dxG\"\"\"%#duG!\"\"*&%%sinhGF&%\"uGF&" }{TEXT -1 5 " and " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(x^2-1),x = \+ 1 .. cosh*t);" "6#-%$IntG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F-!\"\"/F+;F- *&%%coshGF-%\"tGF-" }{TEXT -1 14 " ... " }{XPPEDIT 18 0 "PIEC EWISE([x = cosh*u, x = 1*` implies u =`*0],[dx = sinh*u*du, x = cosh* t*` implies u =`*t]);" "6#-%*PIECEWISEG6$7$/%\"xG*&%%coshG\"\"\"%\"uGF +/F(*(F+F+%.~implies~~u~=GF+\"\"!F+7$/%#dxG*(%%sinhGF+F,F+%#duGF+/F(** F*F+%\"tGF+%-~implies~u~=GF+F9F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ Int(sqrt(cosh^2*u-1)*sinh*u,u = 0 .. t);" "6#/%!G-%$IntG6$*(-%%sqrtG6# ,&*&%%coshG\"\"#%\"uG\"\"\"F1F1!\"\"F1%%sinhGF1F0F1/F0;\"\"!%\"tG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sinh^2*u,u = 0 .. t);" "6#/%!G-%$IntG6$*&%%sinhG\"\"#%\"uG\"\"\"/ F+;\"\"!%\"tG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int((cosh*2*u-1)/2,u = 0 .. t);" "6#/%!G-%$IntG6$* &,&*(%%coshG\"\"\"\"\"#F,%\"uGF,F,F,!\"\"F,F-F//F.;\"\"!%\"tG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/4" "6 #/%!G*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*2*u-u/ 2;" "6#,&*(%%sinhG\"\"\"\"\"#F&%\"uGF&F&*&F(F&F'!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([t, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$% \"tG%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/4" "6#/%!G*&\"\" \"F&\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*2*t-t/2" "6#,&*(% %sinhG\"\"\"\"\"#F&%\"tGF&F&*&F(F&F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*t*cosh*t -t/2" "6#,&**%%sinhG\"\"\"%\"tGF&%%coshGF&F'F&F&*& F'F&\"\"#!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It f ollows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A(t) \+ = t/2" "6#/-%\"AG6#%\"tG*&F'\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 6 "______" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 6 "Note I" }{TEXT -1 1 " " }{TEXT 268 47 ".. alternative method of evaluation of integral" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The variable " }{TEXT 295 1 "u" }{TEXT -1 102 " introduced in t he substitution used to find the integral above, may be identified wit h the parameter " }{TEXT 296 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 "In fact we can obtain our result by the following alterna tive argument" }}{PARA 0 "" 0 "" {TEXT -1 30 "In general, given an int egral " }{XPPEDIT 18 0 "Int(f(x),x=a..g(t))" "6#-%$IntG6$-%\"fG6#%\"xG /F);%\"aG-%\"gG6#%\"tG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 7 " = F '(" }{TEXT 264 2 " x" }{TEXT -1 21 " ) for some function " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" } {TEXT -1 2 ", " }{TEXT 265 1 "a" }{TEXT -1 19 " is a constant and " } {XPPEDIT 18 0 "g( t )" "6#-%\"gG6#%\"tG" }{TEXT -1 28 " is differentia ble, we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dt " "6#*&%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ Int(f( x),x=a..g(t)) ] = d/dt" "6#/7#-%$IntG6$-%\"fG6#%\"xG/F+;%\"aG-%\"gG6#% \"tG*&%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ F(g(t)) -F(a)]" "6#7#,&-%\"FG6#-%\"gG6#%\"tG\"\"\"-F&6#%\"aG!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 9 " = F '( " }{XPPEDIT 18 0 "g(t) " "6#-%\"gG6#%\"tG" }{TEXT -1 8 " ) g '( " }{TEXT 263 1 "t" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= f(g(t)) *g" "6#/%!G*&-%\"fG6#-%\"gG6#%\"tG\"\"\"F*F-" }{TEXT -1 4 " '( " } {TEXT 266 1 "t" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 7 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dt" "6#*&%\"d G\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[Int(sqrt(x^2-1),x \+ = 1 .. cosh*t)] = sqrt(cosh^2*t-1)*sinh*t;" "6#/7#-%$IntG6$-%%sqrtG6#, &*$%\"xG\"\"#\"\"\"F/!\"\"/F-;F/*&%%coshGF/%\"tGF/*(-F)6#,&*&F4F.F5F/F /F/F0F/%%sinhGF/F5F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sinh^2*t;" "6#/%!G*&%%sinhG\"\"#%\"tG\"\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (cosh*2*t-1)/2;" "6#/%!G*&,&*(%%co shG\"\"\"\"\"#F)%\"tGF)F)F)!\"\"F)F*F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 6 " B '( " } {TEXT 267 2 "t " }{TEXT -1 5 ") = " }{XPPEDIT 18 0 "(cosh*2*t-1)/2;" "6#*&,&*(%%coshG\"\"\"\"\"#F'%\"tGF'F'F'!\"\"F'F(F*" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 16 "This means that " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "B(t)=1/4 " "6#/-%\"BG6#%\"tG*&\"\"\"F )\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*2*t+t/2+c;" "6#,(*(% %sinhG\"\"\"\"\"#F&%\"tGF&F&*&F(F&F'!\"\"F&%\"cGF&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "B(0) = 0;" "6#/-% \"BG6#\"\"!F'" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "c=0" "6#/%\"cG\"\"!" }{TEXT -1 14 ", which gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(t)=1/4" "6#/-%\"BG6#%\"tG*&\"\"\"F)\" \"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*2*t+t/2;" "6#,&*(%%sinh G\"\"\"\"\"#F&%\"tGF&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 "sinh*t*cosh*t+t/2;" "6#,&**%%sin hG\"\"\"%\"tGF&%%coshGF&F'F&F&*&F'F&\"\"#!\"\"F&" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 4 "diff" }{TEXT -1 45 " \"knows\" how to handle a situation like this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := 'f ': g := 'g':\nDiff(Int(f(x),x=a..g(t)),t);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%$IntG6$-%\"fG6#%\"xG/F,;%\"aG-%\"gG6#% \"tGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%diffG6$-%\"gG6#%\"tGF* \"\"\"-%\"fG6#F'F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 26 "This formula is used when " }{TEXT 0 4 "diff" }{TEXT -1 52 " is applied to such an integral given in inert form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "In t(sqrt(x^2-1),x=1..cosh(t));\ndiff(%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"F/F/!\"\"F//F-;F/-%%co shG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%sinhG6#%\"tG\"\"\"- %%sqrtG6#,&*$)-%%coshGF&\"\"#F(F(F(!\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The following commands give the same result by the longer method of evaluating the integral first." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Diff(Int(sqrt(x^2-1),x=1..cosh(t)),t);\nvalue(%);\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%$IntG6$*$,&*$)%\"xG\" \"#\"\"\"F/F/!\"\"#F/F./F-;F/-%%coshG6#%\"tGF7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&*&-%%sinhG6#%\"tGF&,&*$)-%%coshGF+F'F &F&F&!\"\"F%F&F&*&F%F&*(F0F'F-#F2F'F)F&F&F&*&#F&F'F&*&,&F)F&*(F-F5F0F& F)F&F&F&,&F0F&*$F-F%F&F2F&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%s inhG6#%\"tG\"\"\",&*$)-%%coshGF&\"\"#F(F(F(!\"\"#F(F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 260 7 "Note II" }{TEXT -1 1 " " }{TEXT 269 44 ".. comparison with area of a circular sector" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 143 "The re sult of this section concerning the area of a hyperbolic sector may be compared with the following analagous result for circular sectors." } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 386 307 307 {PLOTDATA 2 " 62-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O \"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K 5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDh F/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\! 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" }}{PARA 0 "" 0 "" {TEXT -1 74 "As mentioned above, the length of the arc along the recta ngular hyperbola " }{XPPEDIT 18 0 "x^2-y^2=1" "6#/,&*$%\"xG\"\"#\"\"\" *$%\"yGF'!\"\"F(" }{TEXT -1 16 " from the point " }{XPPEDIT 18 0 " S(1 ,0)" "6#-%\"SG6$\"\"\"\"\"!" }{TEXT -1 14 " to the point " }{XPPEDIT 18 0 "P(cosh*t,sinh*t);" "6#-%\"PG6$*&%%coshG\"\"\"%\"tGF(*&%%sinhGF(F )F(" }{TEXT -1 1 " " }{TEXT 260 15 "does not depend" }{TEXT -1 34 " in a simple way on the parameter " }{TEXT 297 1 "t" }{TEXT -1 48 ". This will be investigated in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Arc length along a parametric curve " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "For a curv e given by parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = phi(t), ``],[y = eta(t), ``]);" "6#-%*P IECEWISEG6$7$/%\"xG-%$phiG6#%\"tG%!G7$/%\"yG-%$etaG6#F,F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 " the arc length along the curve from the point given by the parameter v alue " }{XPPEDIT 18 0 "t = a;" "6#/%\"tG%\"aG" }{TEXT -1 43 " to the p oint given by the parameter value " }{XPPEDIT 18 0 "t = b;" "6#/%\"tG% \"bG" }{TEXT -1 6 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sqrt(phi*`'`(t)^2+eta*`'`(t)^2),t = a .. b);" "6#-% $IntG6$-%%sqrtG6#,&*&%$phiG\"\"\"*$-%\"'G6#%\"tG\"\"#F,F,*&%$etaGF,*$- F/6#F1F2F,F,/F1;%\"aG%\"bG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sqrt((dx/dt)^2+(dy/dt)^2),t = a .. b );" "6#/%!G-%$IntG6$-%%sqrtG6#,&*$*&%#dxG\"\"\"%#dtG!\"\"\"\"#F/*$*&%# dyGF/F0F1F2F//%\"tG;%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(ds/dt,t = a .. b);" "6#/%!G-%$ IntG6$*&%#dsG\"\"\"%#dtG!\"\"/%\"tG;%\"aG%\"bG" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dt = sqrt(``(dx/dt)^2+``(dy/dt)^2);" "6#/*&%#dsG\" \"\"%#dtG!\"\"-%%sqrtG6#,&*$-%!G6#*&%#dxGF&F'F(\"\"#F&*$-F/6#*&%#dyGF& F'F(F3F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT 260 38 "Explanation of the arc length formula " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 76 "A rough explanation of the formula for the arc length of \+ a parametric curve " }{XPPEDIT 18 0 "PIECEWISE([x = phi(t), ``],[y = e ta(t), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG-%$phiG6#%\"tG%!G7$/%\"yG-%$et aG6#F,F-" }{TEXT -1 60 " can be obtained by starting with the approxim ate equation: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "delta*s^2" "6#*&%&deltaG\"\"\"*$%\"sG\" \"#F%" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "delta*x^2 + delta*y^2" "6#,&*&%&deltaG\"\"\"*$%\"xG\"\"#F&F&*&F%F&* $%\"yGF)F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "for a small change " }{XPPEDIT 18 0 "delt a;" "6#%&deltaG" }{TEXT 299 1 "t" }{TEXT -1 18 " in the parameter " } {TEXT 300 1 "t" }{TEXT -1 28 ", and corresponding changes " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 301 1 "x" }{TEXT -1 4 " in " }{TEXT 274 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" } {TEXT 302 1 "y" }{TEXT -1 4 " in " }{TEXT 275 1 "y" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 303 1 "s" }{TEXT -1 19 " in the arc length " }{TEXT 273 1 "s" }{TEXT -1 90 " measured from som e specific point along the curve, as suggested by the following pictur e." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 327 215 215 {PLOTDATA 2 "63-%'CURVESG6$7%7$$\"\"!F)F(7$$\"\"$F)F(7$F+$\"\"\"F)-%'C OLOURG6&%$RGBGF)F)F)-F$6$7%7$$\"3#)*************z#!#7$$!3=++D1%\\c&))F>$!3 W')))>!z$4$\\#F>7$$!3e*\\PM2c*fyF>$!3z8#3_B:KA#F>7$$!3')**\\P%y%>SnF>$ !36by!)o,b;>F>7$$!3J+]i!*4,8cF>$!3^?#prOvWg\"F>7$$!3%)*\\(o/X=\"\\%F>$ !3?\\\"=N!)>0H\"F>7$$!3y+vV[x5^MF>$!3ME,&[YZW'**!#>7$$!3#**\\P4^oTP#F> $!3i-.i\\>2*)oF`o7$$!3))*\\P4T'Rg7F>$!3+OtY$[\")fn$F`o7$$!3x0](=#>'>] \"F`o$!3X`#4v6ZFS%!#?7$$\"3c4+]P>\"y\"**F`o$\"3o=G]+*RB#HF`o7$$\"3'4+v o\\Nw*>F>$\"3Dn*y74QH\"fF`o7$$\"3@,+D1#*)*HJF>$\"3ruJ?.h#>J*F`o7$$\"3Q ***\\(oC*pE%F>$\"3k/EpBu#fF\"F>7$$\"3+++v$z0FO&F>$\"3'Q$H%R=09h\"F>7$$ \"3i+D1k%=xN'F>$\"3&*=[Sh[#)=>F>7$$\"3C++voZ)3a(F>$\"3ML3Z*y7yG#F>7$$ \"3M,++D==V&)F>$\"3U=-V4YJ.EF>7$$\"3x+Dcw:44(*F>$\"3o5z%z!))otHF>7$$\" 3i***\\i#>6u5F:$\"3byD=urb/LF>7$$\"3;]i:?7$$ \"3=](o/Ej^H\"F:$\"3R$fe+G0G-%F>7$$\"3F+DcEPm29F:$\"3%*3]0?#\\LR%F>7$$ \"36]7GtU(4^\"F:$\"3w*)3xopfOZF>7$$\"3%)*\\(oa'3Ci\"F:$\"3.>_N*pE+6&F> 7$$\"3u\\i!*ew:Q7$$\"3.]PM2v\"*Q=F:$\"3-65+o./XeF>7 $$\"3++D1/4uZ>F:$\"3\"HYZ\"p#*>>iF>7$$\"3)****\\77m,1#F:$\"3'4;S'[&e!4 mF>7$$\"3i*\\7GE_,<#F:$\"3s.p\"oT@P*pF>7$$\"3)*\\i:5&plF#F:$\"3?9I7$$\"3')*\\(o>xs%R#F:$\"3QOGV\"4k\"*y(F>7$$\"3j***\\i7)*3]#F:$\"3 !e@E*G+!*p\")F>7$$\"3?+](opdUh#F:$\"3/iO_=]tz&)F>7$$\"3w\\iS;$ypr#F:$ \"3#ec#p%)e1a*)F>7$$\"33+]i&3z#HGF:$\"3[#QQT+GlO*F>7$$\"3U\\ild`%\\$HF :$\"3;IFd;Opd(*F>7$$\"3S]7GeqRXIF:$\"3nNnr8d(p,\"F:7$$\"3W**\\P%y$Q`JF :$\"3%*RZOnpdd5F:7$$\"3a\\i!*yeVmKF:$\"3)e_hD!fT+6F:7$$\"3+++]#[=`P$F: $\"3.<\")Q'3(*>9\"F:7$$\"3#)*\\P4nmm[$F:$\"3/x5b;o%[=\"F:7$$\"3I]i:SG4 (f$F:$\"3tf4be#owA\"F:7$$\"3-++D@Ic)p$F:$\"3_wMj)R.tE\"F:7$$\"3$**\\(o *Hf[\"QF:$\"3q')eP9u188F:7$$\"3A++]xg()=RF:$\"3]a$pc%[Ia8F:7$$\"3!*\\i ld'z(HSF:$\"3/h*QUN!f)R\"F:7$$\"3X]P%[`Gf8%F:$\"36JD(\\V%GT9F:7$$\"3++ +++++]UF:$\"3+v$f,++v[\"F:-F16&F3$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\" #-%%TEXTG6%7$$\"$W\"!\"#$!#:Fc]lQ\"d6\"-%%FONTG6$%'SYMBOLG\"#5-F^]l6%7 $$\"$D$Fc]l$\"\"&FEFf]lFh]l-F^]l6%7$$\"$C\"Fc]l$\"#jFc]lFf]lFh]l-F^]l6 %7$$\"$h\"Fc]lFd]lQ\"xFg]l-Fi]l6$%*HELVETICAGF\\^l-F^]l6%7$$\"$U$Fc]lF b^lQ\"yFg]lFa_l-F^]l6%7$$\"$T\"Fc]lFi^lQ\"sFg]lFa_l-F^]l6%7$$\"#UFE$\" #6FEQ*x~=~~~(t)Fg]lFd\\l-F^]l6%7$Fc`l$\"#%*Fc]lQ*y~=~~~(t)Fg]lFd\\l-F^ ]l6&7$$\"#OFE$\"$0\"Fc]lQ\"|frFg]lFd\\l-Fi]l6$Fc_l\"#=-F^]l6&7$$\"$D%F c]l$\"$6\"Fc]lQ\"fFg]lFd\\l-Fi]l6$F[^lFf`l-F^]l6&7$F\\bl$\"#&*Fc]lQ\"h Fg]lFd\\lFabl-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!Fg]lF`cl-Fi]l6#%( DEFAULTG-%%VIEWG6$;FDF\\blFccl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``(delta*s/(delta*t))^2;" "6#*$-%!G6#*(%&deltaG\"\"\"% \"sGF)*&F(F)%\"tGF)!\"\"\"\"#" }{TEXT -1 1 " " }{TEXT 271 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(delta*x/(delta*t))^2+``(delta*y/(del ta*t))^2;" "6#,&*$-%!G6#*(%&deltaG\"\"\"%\"xGF**&F)F*%\"tGF*!\"\"\"\"# F**$-F&6#*(F)F*%\"yGF**&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 "delta*s/(delta*t);" "6#*(%&deltaG\"\"\"%\"sGF%*&F$F%%\" tGF%!\"\"" }{TEXT -1 1 " " }{TEXT 272 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(``(delta*x/(delta*t))^2+``(delta*y/(delta*t))^2);" "6#-%%sq rtG6#,&*$-%!G6#*(%&deltaG\"\"\"%\"xGF-*&F,F-%\"tGF-!\"\"\"\"#F-*$-F)6# *(F,F-%\"yGF-*&F,F-F0F-F1F2F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "In the limit as " } {XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 304 1 "t" }{TEXT -1 39 " te nds to zero we obtain the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ds/dt = sqrt(``(dx/dt)^2+``(dy/dt)^2);" "6#/*&%#dsG \"\"\"%#dtG!\"\"-%%sqrtG6#,&*$-%!G6#*&%#dxGF&F'F(\"\"#F&*$-F/6#*&%#dyG F&F'F(F3F&" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "Example 1 .. spiral" }}{PARA 0 "" 0 "" {TEXT 279 8 "Que stion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 41 "Find the length of the parametric curve: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([x = exp(t)*cos*t, ``],[y = exp(t)*sin*t, ``] );" "6#-%*PIECEWISEG6$7$/%\"xG*(-%$expG6#%\"tG\"\"\"%$cosGF.F-F.%!G7$/ %\"yG*(-F+6#F-F.%$sinGF.F-F.F0" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "The curve is a spiral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 281 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([dx/dt = exp(t)*(cos*t-sin*t) , ``],[dy/dt = exp(t)*(sin*t+cos*t), ``]);" "6#-%*PIECEWISEG6$7$/*&%#d xG\"\"\"%#dtG!\"\"*&-%$expG6#%\"tGF*,&*&%$cosGF*F1F*F**&%$sinGF*F1F*F, F*%!G7$/*&%#dyGF*F+F,*&-F/6#F1F*,&*&F6F*F1F*F**&F4F*F1F*F*F*F7" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``(ds/dt)^2 = ``(dx/dt)^2+``(dy/dt)^2 ;" "6#/*$-%!G6#*&%#dsG\"\"\"%#dtG!\"\"\"\"#,&*$-F&6#*&%#dxGF*F+F,F-F** $-F&6#*&%#dyGF*F+F,F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(2*t)*(c os*t-sin*t)^2+exp(2*t)*(sin*t+cos*t)^2;" "6#/%!G,&*&-%$expG6#*&\"\"#\" \"\"%\"tGF,F,*$,&*&%$cosGF,F-F,F,*&%$sinGF,F-F,!\"\"F+F,F,*&-F(6#*&F+F ,F-F,F,*$,&*&F3F,F-F,F,*&F1F,F-F,F,F+F,F," }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 2*exp(2*t)" "6#/%!G*&\"\"#\"\"\"-%$expG6#*&F&F'%\"tGF'F'" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 32 "and the required arc len gth is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt( 2)*exp(t),t=0..2)" "6#-%$IntG6$*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#%\"tGF+ /F/;\"\"!F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= sqrt(2)*exp(t)" "6#/%!G*&-%%sqrtG6#\"\"#\"\"\"-%$ex pG6#%\"tGF*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ 2,`` ],[0 , ` `])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sqrt(2)*(exp(2)-1)" "6 #/%!G*&-%%sqrtG6#\"\"#\"\"\",&-%$expG6#F)F*F*!\"\"F*" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 280 8 "________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "f := t -> exp(t)*cos(t );\ng := t -> exp(t)*sin(t);\nInt(sqrt(Diff(f(t),t)^2+Diff(g(t),t)^2), t=0..2);\nvalue(%);\nevalf(evalf(%,14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%$expG6#9$\"\"\"-%$c osGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$% )operatorG%&arrowGF(*&-%$expG6#9$\"\"\"-%$sinGF/F1F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&*$)-%%DiffG6$*&-%$expG6#%\"tG\"\"\" -%$cosGF0F2F1\"\"#F2F2*$)-F+6$*&F.F2-%$sinGF0F2F1F5F2F2#F2F5/F1;\"\"!F 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"##\"\"\"F%-%$expG6#F%F'F' *$F%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'y*[N!*!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The follo wing picture shows the curve " }{XPPEDIT 18 0 "PIECEWISE([x = exp(t)*c os*t, ``],[y = exp(t)*sin*t, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*(-%$exp G6#%\"tG\"\"\"%$cosGF.F-F.%!G7$/%\"yG*(-F+6#F-F.%$sinGF.F-F.F0" } {TEXT -1 23 ", with the section for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!% \"tG" }{XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 10 " drawn in " } {TEXT 305 7 "magenta" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "p1 := plot([exp(t)*cos(t) ,exp(t)*sin(t),t=0..2],color=magenta,thickness=2): \np2 := plot([exp(t )*cos(t),exp(t)*sin(t),t=-2..0],color=navy): \np3 := plot([exp(t)*cos( t),exp(t)*sin(t),t=2..2.3],color=navy):\nplots[display]([p1,p2,p3],vie w=[-7..2,-1..8]); " }}{PARA 13 "" 1 "" {GLPLOT2D 331 357 357 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3!y*yl%3mN/\"!#<$ \"37%)eV>$QAb%!#>7$$\"3K6IN$QP83\"F/$\"31l(GW5Q_$))F27$$\"3YGF=7$$\"3\\#eBi,Uk? 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" }}{PARA 0 "" 0 "" {TEXT -1 15 "The curve is a " }{TEXT 260 7 "cycl oid" }{TEXT -1 77 ", that is, the curve traced out by the point on the rim of a wheel of radius " }{TEXT 284 1 "a" }{TEXT -1 40 " units as i t rolls along a flat surface." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 278 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([dx/dt = a-a*cos*t, `` ],[dy/dt = a*sin*t, ``]);" "6#-%*PIECEWISEG6$7$/*&%#dxG\"\"\"%#dtG!\" \",&%\"aGF**(F.F*%$cosGF*%\"tGF*F,%!G7$/*&%#dyGF*F+F,*(F.F*%$sinGF*F1F *F2" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(ds/dt)^2 = ``(dx/dt)^2+``(dy/ dt)^2;" "6#/*$-%!G6#*&%#dsG\"\"\"%#dtG!\"\"\"\"#,&*$-F&6#*&%#dxGF*F+F, F-F**$-F&6#*&%#dyGF*F+F,F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a^2*(1 -cos*t)^2+a^2*sin^2*t;" "6#/%!G,&*&%\"aG\"\"#,&\"\"\"F**&%$cosGF*%\"tG F*!\"\"F(F**(F'F(%$sinGF(F-F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*a^2*(1-cos*t);" "6#/%!G*(\"\"#\" \"\"*$%\"aGF&F',&F'F'*&%$cosGF'%\"tGF'!\"\"F'" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 32 "and the required arc length is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(a*sqrt(2-2*cos*t) ,t = 0 .. 2*Pi);" "6#-%$IntG6$*&%\"aG\"\"\"-%%sqrtG6#,&\"\"#F(*(F-F(%$ cosGF(%\"tGF(!\"\"F(/F0;\"\"!*&F-F(%#PiGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sqrt(2)*a*sqrt(1-c os*t),t = 0 .. 2*Pi);" "6#/%!G-%$IntG6$*(-%%sqrtG6#\"\"#\"\"\"%\"aGF-- F*6#,&F-F-*&%$cosGF-%\"tGF-!\"\"F-/F4;\"\"!*&F,F-%#PiGF-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sqrt(2 )*a*sqrt(2*sin^2*``(t/2)),t = 0 .. 2*Pi);" "6#/%!G-%$IntG6$*(-%%sqrtG6 #\"\"#\"\"\"%\"aGF--F*6#*(F,F-*$%$sinGF,F--F$6#*&%\"tGF-F,!\"\"F-F-/F7 ;\"\"!*&F,F-%#PiGF-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = Int(2*a*sqrt(sin^2*``(t/2)),t = 0 .. 2*Pi);" "6 #/%!G-%$IntG6$*(\"\"#\"\"\"%\"aGF*-%%sqrtG6#*&%$sinGF)-F$6#*&%\"tGF*F) !\"\"F*F*/F4;\"\"!*&F)F*%#PiGF*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(2*a*sin(t/2),t = 0 .. 2*Pi);" "6#/%!G-%$IntG6$*(\"\"#\"\"\"%\"aGF*-%$sinG6#*&%\"tGF*F)!\"\"F*/F0;\" \"!*&F)F*%#PiGF*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "since \+ " }{XPPEDIT 18 0 "sin(t/2)>=0" "6#1\"\"!-%$sinG6#*&%\"tG\"\"\"\"\"#!\" \"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" } {XPPEDIT 18 0 "``<=2*Pi" "6#1%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 3 ", \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -4*a*cos(t/2 );" "6#/%!G,$*(\"\"%\"\"\"%\"aGF(-%$cosG6#*&%\"tGF(\"\"#!\"\"F(F0" } {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([2*Pi, ``],[0, ``]);" "6#-%* PIECEWISEG6$7$*&\"\"#\"\"\"%#PiGF)%!G7$\"\"!F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 8*a;" "6#/%!G*&\"\")\"\"\"%\"aGF'" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 277 5 "_____" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "f := t -> a*t-a*sin(t);\ng \+ := t -> a-a*cos(t);\nInt(sqrt(Diff(f(t),t)^2+Diff(g(t),t)^2),t=0..2*Pi );\nassume(a_>=0);\nsubs(a_=a,value(subs(a=a_,%)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%\"aG\" \"\"9$F/F/*&F.F/-%$sinG6#F0F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&%\"aG\"\"\"*&F-F.-%$c osG6#9$F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%% sqrtG6#,&*$)-%%DiffG6$,&*&%\"aG\"\"\"%\"tGF3F3*&F2F3-%$sinG6#F4F3!\"\" F4\"\"#F3F3*$)-F.6$,&F2F3*&F2F3-%$cosGF8F3F9F4F:F3F3F3/F4;\"\"!,$%#PiG F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"aG\"\")" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The following picture s hows the curve " }{XPPEDIT 18 0 "PIECEWISE([x = t-sin*t, ``],[y = 1-co s*t, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,&%\"tG\"\"\"*&%$sinGF+F*F+!\"\" %!G7$/%\"yG,&F+F+*&%$cosGF+F*F+F.F/" }{TEXT -1 23 ", with the section \+ for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "`` <= 2*Pi ;" "6#1%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 10 " drawn in " }{TEXT 305 7 "magenta" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "p1 := plot([t-sin(t),1-cos(t),t=0. .2*Pi],color=magenta,thickness=2): \np2 := plot([t-sin(t),1-cos(t),t=- Pi..0],color=navy): \np3 := plot([t-sin(t),1-cos(t),t=2*Pi..3*Pi],colo r=navy):\nplots[display]([p1,p2,p3]); " }}{PARA 13 "" 1 "" {GLPLOT2D 717 150 150 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3e;cW/cSxU!# @$\"3%R#3*QgiPO*!#?7$$\"3!e+5`Gi4z#F0$\"3Sn)o<,z>E$!#>7$$\"3=%\\Q;ZV:# )*F0$\"3!)\\'*Q6_89vF67$$\"3s^>#3I_\"zBF6$\"3#)RJ?\"3HpM\"!#=7$$\"3]e/ 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do$\"33@T_xm:H6Fdo7$$\"3:mWQ=pK4dFdo$\"3!zOHMQdI+\"Fdo7$$\"3$o77H1%>Le Fdo$\"3spxOYnF7()FA7$$\"3(z@TH0mv$fFdo$\"3ungHa'e\\W(FA7$$\"35XGdRicHg Fdo$\"3C:@W9;?khFA7$$\"3Z/qowO?-hFdo$\"3k(Q-RU*p'*\\FA7$$\"31UI8X3JhhF do$\"3!3v60g\\1*QFA7$$\"3H)*HI_+11iFdo$\"3#fRKNh*p+HFA7$$\"3]'=)=I4JOi Fdo$\"3]S\"yig@)*4#FA7$$\"35r[&G$G'*fiFdo$\"32g)*y$[icK\"FA7$$\"3/L^'= )R*GF'Fdo$\"3-s;%yY!**\\xF67$$\"3+\"*=`-Y9!G'Fdo$\"3OR8x286`MF67$$\"3] w>MG;w#G'Fdo$\"3Hz?W.kW/$*F07$$\"3C'ezrI&=$G'FdoF(-%'COLOURG6&%$RGBG$ \"*++++\"!\")F(F[[l-%*THICKNESSG6#\"\"#-F$6$7S7$$!332-TaEfTJFdo$Fa[lF) 7$$!3qRay+1p/IFdo$\"3+=D%HJcw*>Fdo7$$!3=-YxDD#e)GFdo$\"3!\\Jxp_6=*>Fdo 7$$!3)=Fdo7$$!3jZ:K#*Qc>EFdo$\"3Y$Q*p&*)Rd' >Fdo7$$!3I)>F()3P#)[#Fdo$\"3I(=ULvch%>Fdo7$$!3CrnraoxnBFdo$\"3,s+**R3> C>Fdo7$$!3k6-xH&3YC#Fdo$\"3W[%onKww*=Fdo7$$!3P2)em'[:>@Fdo$\"3%)*zKiHO j'=Fdo7$$!3)*z,so.J'*>Fdo$\"3HF2h(Ho7$=Fdo7$$!3a'3%))Q3^s=Fdo$\"3eLPl# zr8z\"Fdo7$$!3!fgW(eO#ew\"Fdo$\"3G.VV^X;`t4WO6Fdo$\"3Q6He\"f+]W\"Fdo7$$!3kMPdp,$y.\" 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$\"3/gm3i,>'G'Fdo$\"3_Ix*>Y5gU$F67$$\"3IQHTQK7*G'Fdo$\"3Ggvh0ZK%Q&F67$ $\"3')QXb*4RJH'Fdo$\"33CS*z/;4e(F67$$\"3GFd?h&e))H'Fdo$\"396$QxtOK-\"F A7$$\"3&4%**\\bFA7$$\"3>:\\W(Q,#QlFdo$\"3MLM(3u0^='FA7$$\"37FOu#\\v\"y lFdo$\"3ak/i)*[(Gw'FA7$$\"3n\"f;WA'=EmFdo$\"3kz?&*)p%*3T(FA7$$\"39I1)p &G-wmFdo$\"3%>$pH1;0R!)FA7$$\"3&eEe[)HPKnFdo$\"3,Zrv**e7.()FA7$$\"3kQM ]cd2)y'Fdo$\"3[83u))4?=$*FA7$$\"3!)y#e)z@V_oFdo$\"3E5'yBW$\\%)**FA7$$ \"3mU0B&p')R#pFdo$\"3Wt<<_;mn5Fdo7$$\"3+![/nSr,*pFdo$\"3'GXSSjew7\"Fdo 7$$\"3kchu-1qlqFdo$\"3a#okrQH>>\"Fdo7$$\"3CtX7&3)4[rFdo$\"3AT4-#zpuD\" Fdo7$$\"3l'*zo$**QHB(Fdo$\"3!3M?57j/K\"Fdo7$$\"3V%*f$>>K*=tFdo$\"3o:OH NV4!Q\"Fdo7$$\"3!)Q?>K9!)=uFdo$\"3l8i+M#*[W9Fdo7$$\"3z\"oC()GhB^(Fdo$ \"3M(GNKOl/]\"Fdo7$$\"3E^FBc#4hh(Fdo$\"3qSm6Ys*zb\"Fdo7$$\"3y6e`X;U8xF do$\"3>#)o:'**>zg\"Fdo7$$\"3!Q+AOFOK#yFdo$\"3+8Dm\")[')f;Fdo7$$\"3ec'o ,?k'HzFdo$\"3jnf+1]-1HP4JWe\")Fdo$\"3OCJ(Q13Bz\"Fdo7$$\"3W>.^ay/\"G)Fdo$\"3WY!GP4R<$=Fdo7$ $\"3*)eYdA[`,%)Fdo$\"3kD(y$y17m=Fdo7$$\"3!eW'>.?%p_)Fdo$\"3[el[B%yu*=F do7$$\"3q$pyukWKl)Fdo$\"3&[t!=&=VY#>Fdo7$$\"3gY,.vbxq()Fdo$\"3G(*e(36 \\g%>Fdo7$$\"3FqO*\\GNp!*)Fdo$\"3%\\?#*\\E!Hm>Fdo7$$\"3>T)4^z$yH!*Fdo$ \"3y`Q--OV!)>Fdo7$$\"3)=tWL&**fh\"*Fdo$\"3/Cg@M'H8*>Fdo7$$\"3=q#R)\\0J )G*Fdo$\"3G$4YnFdo7$$\"3A@1BjzxC%*FdoFh[lFejl-%+AXESLABELSG6%Q!6 \"F_jm-%%FONTG6#%(DEFAULTG-%%VIEWG6$FdjmFdjm" 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" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Example 3 .. rectangula r hyperbola" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 282 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "Find the length of the rectangular hyperbola: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = cosh*t, ``], [y = sinh*t, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*&%%coshG\"\"\"%\"tGF+%! G7$/%\"yG*&%%sinhGF+F,F+F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "`` \+ <= 2;" "6#1%!G\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "N ote" }{TEXT -1 30 ": This curve has the equation " }{XPPEDIT 18 0 "x^2 -y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution " }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " PIECEWISE([dx/dt = sinh*t, ``],[dy/dt = cosh*t, ``]);" "6#-%*PIECEWISE G6$7$/*&%#dxG\"\"\"%#dtG!\"\"*&%%sinhGF*%\"tGF*%!G7$/*&%#dyGF*F+F,*&%% coshGF*F/F*F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(ds/dt)^2 = ``(dx/d t)^2+``(dy/dt)^2;" "6#/*$-%!G6#*&%#dsG\"\"\"%#dtG!\"\"\"\"#,&*$-F&6#*& %#dxGF*F+F,F-F**$-F&6#*&%#dyGF*F+F,F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = cosh^2*t+sinh^2*t;" "6#/%!G,&*&%%coshG\"\"#%\"tG\"\"\"F**&%%sinhG F(F)F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = cosh*2*t;" "6#/%!G*(%%cosh G\"\"\"\"\"#F'%\"tGF'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 32 " and the required arc length is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sqrt(cosh*2*t),t = 0 .. 2);" "6#-%$IntG6$-%%sqrtG6# *(%%coshG\"\"\"\"\"#F+%\"tGF+/F-;\"\"!F," }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 22 "Consider the integral " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(cosh*2*t),t = 0 .. a);" "6#-%$IntG6$ -%%sqrtG6#*(%%coshG\"\"\"\"\"#F+%\"tGF+/F-;\"\"!%\"aG" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a>=0" "6#1\"\" !%\"aG" }{TEXT -1 32 ", which gives the arclength for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<=a" "6#1%!G%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 122 "Although this integral cannot be expressed analytically in terms \+ of elementary functions, it can be expressed in terms an " }{TEXT 260 17 "elliptic function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "To achieve this, make the substitutio n " }{XPPEDIT 18 0 "cosh*2*t = 1/(1-u^2);" "6#/*(%%coshG\"\"\"\"\"#F&% \"tGF&*&F&F&,&F&F&*$%\"uGF'!\"\"F-" }{TEXT -1 25 " in the integral, w here " }{XPPEDIT 18 0 "t>=0" "6#1\"\"!%\"tG" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "u = sqrt((cosh*2*t-1)/(cosh*2*t));" "6#/%\"uG-%%sqrtG6#*&,&*(%%cosh G\"\"\"\"\"#F,%\"tGF,F,F,!\"\"F,*(F+F,F-F,F.F,F/" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "t=1/2" "6#/%\"tG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "arccosh(1/(1-u^2)" "6#-%(arccoshG6#*&\"\"\"F',&F'F'* $%\"uG\"\"#!\"\"F," }{TEXT -1 9 " so that:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dt/du=1/2" "6#/*&%#dtG\"\"\"%#duG!\"\"*&F&F& \"\"#F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/sqrt(1/((1-u^2)^2)-1)); " "6#-%!G6#*&\"\"\"F'-%%sqrtG6#,&*&F'F'*$,&F'F'*$%\"uG\"\"#!\"\"F1F2F' F'F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/du" "6#*&%\"dG\"\"\"%#duG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1/(1-u^2)]" "6#7#*&\"\"\"F%,&F%F% *$%\"uG\"\"#!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\" \"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sqrt((1-u^2)^2/(1 -(1-2*u^2+u^4))))*``(2*u/((1-u^2)^2));" "6#*&-%!G6#-%%sqrtG6#*&,&\"\" \"F,*$%\"uG\"\"#!\"\"F/,&F,F,,(F,F,*&F/F,*$F.F/F,F0*$F.\"\"%F,F0F0F,-F %6#*(F/F,F.F,*$,&F,F,*$F.F/F0F/F0F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/((1-u^2)*sqrt(2-u^2));" "6#/% !G*&\"\"\"F&*&,&F&F&*$%\"uG\"\"#!\"\"F&-%%sqrtG6#,&F+F&*$F*F+F,F&F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Diff(1/2*arccosh(1/(1-u^2)),u);\nassume(u_>=0,u_ <1);\nsubs(u_=u,simplify(subs(u=u_,value(%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,$*&#\"\"\"\"\"#F)-%(arccoshG6#*&F)F),&F)F)*$ )%\"uGF*F)!\"\"F3F)F)F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F %*&,&F%!\"\"*$)%\"uG\"\"#F%F%F%,&F,F%F)F(#F%F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "t=0, u=0" "6$/%\"tG\"\"!/%\"uGF%" }{TEXT -1 10 " and wh en " }{XPPEDIT 18 0 "t=a, u=b" "6$/%\"tG%\"aG/%\"uG%\"bG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "b = sqrt((cosh*2*a-1)/(cosh*2*a));" "6#/%\"b G-%%sqrtG6#*&,&*(%%coshG\"\"\"\"\"#F,%\"aGF,F,F,!\"\"F,*(F+F,F-F,F.F,F /" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The integral giving the arc length along the rectangular \+ hyperbola now becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((1-u^2)^(3/2)*sqrt(2-u^2)),u = 0 .. b);" "6#-%$IntG6$*&\" \"\"F'*&),&F'F'*$%\"uG\"\"#!\"\"*&\"\"$F'F-F.F'-%%sqrtG6#,&F-F'*$F,F-F .F'F./F,;\"\"!%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "Int(sqrt(cosh(2*t)),t=0.. a);\nstudent[changevar](cosh(2*t)=1/(1-u^2),%,u):\nassume(u_>=0,u_<1); \nsubs(u_=u,simplify(subs(u=u_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$*$-%%coshG6#,$*&\"\"#\"\"\"%\"tGF-F-#F-F,/F.;\"\"!%\"aG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&),&F'F'*$)%\"uG\" \"#F'!\"\"#\"\"$F.F',&F.F'F+F/#F'F.F//F-;\"\"!*$*&,&-%%coshG6#,$*&F.F' %\"aGF'F'F'F'F/F'F:F/#F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " Maple can evaluate the function " }{TEXT 306 10 "EllipticPi" }{TEXT -1 8 ", where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Ell ipticPi(x,nu,k) = Int(1/((1-nu*u^2)*sqrt(1-u^2)*sqrt(1-k^2*u^2)),u = 0 .. x);" "6#/-%+EllipticPiG6%%\"xG%#nuG%\"kG-%$IntG6$*&\"\"\"F.*(,&F.F .*&F(F.*$%\"uG\"\"#F.!\"\"F.-%%sqrtG6#,&F.F.*$F3F4F5F.-F76#,&F.F.*&F)F 4F3F4F5F.F5/F3;\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 285 0 "" }{TEXT 286 7 "Taking " }{XPPEDIT 18 0 "k=1/sqrt(2)" "6#/%\"kG*&\"\" \"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT 287 5 " and " }{XPPEDIT 18 0 "nu=1" " 6#/%#nuG\"\"\"" }{TEXT 288 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "EllipticPi(x,1,1/sqrt(2)) = Int(sqrt(2)/((1-u ^2)^(3/2)*sqrt(2-u^2)),u = 0 .. x);" "6#/-%+EllipticPiG6%%\"xG\"\"\"*& F(F(-%%sqrtG6#\"\"#!\"\"-%$IntG6$*&-F+6#F-F(*&),&F(F(*$%\"uGF-F.*&\"\" $F(F-F.F(-F+6#,&F-F(*$F9F-F.F(F./F9;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(cosh*2*t),t = 0 .. a) = 1/sqrt(2);" "6# /-%$IntG6$-%%sqrtG6#*(%%coshG\"\"\"\"\"#F,%\"tGF,/F.;\"\"!%\"aG*&F,F,- F(6#F-!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "EllipticPi(b,1,1/sqrt(2)) " "6#-%+EllipticPiG6%%\"bG\"\"\"*&F'F'-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 289 23 "_____________ __________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "b = sqrt((cosh*2*a-1)/(cosh*2*a));" "6#/%\"bG-%%sqrtG6# *&,&*(%%coshG\"\"\"\"\"#F,%\"aGF,F,F,!\"\"F,*(F+F,F-F,F.F,F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "a = 2,b = sqrt((cosh*4-1)/(cosh*4));" "6$/% \"aG\"\"#/%\"bG-%%sqrtG6#*&,&*&%%coshG\"\"\"\"\"%F/F/F/!\"\"F/*&F.F/F0 F/F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "The arc length a long the rectangular hyperbola " }{XPPEDIT 18 0 "x^2-y^2 = 1;" "6#/,&* $%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(" }{TEXT -1 15 " from the point" } {XPPEDIT 18 0 " ``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 13 " to the p oint" }{XPPEDIT 18 0 "``(cosh*2,sinh*2);" "6#-%!G6$*&%%coshG\"\"\"\"\" #F(*&%%sinhGF(F)F(" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(cosh*2*t),t = 0 .. 2) = 1/sqrt(2);" "6# /-%$IntG6$-%%sqrtG6#*(%%coshG\"\"\"\"\"#F,%\"tGF,/F.;\"\"!F-*&F,F,-F(6 #F-!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "EllipticPi(sqrt((cosh*4-1)/( cosh*4)),1,1/sqrt(2));" "6#-%+EllipticPiG6%-%%sqrtG6#*&,&*&%%coshG\"\" \"\"\"%F-F-F-!\"\"F-*&F,F-F.F-F/F-*&F-F--F'6#\"\"#F/" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(sqrt(cosh(2*t)),t=0..2) ;\nvalue(%);\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*$-%%coshG6#,$*&\"\"#\"\"\"%\"tGF-F-#F-F,/F.;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&F'F%-%+EllipticPiG6%*$*& ,&-%%coshG6#\"\"%F&F&!\"\"F&F/F3F%F&,$*&F'F3F'F%F&F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+W%*[DY!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Alternatively, we could evaluate " } {XPPEDIT 18 0 "Int(sqrt(cosh*2*t),t = 0 .. 2);" "6#-%$IntG6$-%%sqrtG6# *(%%coshG\"\"\"\"\"#F+%\"tGF+/F-;\"\"!F," }{TEXT -1 13 " numerically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(sqrt(cosh(2*t)),t=0..2);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#-%%coshG6#,$%\"tG\"\"#\"\"\"/F.;\" \"!F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W%*[DY!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Letting Maple do a ll the work gives a more complicated analytical expression for the arc length." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f := t -> cosh(t);\ng := t -> sinh(t);\nInt(sqrt(Dif f(f(t),t)^2+Diff(g(t),t)^2),t=0..2);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%%coshG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG%%sinhG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%s qrtG6#,&*$)-%%DiffG6$-%%coshG6#%\"tGF3\"\"#\"\"\"F5*$)-F.6$-%%sinhGF2F 3F4F5F5F5/F3;\"\"!F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&-%%sqrtG6# \"\"#\"\"\"-%*EllipticKG6#,$*$F%F)#F)F(F)F/*&#F)F(F)*&F%F)-%*EllipticF G6$*&F)F)-%%coshG6#F(!\"\"F-F)F)F:*&F%F)-%*EllipticEGF,F)F:*&F%F)-F=F5 F)F)*&F7F:*&,&F:F)*$)F7F(F)F)F),&FCF(F)F:F)F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W%*[DY!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The following graph shows the curve " } {XPPEDIT 18 0 "PIECEWISE([x = cosh*t, ``],[y = sinh*t, ``]);" "6#-%*PI ECEWISEG6$7$/%\"xG*&%%coshG\"\"\"%\"tGF+%!G7$/%\"yG*&%%sinhGF+F,F+F-" }{TEXT -1 23 ", with the section for " }{XPPEDIT 18 0 "0<=t" "6#1\"\"! %\"tG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 10 " drawn in " }{TEXT 305 7 "magenta" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "p1 := plot([cosh(t), sinh(t),t=0..2],color=magenta,thickness=2): \np2 := plot([cosh(t),sinh (t),t=-1.5..0],color=navy): \np3 := plot([cosh(t),sinh(t),t=2..3],colo r=navy):\nplots[display]([p1,p2,p3],view=[0..5,-2..5]); " }}{PARA 13 " " 1 "" {GLPLOT2D 393 332 332 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"\"\" \"!$F*F*7$$\"3Mi\"fBQ]4+\"!#<$\"3'oD0zy63O%!#>7$$\"3;_B?V]K.5F/$\"3OqC q[@eh\")F27$$\"3dHE'*H1s25F/$\"3jL#H6#[-X7!#=7$$\"33pgA[w*R,\"F/$\"3WX `APW-z;F=7$$\"3CXv')p95A5F/$\"33Vj=1W.9@F=7$$\"3QdYl'G#GJ5F/$\"3^+qfgn x?DF=7$$\"3h))e`;]\\U5F/$\"3_Yi'Gq8h%HF=7$$\"3[4CMuu$f0\"F/$\"3#z:;lAB 7R$F=7$$\"3;*Q.%**)G72\"F/$\"3n!p#\\e5)4%QF=7$$\"3C'F=7$$\"3Bpuk@N_+7F/$\"36Ns3bdrUmF=7$$\"3=gO:eSpJ7F/$\"3wp (F=7$$\"3%*G>l!*\\0g7F/$\"3mvgC@5TmwF=7$$\"3i2M/3,O&H\"F/$\"3ahclT$>QB )F=7$$\"3zq:dM7uG8F/$\"3C;=$e4!f\\()F=7$$\"3Q_5i$yIxO\"F/$\"3M2_>$pj5L *F=7$$\"3f2$\\$HFA29F/$\"3DDQY*)y)3!**F=7$$\"3IgBhG3'4X\"F/$\"3Y!G>.YE 80\"F/7$$\"3Mz3&\\zrM\\\"F/$\"3a#[7[6g#46F/7$$\"35FaP_7#>a\"F/$\"3]\"Q r*o%yO<\"F/7$$\"3Sg.ow/>&f\"F/$\"3oQAh8D$GC\"F/7$$\"3YrDIgK3W;F/$\"3T& e9N;%*\\I\"F/7$$\"3cWp%**G7'*p\"F/$\"3G_#)o\\KHu8F/7$$\"3)HwU$[%[+w\"F /$\"3G#HUPBo$[9F/7$$\"3]KU\"HD(HA=F/$\"3[.torkSB:F/7$$\"3M(HDdeqb)=F/$ \"3/SCd#[a&)f\"F/7$$\"3cf8`)fg%f>F/$\"3s-T)zRx]o\"F/7$$\"3rjpBIlBH?F/$ \"37wypk&Hdw\"F/7$$\"3mt;V?]S2@F/$\"3Gg>6)GN]&=F/7$$\"3u15w$*yi\"=#F/$ \"3t\")R$)zI%*Q>F/7$$\"3Ec'z#\\.gmAF/$\"3HOAEU*yS.#F/7$$\"39#Gf._Q.N#F /$\"3n<3*[5\"*p7#F/7$$\"3H=we*)G%>W#F/$\"3e[ZZd#)zFAF/7$$\"36&\\J_-#oN DF/$\"3/!pk64o,L#F/7$$\"34hz6Y%>%QEF/$\"3rWpw)eo:W#F/7$$\"3goYUqD*>u#F /$\"3GIV%=))RJb#F/7$$\"3ie5=yAz_GF/$\"3#G\\,Y)GyrEF/7$$\"3E1,V.UunHF/$ \"3gTq!QT\">%z#F/7$$\"3U@2xd*)*z2$F/$\"3cJs5&*o-6HF/7$$\"3#4%Q0*fI+@$F /$\"3y:*o_[%H]IF/7$$\"3\"H'[Xw>XLLF/$\"37pGi4<#*zJF/7$$\"3'o6Gb\"f\"3Z $F/$\"3l*y\\=.POK$F/7$$\"3[!=M2\"343OF/$\"3m1ytpZumMF/7$$\"3SJO3\"p&>i PF/$\"3->q%ySgoi$F/-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F][l-%*THICKNESSG 6#\"\"#-F$6$7S7$$\"3?ZKC:'4CN#F/$!3J<[4b%z#H@F/7$$\"3/RMn\\j.%G#F/$!3s n%*)4@!\\`?F/7$$\"3YUwH2V`EAF/$!3'ou^:KN$*)>F/7$$\"3GH!QP)y,k@F/$!3:oh 3\\u5>>F/7$$\"3CL5#F/$!3/xAuvzR]=F/7$$\"3o7)*4]_3X?F/$!3IPRts3#R y\"F/7$$\"3m0z6V'oH*>F/$!3M$>:i\"p#Rs\"F/7$$\"3I9`D\"4f3%>F/$!3k2z*eN5 Mm\"F/7$$\"37GG-&z**)))=F/$!34+()**)3![-;F/7$$\"3uW7XQ4,R=F/$!3V)R=+Pg La\"F/7$$\"3oTimJVi*y\"F/$!3c9h(3?oT[\"F/7$$\"3%p>$G\"\\/xu\"F/$!3**)G !)HfULV\"F/7$$\"3$zJ0VFQAq\"F/$!3[7\\%[_RvP\"F/7$$\"3_f$yv?y$e;F/$!3#y <2[:eHK\"F/7$$\"3#eK5%>vw<;F/$!3/(yB!y1or7F/7$$\"3kz>U6RE#e\"F/$!3`\"y ^5+(>E7F/7$$\"32Z\"F/$!37()eosO3!3\"F/7$$\"3tEAb#Q=2W\"F/$!3q \"*)\\'e[9P5F/7$$\"3Ku(pVZ8zS\"F/$!3pR'z$fIq5**F=7$$\"3UhNY,d/y8F/$!3k P.4^Nh\"[*F=7$$\"3#e)*yAlv#[8F/$!3)[\">[n1\\V!*F=7$$\"3-p4KL_;A8F/$!3_ *\\f#p#)R\\')F=7$$\"3cYZ/qCH&H\"F/$!3*HWAo?bFB)F=7$$\"31(>t!R#p(o7F/$! 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\"\"F&F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 51 "The right-han d branch of the rectangular hyperbola " }{XPPEDIT 18 0 "x^2-y^2=1" "6# /,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(" }{TEXT -1 31 " has the parametr ic equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([x=sqrt((2-u^2)/(2*(1-u^2))) ,`` ],[ y=u/sqrt(2*(1-u^2)),`` ]) " "6#-%*PIECEWISEG6$7$/%\"xG-%%sqrtG6#*&,&\"\"#\"\"\"*$%\"uGF.!\"\"F/* &F.F/,&F/F/*$F1F.F2F/F2%!G7$/%\"yG*&F1F/-F*6#*&F.F/,&F/F/*$F1F.F2F/F2F 6" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 67 "We can also compute the arclength using these parametri c equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f := u -> sqrt((2-u^2)/(2*(1-u^2)));\ng := u -> u /sqrt(2*(1-u^2));\nInt(sqrt(Diff('f(u)',u)^2+Diff('g(u)',u)^2),u=0..sq rt((cosh(4)-1)/cosh(4)));\nvalue(%):\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"uG6\"6$%)operatorG%&arrowGF(-%%sqr tG6#*&,&\"\"#\"\"\"*$)9$F1F2!\"\"F2,&F1F2*&F1F2F4F2F6F6F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"uG6\"6$%)operatorG%&arrow GF(*&9$\"\"\"-%%sqrtG6#,&\"\"#F.*&F3F.)F-F3F.!\"\"F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&*$)-%%DiffG6$-%\"fG6#%\"uGF0\" \"#\"\"\"F2*$)-F+6$-%\"gGF/F0F1F2F2#F2F1/F0;\"\"!*$*&,&-%%coshG6#\"\"% F2F2!\"\"F2F@FDF9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W%*[DY!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "A remark concerning hyperbolic arcs" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 114 "In the 3rd example of the previous section it is shown that th e length of the arc along the rectangular hyperbola " }{XPPEDIT 18 0 " x^2-y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"F(" }{TEXT -1 15 " fr om the point" }{XPPEDIT 18 0 " ``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 13 " to the point" }{XPPEDIT 18 0 "``(cosh*a,sinh*a);" "6#-%!G6$*&% %coshG\"\"\"%\"aGF(*&%%sinhGF(F)F(" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Int(sqrt(cosh*2*t),t = 0 .. a);" "6#-%$IntG6$-%%sqrtG6#*(%%coshG\" \"\"\"\"#F+%\"tGF+/F-;\"\"!%\"aG" }{TEXT -1 54 ", and an analytic form ula for this integral is given. 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" }}{PARA 0 "" 0 "" {TEXT -1 61 "The distance along the asympote from the origin to the point " }{XPPEDIT 18 0 "Q(cosh*a,cosh*a)" "6#-%\"QG6$*&%%coshG\"\"\"%\"aGF(*&F 'F(F)F(" }{TEXT -1 5 " is s" }{XPPEDIT 18 0 "qrt(2)*cosh*a" "6#*(-%$qr tG6#\"\"#\"\"\"%%coshGF(%\"aGF(" }{TEXT -1 11 ", hence as " }{TEXT 307 1 "x" }{TEXT -1 27 " increases the difference: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2)*cosh*a-Int(sqrt(cosh*2*t),t \+ = 0 .. a)" "6#,&*(-%%sqrtG6#\"\"#\"\"\"%%coshGF)%\"aGF)F)-%$IntG6$-F&6 #*(F*F)F(F)%\"tGF)/F2;\"\"!F+!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "between the corresponding distances travelled becomes app roximately constant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "L := a -> Int(sqrt(cosh(2*t)),t=0..a);\np lot([L(a),sqrt(2)*cosh(a)],a=0..3,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%$IntG6$-% %sqrtG6#-%%coshG6#,$*&\"\"#\"\"\"%\"tGF8F8/F9;\"\"!9$F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"! F)F(7$$\"+]i9Rl!#6$\"+QGY[lF-7$$\"+WA)GA\"!#5$\"+x!p*G7F37$$\"+Qeui=F3 $\"+?!=U)=F37$$\"+i3&o]#F3$\"+(R^!fDF37$$\"+pX*y9$F3$\"+Ts#4D$F37$$\"+ WTAUPF3$\"+\"RYZ\"RF37$$\"+%*zhdVF3$\"++?)*GYF37$$\"+%>fS*\\F3$\"+cv$3 S&F37$$\"+>$f%GcF3$\"+uRI3iF37$$\"+Dy,\"G'F3$\"+KCH$3(F37$$\"+7\"F\\p7$$\"+siL-5F\\p$ \"+7gs@8F\\p7$$\"+!R5'f5F\\p$\"+d#3iV\"F\\p7$$\"+/QBE6F\\p$\"+P![wd\"F \\p7$$\"+:o?&=\"F\\p$\"+^*p2r\"F\\p7$$\"+a&4*\\7F\\p$\"+&\\\")f'=F\\p7 $$\"+j=_68F\\p$\"+%eyK-#F\\p7$$\"+Wy!eP\"F\\p$\"+'=_z>#F\\p7$$\"+UC%[V \"F\\p$\"+,#o%oBF\\p7$$\"+J#>&)\\\"F\\p$\"+(zeRc#F\\p7$$\"+>:mk:F\\p$ \"+D\"p0y#F\\p7$$\"+w&QAi\"F\\p$\"+GZ5\")HF\\p7$$\"+uLU%o\"F\\p$\"+bw, 6KF\\p7$$\"+bjm[F\\p$\"+rWR?VF\\p7$$\"+:K^+?F\\p$\"+` ?JOlF\\p7$$\"+i@OtBF\\p$\"+l!30*pF\\p7$$\"+fL'zV#F\\p$\"+10,( \\(F\\p7$$\"+!*>=+DF\\p$\"+#pcn,)F\\p7$$\"+E&4Qc#F\\p$\"+4Cz#e)F\\p7$$ \"+%>5pi#F\\p$\"+rr&3=*F\\p7$$\"+bJ*[o#F\\p$\"+o')pk(*F\\p7$$\"+r\"[8v #F\\p$\"+\\KoZ5!\")7$$\"+Ijy5GF\\p$\"+5H^:6Fcy7$$\"+/)fT(GF\\p$\"+')\\ T#>\"Fcy7$$\"+1j\"[$HF\\p$\"+$\"3a)Q9Fd[l7$$\"3A++]i3&o]#F^\\l$\"3+!Q5@[$))e9 Fd[l7$$\"3%)***\\(oX*y9$F^\\l$\"3i'RDX*H'[[\"Fd[l7$$\"3z***\\P9CAu$F^ \\l$\"3ro$3Jb*R9:Fd[l7$$\"3!)***\\P*zhdVF^\\l$\"3Y;\"or5B1b\"Fd[l7$$\" 31++v$>fS*\\F^\\l$\"3=N&fGTmUf\"Fd[l7$$\"3$)***\\(=$f%GcF^\\l$\"35l'*3 x$)>W;Fd[l7$$\"3Q+++Dy,\"G'F^\\l$\"3]=^:v\"oCq\"Fd[l7$$\"33++]7Fd[l7$$\"3[+++D!*oy()F^\\l$\"3q//I+f0&*> Fd[l7$$\"3))***\\PpnsM*F^\\l$\"3cwhRLeLy?Fd[l7$$\"3,++]siL-5Fd[l$\"3?9 '*frV8'=#Fd[l7$$\"3-+++!R5'f5Fd[l$\"3;5$H(*=a_G#Fd[l7$$\"3)***\\P/QBE6 Fd[l$\"3*)\\Csc?,5CFd[l7$$\"3!******\\\"o?&=\"Fd[l$\"3[3[IvONHDFd[l7$$ \"31+]Pa&4*\\7Fd[l$\"3kRx&*o(H/n#Fd[l7$$\"33+]7j=_68Fd[l$\"3#zB,4M`^\" GFd[l7$$\"33++vVy!eP\"Fd[l$\"3W\"eE.(>cxHFd[l7$$\"34+](=WU[V\"Fd[l$\"3 ]>6;O\"Hv8$Fd[l7$$\"3)****\\7B>&)\\\"Fd[l$\"3\"\\V-UDaBK$Fd[l7$$\"3)** *\\P>:mk:Fd[l$\"3i3$[B)\\hGNFd[l7$$\"3'***\\iv&QAi\"Fd[l$\"3u9cx\"f/2s $Fd[l7$$\"31++vtLU%o\"Fd[l$\"3$)fjr1_/URFd[l7$$\"3!******\\Nm'[Fd[l$\"3wS?>@VW@]Fd[ l7$$\"3z*****\\@80+#Fd[l$\"3Lj2\"3B\"=B`Fd[l7$$\"31++]7,Hl?Fd[l$\"3;=b H0N,ncFd[l7$$\"3()**\\P4w)R7#Fd[l$\"3iQ[sO\"z!**fFd[l7$$\"3;++]x%f\")= #Fd[l$\"3%*HaQiY\"eQ'Fd[l7$$\"3!)**\\P/-a[AFd[l$\"3]D([jL$ptnFd[l7$$\" 3/+](=Yb;J#Fd[l$\"3j6b*o'*ob?(Fd[l7$$\"3')****\\i@OtBFd[l$\"3/5@AyCbbw Fd[l7$$\"3')**\\PfL'zV#Fd[l$\"3#R+=.#o\"z:)Fd[l7$$\"3>+++!*>=+DFd[l$\" 3)4C%y^U#Rn)Fd[l7$$\"3-++DE&4Qc#Fd[l$\"3X8!))p@rjB*Fd[l7$$\"3=+]P%>5pi #Fd[l$\"3RLisDn4J)*Fd[l7$$\"39+++bJ*[o#Fd[l$\"3-*oGR507/\"!#;7$$\"33++ Dr\"[8v#Fd[l$\"3Vddt>w576F^il7$$\"3++++Ijy5GFd[l$\"3s8`0xinz6F^il7$$\" 31+]P/)fT(GFd[l$\"3!4f!z'p;jD\"F^il7$$\"31+]i0j\"[$HFd[l$\"3%*eCL'R%RM 8F^il7$Fdz$\"3z4 " 0 "" {MPLTEXT 1 0 91 "g := a -> \+ sqrt((cosh(2*a)-1)/cosh(2*a));\nL2 := a -> 1/sqrt(2)*EllipticPi(g(a),1 ,1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"aG6\"6$%) operatorG%&arrowGF(-%%sqrtG6#*&,&-%%coshG6#,$*&\"\"#\"\"\"9$F7F7F7F7! \"\"F7F1F9F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L2Gf*6#%\"aG6\" 6$%)operatorG%&arrowGF(*&-%%sqrtG6#\"\"#!\"\"-%+EllipticPiG6%-%\"gG6#9 $\"\"\"*&F9F9F-F1F9F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "aa := 50;\nevalf(evalf(sqrt(2)*cosh (aa)-L2(aa),80),20);\nevalf(evalf(sqrt(2)*cosh(aa)-L2(aa),90),20);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5s.hznt6q!*f!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"5s.hznt6q!*f!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 101 "aa := 60;\nevalf(evalf(sqrt(2)*cosh(aa)-L2( aa),100),20);\nevalf(evalf(sqrt(2)*cosh(aa)-L2(aa),110),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"5s.hznt6q!*f!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5s.hznt6q!* f!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "A s " }{XPPEDIT 18 0 "a->infinity" "6#f*6#%\"aG7\"6$%)operatorG%&arrowG6 \"%)infinityGF*F*F*" }{TEXT -1 17 ", the difference " }{XPPEDIT 18 0 " sqrt(2)*cosh*a-Int(sqrt(cosh*2*t),t = 0 .. a);" "6#,&*(-%%sqrtG6#\"\"# \"\"\"%%coshGF)%\"aGF)F)-%$IntG6$-F&6#*(F*F)F(F)%\"tGF)/F2;\"\"!F+!\" \"" }{TEXT -1 22 " approaches a number " }{XPPEDIT 18 0 "kappa" "6#%& kappaG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "kappa" "6#%&kappaG" } {TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 25 " 0.59907011736779610372. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "g := a -> sqrt((cosh(2*a)-1)/cosh(2*a)):\nL2 := a -> 1/sqrt(2)*EllipticPi(g(a),1,1/sqrt(2)):\nkappa := .599070117367796103 72;\nevalf(plot([sqrt(2)*cosh(t)-L2(t),kappa],t=0..8,0..1.4,\n \+ color=[blue,grey],linestyle=[1,3]),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&kappaG$\"5s.hznt6q!*f!#?" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7Y7$$\"\"!F)$\"5)[]4t Bc8UT\"!#>7$$\"5NLLLLL3VfV!#@$\"5&[J\"QV'oN>P\"F,7$$\"5qmmmmm;')=()F0$ \"5L*yOo*4H=K8F,7$$\"5+++++]#HyI\"!#?$\"5zIMaPuzz%H\"F,7$$\"5MLLLLLBxV 5E$F;$\"5-tp5Q!RGD:\"F,7$$\"5,++++v2<9TF;$\"5hL+08D#485\"F,7$$\"5MLLLL LAKn\\F;$\"5Yi6!o[%p`b5F,7$$\"5NLLLLL*Gh#eF;$\"5qZ*[KN=hU,\"F,7$$\"5NL LLLLc$\\o'F;$\"5B/boc\"=d>x*F;7$$\"5ommmm;bQ%R)F;$\"5@xfpkJ>qR\"*F;7$$ \"5NLLLL$Qk#z**F;$\"59eOiZ8eKc')F;7$$\"5+++++l9.i6F,$\"5,ZDJLOS$*Q#)F; 7$$\"5MLLLL=\"\\8(F;7$$\"5+++++g-w+?F,$\"5ojCw+^L&)\\pF;7$$\"5++++++z,u@F,$\"5 4L+Wq7Yb'z'F;7$$\"5+++++SP)4M#F,$\"5w))fD\"4(4AsmF;7$$\"5MLLLL=Zg#\\#F ,$\"5&GMsB[a/hd'F;7$$\"5nmmmmEn*Gn#F,$\"5muwDjU*f$zkF;7$$\"5nmmmm1xiDG F,$\"5#ptWSH%*e+T'F;7$$\"5,++++X,H.IF,$\"5(*=mH!o`O[?'F;7$$\"5nmmmm;4#)oOF,$\"5'H8Pg)3\"z5<'F;7$$\"5 nmmmm6lCEQF,$\"5%\\4\">F93![9'F;7$$\"5MLLLL$G^g*RF,$\"5.Fx&[$R?t?hF;7$ $\"5MLLLL=2VsTF,$\"5Sn)RR??/(*4'F;7$$\"5,++++N&pfK%F,$\"5**z%)o`Y&)=%3 'F;7$$\"5MLLLLjcz\"\\%F,$\"5O-1!Q$>C!*pgF;7$$\"5,++++!G5Jm%F,$\"5'yddf #*GKu0'F;7$$\"5,++++5#32$[F,$\"5z)pofa-Nr/'F;7$$\"5,++++Dy'G*\\F,$\"5* [1Lrb/(oQgF;7$$\"5,++++I%=H<&F,$\"5qdk.w@-yIgF;7$$\"5ommmm1>qM`F,$\"5i boW6yLzCgF;7$$\"5,+++++.W2bF,$\"5^:Ohhz[Q>gF;7$$\"5MLLLLep'Rm&F,$\"5rU gQrU)G_,'F;7$$\"5,++++S>4NeF,$\"5hk2'en-r8,'F;7$$\"5ommmm6s5'*fF,$\"5` =!p(4!)pH3gF;7$$\"5,++++lXTkhF,$\"54li@k!Grb+'F;7$$\"5ommmmmd'*GjF,$\" 5+:xJ)45:L+'F;7$$\"5,++++DcB,lF,$\"56&G\"fVq*=8+'F;7$$\"5NLLLLt>:nmF,$ \"5*3?-[)*o&p**fF;7$$\"5NLLLL.a#o$oF,$\"5VLFqX3?H)*fF;7$$\"5ommmm^Q40q F,$\"5*3$HtC#R;r*fF;7$$\"5,++++!3:(frF,$\"5Psc\\XXt>'*fF;7$$\"5ommmmc% GpL(F,$\"5hw0UpsZI&*fF;7$$\"5NLLLL8-V&\\(F,$\"5^'zKlP)*HY*fF;7$$\"5-++ ++XhUkwF,$\"5fvGEJ-\">S*fF;7$$\"5-++++:o " 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 "Code f or 1st picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 795 "f := x->sqrt(x^2-1):\ndarkgrey := COLOR(RGB,. 01,.01,.01):\np1 := plot([cosh(t),sinh(t),t=0..1.5],color=red,thicknes s=2):\npt := [1.5,f(1.5)]:\np2 := plot([[0,0],pt],color=COLOR(RGB,.5,. 2,.6)):\nt0 := arccosh(1.5):\nsect := [[0,0],op(op(1,op(1,plot([cosh(t ),sinh(t),t=0..t0])))),pt]:\np3:=plots[polygonplot](sect,color=COLOR(R GB,.88,.88,.93),style=patchnogrid):\nall := [circle,diamond,cross]:\np 4 := plot([[pt]$3],style=point,symbol=all,color=red):\nt1 := plots[tex tplot]([1.85,1.1,`P(cosh t,sinh t)`],color=COLOR(RGB,.85,.2,.2)):\nt2 \+ := plots[textplot]([.7,.3,`A(t)`],color=COLOR(RGB,.4,.1,.5)):\nt3 := p lots[textplot]([[-.1,-.1,`O`],[1,-.08,`S(1,0)`],\n [2.15,-.05,`x`] ,[-.06,1.93,`y`]],color=darkgrey):\nplots[display]([p1,p2,p3,p4,t1,t2, t3],view=[-.1..2.2,-.1..2],\n tickmarks=[0,0],labels=[`x`,`y`]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for 2nd picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1013 "f := x->sqrt(x^2-1):\np1 := plot([cosh(t),sinh(t),t=0..1.4],co lor=red,thickness=2):\np2 := plot([[1.5,0],pt],color=COLOR(RGB,.6,.2,. 5)):\npt := [1.5,f(1.5)]:\np3 := plot([[0,0],pt],color=COLOR(RGB,.5,.2 ,.6)):\nt0 := arccosh(1.5):\nsect := op(1,op(1,plot([cosh(t),sinh(t),t =0..t0]))):\np4 := plots[polygonplot]([[0,0],op(sect),pt],\n color= COLOR(RGB,.88,.88,.93),style=patchnogrid):\np5 := plots[polygonplot]([ op(sect),pt,[1.5,0]],\n color=COLOR(RGB,.92,.86,.88),style=patchnog rid):\nall := [circle,diamond,cross]:\np6 := plot([[pt]$3],style=point ,symbol=all,color=red):\nt1 := plots[textplot]([1.8,1.1,`P(cosh t,sinh t)`],color=COLOR(RGB,.85,.2,.2)):\nt2 := plots[textplot]([.7,.3,`A(t) `],color=COLOR(RGB,.4,.1,.5)):\nt3 := plots[textplot]([1.25,.3,`B(t)`] ,color=COLOR(RGB,.5,.1,.4)):\nt4 := plots[textplot]([[-.08,-.08,`O`],[ 1,-.08,`S(1,0)`],\n [1.5,-.08,`T`],[1.96,-.05,`x`],[-.06,1.71,`y`]] ,color=darkgrey):\nplots[display]([p1,p2,p3,p4,p5,p6,t1,t2,t3,t4],view =[-.1..2,-.1..1.8],\n tickmarks=[0,0],labels=[`x`,`y`]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for 3rd picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "f := x->sqrt(1-x^2):\np1 := plot([cos(t),sin(t),t=0..2*Pi],colo r=red,thickness=2):\npt := [.55,f(.55)]:\nt0 := arccos(.55):\nsect := \+ [[0,0],op(op(1,op(1,plot([cos(t),sin(t),t=0..t0])))),pt]:\np2:=plots[p olygonplot](sect,color=COLOR(RGB,.88,.88,.93),style=patchnogrid):\np3 \+ := plot([[0,0],pt],color=COLOR(RGB,.5,.2,.6)):\nall := [circle,diamond ,cross]:\np4 := plot([[pt,[1,0]]$3],style=point,symbol=all,color=red): \nt1 := plots[textplot]([.57,.33,`A(t)`],color=COLOR(RGB,.4,.1,.5)):\n t2 := plots[textplot]([[-.08,-.08,`O`],[1.18,.1,`S(1,0)`],\n [.91,.9, `P(cos t,sin t)`],[1.3,-.05,`x`],[-.06,1.16,`y`]],color=darkgrey):\npl ots[display]([p1,p2,p3,p4,t1,t2],tickmarks=[0,0],\n view=[-1.1..1.33, -1.1..1.2],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 21 "Code for 4th picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 666 "p1 := plot([[[0,0],[3,0], [3,1]],\n [[2.8,0],[2.8,.1],[3,.1]]],color=black):\np2 := \+ plot(.1333333335e-1*x^2+.2933333333*x,x=-1..4.25,\n \+ color=red,thickness=2):\nt1 := plots[textplot]([[1.44,-.15,`d`],[3.25 ,.5,`d`],\n [1.24,.63,`d`]],font=[SYMBOL,10]):\nt2 := plots[t extplot]([[1.61,-.15,`x`],[3.42,.5,`y`],\n [1.41,.63,`s`]],font=[ HELVETICA,10]):\nt3 := plots[textplot]([[4.2,1.1,`x = (t)`],[4.2,.94 ,`y = (t)`]],color=red):\nt4 := plots[textplot]([3.6,1.05,`\{`],font =[HELVETICA,18],color=red):\nt5 := plots[textplot]([[4.25,1.11,`f`],[4 .25,.95,`h`]],font=[SYMBOL,11],color=red):\nplots[display]([p1,p2,t1,t 2,t3,t4,t5],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for 5th picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 735 "f := x->sqrt(x^2-1):\npt := [1.5,f(1.5)]:\nt0 := arccosh(1.5):\np1 := plot([cosh(t),sinh(t),t=0.. t0],color=magenta,thickness=2):\np2 := plot([cosh(t),sinh(t),t=-1.4..0 ],color=navy):\np3 := plot([cosh(t),sinh(t),t=t0..1.8],color=navy):\np 4 := plot(-x,x=0..2,color=blue):\np5 := plot(x,x=1.5..3,color=blue):\n p6 := plot(x,x=0..1.5,color=coral,thickness=2):\np7 := plot([[1.5,f(1. 5)],[1.5,1.5]],color=black,linestyle=2):\np8 := plot([[[1.5,f(1.5)],[1 .5,1.5]]$3],style=point,\n symbol=[circle,diamond,cross],color=n avy):\nt1 := plots[textplot]([[1.93,1,`P(cosh a,sinh a)`],\n [1. 15,1.75,`Q(cosh a,cosh a)`]],color=navy):\nt2 := plots[textplot]([2.3, 2.6,`y = x`],color=blue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,t1, t2],view=[0..3,-2..3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }