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0 0 0 0 }{CSTYLE "" -1 376 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 377 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 378 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "Calculus Single Variable " }{TEXT 336 1 " " }{TEXT 259 37 " \nBrian E. Blank and Steven G. Kran tz" }{TEXT 258 43 "\n\nSection 8.2\nArc Length and Surface Area\n" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "1. Parameterized Curves\n" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" }{TEXT 270 13 "If to eve ry " }{XPPEDIT 272 0 "t;" "6#%\"tG" }{TEXT 271 18 " in an interval \+ " }{XPPEDIT 274 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT 273 25 " there ar e two numbers " }{XPPEDIT 277 0 "x(t);" "6#-%\"xG6#%\"tG" }{TEXT 275 7 " and " }{XPPEDIT 278 0 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT 276 59 ", then the curve obtained by plotting the ordered pairs " }{TEXT 283 1 "\{" }{TEXT 284 1 " " }{TEXT 281 1 "(" }{TEXT 282 1 " " } {XPPEDIT 257 0 "x(t),y(t);" "6$-%\"xG6#%\"tG-%\"yG6#F&" }{TEXT 279 1 " " }{TEXT 280 5 ") : " }{XPPEDIT 287 0 "a <= t;" "6#1%\"aG%\"tG" } {TEXT 285 1 " " }{XPPEDIT 288 0 "`` <= b;" "6#1%!G%\"bG" }{TEXT 286 4 " \} " }{TEXT 291 11 "is called a" }{TEXT 292 1 " " }{TEXT 289 19 "pa rameterized curve" }{TEXT 290 3 ". " }{TEXT 294 13 "The variable " } {XPPEDIT 295 0 "t;" "6#%\"tG" }{TEXT 293 13 " is called a " }{TEXT 297 9 "parameter" }{TEXT 298 18 " and the interval " }{XPPEDIT 256 0 " [a, b];" "6#7$%\"aG%\"bG" }{TEXT 296 32 " is the range of the paramete r. " }}{PARA 3 "" 0 "" {TEXT -1 1 "\n" }{TEXT 267 8 "Example:" }} {PARA 0 "" 0 "" {TEXT -1 55 "Find thee different parameterizations for the curve " }{XPPEDIT 18 0 "y = x^(2/3);" "6#/%\"yG)%\"xG*&\"\"#\" \"\"\"\"$!\"\"" }{TEXT -1 43 " between the points (8,4) and (27,9) .\n\n" }{TEXT 268 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "Parameterization 1: " }{XPPEDIT 18 0 "x = t*` `,` `*y = t^(2/3);" "6$/%\"xG*&%\"tG\"\"\"%#~~GF'/*&F( F'%\"yGF')F&*&\"\"#F'\"\"$!\"\"" }{TEXT -1 59 " Parameter range \+ [8,27]\nParameterization 2: " }{XPPEDIT 18 0 "x = t^(3/2)*` \+ `,` `*y = t;" "6$/%\"xG*&)%\"tG*&\"\"$\"\"\"\"\"#!\"\"F*%#~~GF*/*&F-F *%\"yGF*F'" }{TEXT -1 29 " Parameter range [4,9]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Parameterization 3: \+ " }{XPPEDIT 18 0 "x = t^3*` `,` `*y = t^2;" "6$/%\"xG*&%\"tG \"\"$%#~~G\"\"\"/*&F(F)%\"yGF)*$F&\"\"#" }{TEXT -1 32 " Param eter range [2,3]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Since Parameterization 1 is just the parameterization of the graph of a function, we can plot \+ the curve as follows in Maple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(x^(2/3), x = 8..27, vie w = [0..27,0..9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\")\"\"!$\"3c**************R!#<7$$\" 3qmm;Hf99%)F-$\"3g#\\'[eT)o8%F-7$$\"3WL$3x3#\\u()F-$\"3)>ug'3#oTD%F-7$ $\"3;mmTI!R(z\"*F-$\"3Ew\"3E*\\;%Q%F-7$$\"3%pm;H@swe*F-$\"3s)*))fY^58X F-7$$\"3KK$3-cmO***F-$\"3bSty6)G'RYF-7$$\"3amT5Hv+P5!#;$\"33dSG/(4av%F -7$$\"3'**\\PHZ#)f2\"FJ$\"3KluzWs#Q([F-7$$\"3gmTg:/H;6FJ$\"3u+/;<.![* \\F-7$$\"3-+v=!4pk:\"FJ$\"3ab(H)3Y%R6&F-7$$\"3MLLe&z(z(>\"FJ$\"3w@]c\\ \"o]B&F-7$$\"3km;zT,?M7FJ$\"3@kCu?UgS`F-7$$\"31++vh0=v7FJ$\"3?ac$)*=z \"eaF-7$$\"3+++D^#HjJ\"FJ$\"3#3=5([i(\\d&F-7$$\"3/++DQO)fN\"FJ$\"3m=cD 5bQ'o&F-7$$\"3amTg?O*>R\"FJ$\"3a'H=w@=my&F-7$$\"3KLLesH\"[V\"FJ$\"3$)e K8ano/fF-7$$\"3ULLL!e'3r9FJ$\"3)*=A@X**y.gF-7$$\"3++vV49G8:FJ$\"3kH9GR :0=hF-7$$\"3ELL$G)4j]:FJ$\"3W%Qn434$=iF-7$$\"3-+vV%Q4;f\"FJ$\"36yqltfQ FjF-7$$\"34+DJ80jI;FJ$\"3i`))f)*fQIkF-7$$\"3CL$3x'\\Mr;FJ$\"3kX#)y\\O) p`'F-7$$\"3OL3_YNt3c\"=PqF-7$$\"30++]\"p)[2>FJ$\"3XyT *fkk!RrF-7$$\"3#***\\()*>$HZ>FJ$\"3CYLZb'Q!QsF-7$$\"3))*\\P%3h!e)>FJ$ \"3ReB5P6;LtF-7$$\"3))**\\7F\"o&G?FJ$\"3J;nVJ41QuF-7$$\"3-LL$Gq\"*p1#F J$\"3?_v/54pJvF-7$$\"3$****\\72X@FJ$ \"3F2a0#*pZ?xF-7$$\"33++vNV$e=#FJ$\"3'*f#QH\"\\oD,'3zF-7$$\"39+v=f%[SE#FJ$\"3&[pN]wzI+)F-7$$\"3RLLep$HJI#FJ$\" 3m9*>eo8\\4)F-7$$\"3l*\\P4YVSM#FJ$\"3k#y\\RZ+0>)F-7$$\"3ommm$f[MQ#FJ$ \"3A2hO2v.#G)F-7$$\"3am;HLguBCFJ$\"3IrU&*=r7v$)F-7$$\"3ML3x*y4PY#FJ$\" 3%=Lr[$\\*oJ()F-7$$\"39+vV47I?EFJ$\"3RXk KG'3?#))F-7$$\"3#**\\iN*preEFJ$\"3'H27&)fC!3*)F-7$$\"#FF*$\"3A)******* *******)F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F`[l-%+AXESLABELSG6$Q\"x6\" Q!Fe[l-%%VIEWG6$;F`[lFez;F`[l$\"\"*F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The other parameterizations can be plotted as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([t^(3/2), t, t = 4..9], view = [0..27,0. .9]); 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In the " }{TEXT 333 5 "Maple" }{TEXT -1 106 " file, click on the graphic. That will bring up the animation controls on the toolbar at the top. 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Arc Length of Parameterized Curves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the parameterize d curve " }{XPPEDIT 18 0 "x(t) = t+sin(3*t),` `*y(t) = cos(2*t)+t/5 ,` `*t = 0 .. 2*Pi;" "6%/-%\"xG6#%\"tG,&F'\"\"\"-%$sinG6#*&\"\"$F)F'F )F)/*&%#~~GF)-%\"yG6#F'F),&-%$cosG6#*&\"\"#F)F'F)F)*&F'F)\"\"&!\"\"F)/ *&F1F)F'F);\"\"!*&F:F)%#PiGF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := t -> t+ sin(3*t):\ng := t -> cos(2*t)+t/5:\nparameterRange := 0 .. 2*Pi:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plot([f(t),g(t), t = paramet erRange], axes=NONE, color=COLOR(RGB,.30,.70,.99), thickness = 2);" }} {PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6(-%'CURVESG6#7gp7 $$\"\"!F)$\"\"\"F)7$$\"3%*p\\:Sefj`!#=$\"3+I)yW@96!**F/7$$\"3*[@=8?sd_ (F/$\"3t6)))[\"*H/j*F/7$$\"3swW^>[w5&*F/$\"3f(\\Gf7H(G#*F/7$$\"3C3:5B0 mZ6!#<$\"3yn$HMA9(H')F/7$$\"3N.G1v#Q5J\"F?$\"3;v(3L_Nv)yF/7$$\"3@****z 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-u^=%33vv&F?$!3Jx!=BY`qq$F_o7$$\"3QLlTs\"Hyw&F?$!33_![(H#Qis*!#@7$$\"3 Q#4-^Y7_u&F?$\"31)RwJ?U+/&F_o7$$\"3C&G$4?]o#p&F?$\"3C5^!\\)fOk6F/7$$\" 3MUh.)\\LVh&F?$\"3OkytZl#H'>F/7$$\"3EQ0Q!Rf^^&F?$\"3!R\"oTBR2*)GF/7$$ \"32:@0R=!4S&F?$\"3)ei(G1S=IRF/7$$\"3')p$)[CV$yF&F?$\"3AY\"G,m>=2&F/7$ $\"3kZlYCCa_^F?$\"3)*y4T+%ezH'F/7$$\"3&3qFn^!pJ]F?$\"3'Q#eXy]C\"f(F/7$ $\"3fGF2!p&y@\\F?$\"3gR;\\#elK$*)F/7$$\"3!4b(\\Fs%*G[F?$\"3[M*>Rs$[I5F ?7$$\"3oYAaF)*oeZF?$\"3[?HfL6jo6F?7$$\"3jofMV#Gdr%F?$\"3*3pj#*H)y08F?7 $$\"3(RZm,E5Qq%F?$\"3-tsm\\.**R9F?7$$\"3YY*=#GReDZF?$\"3Gd$p:F?7$ $\"3uZpp\"F?7$$\"3#[Q[XF?7$$\"3[fcWZ!4+;&F?$\"3vo!RF?AG+#F?7$ $\"399P'*)y5tM&F?$\"3/WB-f#yE3#F?7$$\"3G-_)H&H')ebF?$\"3?\"4\"[NWr[@F? 7$$\"3'oJ^$[qR*y&F?$\"3wHeL\"[(3+AF?7$$\"3!f1A&R@(H.'F?$\"3'f.O@R&=OAF ?7$$\"3C'ezrI&=$G'F?$\"3E6 #&F06#,&F2F3F3F9F9F:F3/F2;F3%\"NG" }{TEXT -1 61 " \+ \nand define the arc length to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ " }{XPPEDIT 18 0 "limit(Sum(sqrt((x(t[j])-x(t[j-1]))^2+(y(t[j])-y(t[j- 1]))^2),j = 1 .. N),N = infinity);" "6#-%&limitG6$-%$SumG6$-%%sqrtG6#, &*$,&-%\"xG6#&%\"tG6#%\"jG\"\"\"-F06#&F36#,&F5F6F6!\"\"F<\"\"#F6*$,&-% \"yG6#&F36#F5F6-FA6#&F36#,&F5F6F6F " 0 "" {MPLTEXT 1 0 65 "`arc length` = int(sqrt( D(f)(t)^2 + D(g)(t)^2 ), \+ t = 0 .. 2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+arc~lengthG-%$in tG6$*$,&*$),&\"\"\"F-*&\"\"$F--%$cosG6#,$*&F/F-%\"tGF-F-F-F-\"\"#F-F-* $),&*&F6F--%$sinG6#,$*&F6F-F5F-F-F-!\"\"#F-\"\"&F-F6F-F-#F-F6/F5;\"\"! ,$*&F6F-%#PiGF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The integral cannot be evaluated in closed form, but we c an evaluate it numerically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "(Execute the code for that is found below befor e executing the next line.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "arcLength(f,g,0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;=f)f\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "3. Arc Length Exercises from the Text" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 20 "Exercise 21 Page 581" } {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 38 "Calculate the length of the graph of " }{XPPEDIT 18 0 "f(x) = (x^2+2)^(3/2)/3;" "6#/-%\"fG6# %\"xG*&),&*$F'\"\"#\"\"\"F,F-*&\"\"$F-F,!\"\"F-F/F0" }{TEXT -1 7 " f or " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 11 " in [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f \+ := x -> (x^2 + 2)^(3/2)/3; paramRange := 0 .. 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\" \"$F/*$),&*$)9$\"\"#F/F/F7F/#F0F7F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+paramRangeG;\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "1 + D(f)(x)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&\"\"\"F$*&,&*$)%\"xG\"\"#F$F$F*F$F$F(F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "factor( 1 + D(f)(x)^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&\"\"\"F&*$)%\"xG\"\"#F&F&F*F&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 90 "Int( sqrt(1+D(f)(x)^2), x = parameterRange) \+ = int( sqrt(1+D(f)(x)^2), x = parameterRange);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$*$),&\"\"\"F+*$)%\"xG\"\"#F+F+F/F+#F+F//F.; \"\"!,$*&F/F+%#PiGF+F+,&*&F/F+F6F+F+*(\"\")F+\"\"$!\"\"F6F;F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 20 "Exercise 23 Page 581" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 38 "Calculate the length of the graph of " } {XPPEDIT 18 0 "f(x) = exp(x)/2+exp(-x)/2+7;" "6#/-%\"fG6#%\"xG,(*&-%$e xpG6#F'\"\"\"\"\"#!\"\"F-*&-F+6#,$F'F/F-F.F/F-\"\"(F-" }{TEXT -1 7 " \+ for " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 11 " in [0,1]." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f := x -> exp(x)/2+exp(-x)/2+7; paramRange := 0 .. 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(* &#\"\"\"\"\"#F/-%$expG6#9$F/F/*&F.F/-F26#,$F4!\"\"F/F/\"\"(F/F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+paramRangeG;\"\"!\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "1 + D(f)(x)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$),&*&#F$\"\"#F$-%$expG6#%\"xGF$F$*& #F$F*F$-F,6#,$F.!\"\"F$F4F*F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "J := simplify( 1 + D(f)(x)^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG,(#\"\"\"\"\"#F'*&#F'\"\"%F'-%$expG6#,$*&F(F'%\"xGF'F'F'F' *&F*F'-F-6#,$*&F(F'F1F'!\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "testeq( J = (exp(x)/2 + exp(-x)/2)^2 );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Int( sqrt(1+D(f)(x)^2), x = paramRange) = Int( exp(x)/2 + exp(-x )/2, x = paramRange);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$, &\"\"\"F)*$),&*&#F)\"\"#F)-%$expG6#%\"xGF)F)*&#F)F/F)-F16#,$F3!\"\"F)F 9F/F)F)F./F3;\"\"!F)-F%6$,&*&F.F)F0F)F)*&F.F)F6F)F)F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Int( sqrt(1+D(f)(x)^2), x = paramRa nge) = int( exp(x)/2 + exp(-x)/2, x = paramRange);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*$,&\"\"\"F)*$),&*&#F)\"\"#F)-%$expG6#%\"xG F)F)*&#F)F/F)-F16#,$F3!\"\"F)F9F/F)F)F./F3;\"\"!F),&*&F.F)-F16#F)F)F)* &#F)F/F)-F16#F9F)F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 20 "Exercise 25 Page 581" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 " " {TEXT -1 38 "Calculate the length of the graph of " }{XPPEDIT 18 0 "f(x) = 2*x^4+1/64/(x^2);" "6#/-%\"fG6#%\"xG,&*&\"\"#\"\"\"*$F'\"\"%F+ F+*(F+F+\"#k!\"\"*$F'F*F0F+" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "x; " "6#%\"xG" }{TEXT -1 13 " in [-2,-1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := x -> 2*x^4+1/64/(x ^2); paramRange := -2 .. -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"#\"\"\")9$\"\"%F/F/*&#F/\" #kF/*&F/F/*$)F1F.F/!\"\"F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%+paramRangeG;!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "1 + D(f)(x)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$),&*& \"\")F$)%\"xG\"\"$F$F$*&F$F$*&\"#KF$F*F$!\"\"F0\"\"#F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J := factor( 1 + D(f)(x)^2 );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG,$*(\"%C5!\"\",&\"\"\"F**&\"$c#F *)%\"xG\"\"'F*F*\"\"#F.!\"'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "testeq( J = (1/32*(1+256*x^6)/x^3)^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "I nt( sqrt(1+D(f)(x)^2), x = paramRange) = Int( abs(1/32*(1+256*x^6)/x^3 ), x = paramRange);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$,& \"\"\"F)*$),&*&\"\")F))%\"xG\"\"$F)F)*&F)F)*&\"#KF)F/F)!\"\"F5\"\"#F)F )#F)F6/F0;!\"#F5-F%6$,$*&#F)F4F)-%$absG6#*&,&F)F)*&\"$c#F))F0\"\"'F)F) F)F0!\"$F)F)F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Int( sqrt (1+D(f)(x)^2), x = paramRange) = int( -1/32*(1+256*x^6)/x^3, x = param Range);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$,&\"\"\"F)*$),& *&\"\")F))%\"xG\"\"$F)F)*&F)F)*&\"#KF)F/F)!\"\"F5\"\"#F)F)#F)F6/F0;!\" #F5#\"%$o(\"$c#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "4. How to Make An Arc Length Exercise" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "In this section we will \+ consider how to devise an exercise in which a \"miracle\" happens to e nable the easy evaluation of arc length.\n\nThe idea is to find number s " }{XPPEDIT 18 0 "p,q,a;" "6%%\"pG%\"qG%\"aG" }{TEXT -1 11 " so t hat " }{XPPEDIT 18 0 "1+diff(x^p+a*x^q,x)^2;" "6#,&\"\"\"F$*$-%%diffG 6$,&)%\"xG%\"pGF$*&%\"aGF$)F+%\"qGF$F$F+\"\"#F$" }{TEXT -1 72 " is a perfect square. The key is that the cross term of the summand " } {XPPEDIT 18 0 "diff(x^p+a*x^q,x)^2;" "6#*$-%%diffG6$,&)%\"xG%\"pG\"\" \"*&%\"aGF+)F)%\"qGF+F+F)\"\"#" }{TEXT -1 14 " should be " } {XPPEDIT 18 0 "-1/2;" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 71 ". \+ The first requirement for this to happen is that the derivatives " } {XPPEDIT 18 0 "diff(x^p,x);" "6#-%%diffG6$)%\"xG%\"pGF'" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "diff(x^q,x);" "6#-%%diffG6$)%\"xG%\"qGF'" } {TEXT -1 25 " should be powers of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 87 " that are equal in absolute value but opposite in sign. \+ We can achieve this by taking " }{XPPEDIT 18 0 "p = n+1;" "6#/%\"pG,& %\"nG\"\"\"F'F'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q = 1-n;" "6#/% \"qG,&\"\"\"F&%\"nG!\"\"" }{TEXT -1 9 ". Then " }{XPPEDIT 18 0 "Diff (x^(n+1)+a*x^(1-n),x)^2 = (n+1)^2*x^(2*n)-2*(n^2-1)*a+a^2*(1-n)^2*x^(- 2*n);" "6#/*$-%%DiffG6$,&)%\"xG,&%\"nG\"\"\"F-F-F-*&%\"aGF-)F*,&F-F-F, !\"\"F-F-F*\"\"#,(*&,&F,F-F-F-F3)F**&F3F-F,F-F-F-*(F3F-,&*$F,F3F-F-F2F -F/F-F2*(F/F3,&F-F-F,F2F3)F*,$*&F3F-F,F-F2F-F-" }{TEXT -1 19 " and s o we choose" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " " }{XPPEDIT 18 0 "a = 1/4/(n ^2-1);" "6#/%\"aG*(\"\"\"F&\"\"%!\"\",&*$%\"nG\"\"#F&F&F(F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "n = 3;" "6#/%\"nG\"\"$" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "a = 1/32;" "6#/%\"aG*&\"\"\"F&\"#K!\"\" " }{TEXT -1 24 ". With this choice of " }{XPPEDIT 18 0 "a;" "6#%\"aG " }{TEXT -1 9 " we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Diff(x^(n+1)+a*x^(1-n),x)^2 = \+ (n+1)^2*x^(2*n)-1/2+a^2*(1-n)^2*x^(-2*n);" "6#/*$-%%DiffG6$,&)%\"xG,&% \"nG\"\"\"F-F-F-*&%\"aGF-)F*,&F-F-F,!\"\"F-F-F*\"\"#,(*&,&F,F-F-F-F3)F **&F3F-F,F-F-F-*&F-F-F3F2F2*(F/F3,&F-F-F,F2F3)F*,$*&F3F-F,F-F2F-F-" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "1+Diff(x^(n+1)+a*x^(1-n),x)^2 = (n+1)^2*x^(2*n)+1/2+a^2 *(1-n)^2*x^(-2*n);" "6#/,&\"\"\"F%*$-%%DiffG6$,&)%\"xG,&%\"nGF%F%F%F%* &%\"aGF%)F,,&F%F%F.!\"\"F%F%F,\"\"#F%,(*&,&F.F%F%F%F4)F,*&F4F%F.F%F%F% *&F%F%F4F3F%*(F0F4,&F%F%F.F3F4)F,,$*&F4F%F.F%F3F%F%" }{TEXT -1 5 " , o r" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }{XPPEDIT 18 0 "1+Diff(x^(n+1)+a*x^(1-n),x)^2 = ((n+1)*x^n+a *(1-n)*x^(-n))^2;" "6#/,&\"\"\"F%*$-%%DiffG6$,&)%\"xG,&%\"nGF%F%F%F%*& %\"aGF%)F,,&F%F%F.!\"\"F%F%F,\"\"#F%*$,&*&,&F.F%F%F%F%)F,F.F%F%*(F0F%, &F%F%F.F3F%)F,,$F.F3F%F%F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Calculate the arc length of " }{XPPEDIT 18 0 "y = x^4+1/32/(x^2); " "6#/%\"yG,&*$%\"xG\"\"%\"\"\"*(F)F)\"#K!\"\"*$F'\"\"#F,F)" }{TEXT -1 8 " for " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 12 " in [1,2 ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 8 "Sol ution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x -> x^4+1/32/(x^2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%& arrowGF(,&*$)9$\"\"%\"\"\"F1*&#F1\"#KF1*&F1F1*$)F/\"\"#F1!\"\"F1F1F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "D(f)(x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&*&\"\"%\"\"\")%\"xG\"\"$F(F(*&F(F(*&\"# ;F(F)F(!\"\"F/\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "J \+ := expand(1 + D(f)(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG,(# \"\"\"\"\"#F'*&\"#;F')%\"xG\"\"'F'F'*&F'F'*&\"$c#F'F+F'!\"\"F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "testeq( J = (4*x^3 + 1/16/x^ 3)^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int(sqrt( 1 + D(f)(x)^2 ), x= 1..2) = Int(4 *x^3 + 1/16/x^3, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I ntG6$*$,&\"\"\"F)*$),&*&\"\"%F))%\"xG\"\"$F)F)*&F)F)*&\"#;F)F/F)!\"\"F 5\"\"#F)F)#F)F6/F0;F)F6-F%6$,&*&F.F)F/F)F)*&F)F)*&F4F)F/F)F5F)F8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int(sqrt( 1 + D(f)(x)^2 ), x = 1..2) = int(4*x^3 + 1/16/x^3, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$,&\"\"\"F)*$),&*&\"\"%F))%\"xG\"\"$F)F)*&F) F)*&\"#;F)F/F)!\"\"F5\"\"#F)F)#F)F6/F0;F)F6#\"%B>\"$G\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 306 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "(With " }{XPPEDIT 18 0 "n = 1/6;" "6#/%\"nG*&\"\"\"F &\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "p = 7/6;" "6#/%\"pG*&\" \"(\"\"\"\"\"'!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "q = 5/6;" "6#/% \"qG*&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "a = 1/4/(n^2-1);" "6#/%\"aG*(\"\"\"F&\"\"%!\"\",&*$%\"nG\"\"#F&F&F(F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -9/35;" "6#/%!G,$*&\"\"*\"\"\"\"#N !\"\"F*" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 29 "Calculate the arc length of " }{XPPEDIT 18 0 "y = x^(7/6)-9*x^(5/6)/35;" "6#/%\"yG,&)%\"xG*&\"\"(\"\"\"\"\"'!\"\"F**(\" \"*F*)F'*&\"\"&F*F+F,F*\"#NF,F," }{TEXT -1 8 " for " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 12 " in [1,2]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x -> x^(7/6)-9*x^(5/6)/35;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$#\"\" (\"\"'\"\"\"F3*&#\"\"*\"#NF3*$)F/#\"\"&F2F3F3!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "D(f)(x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&*(\"\"(\"\"\"\"\"'!\"\"%\"xG#F(F)F(*(\"\"$F(\"#9F* F+#F*F)F*\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "J := ex pand(1 + D(f)(x)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG,(#\"\" \"\"\"#F'*(\"#\\F'\"#O!\"\"%\"xG#F'\"\"$F'*(\"\"*F'\"$'>F,F-#F,F/F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "expand( J - (7*x^(1/6)/6 + 3/x^(1/6)/14)^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Int(sqrt( 1 + D(f)(x)^2 ), x = 1..2) = Int(7*x^(1/6)/6 + 3/x^(1/6)/14, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$,&\"\"\"F)*$),&*(\"\"(F)\"\"'!\"\"%\"x G#F)F/F)*(\"\"$F)\"#9F0F1#F0F/F0\"\"#F)F)#F)F7/F1;F)F7-F%6$,&*(F.F)F/F 0F1F2F)*(F4F)F5F0F1F6F)F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Int(sqrt( 1 + D(f)(x)^2 ), x= 1..2) = int(7*x^(1/6)/6 + 3/x^(1/6)/ 14, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$,&\"\" \"F)*$),&*(\"\"(F)\"\"'!\"\"%\"xG#F)F/F)*(\"\"$F)\"#9F0F1#F0F/F0\"\"#F )F)#F)F7/F1;F)F7,(*&F7F))F7F2F)F)*(\"\"*F)\"#NF0F7#\"\"&F/F)#\"#WF@F0 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "5. Surface Area of a Cone" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "A cone of height " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 21 " and base ra dius " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 8 " has " }{TEXT 308 12 "slant height" }{TEXT -1 4 " " }{XPPEDIT 18 0 "S = sqrt(h^2+ R^2);" "6#/%\"SG-%%sqrtG6#,&*$%\"hG\"\"#\"\"\"*$%\"RGF+F," }{TEXT -1 56 ".\n\n " }{OLE 1 96268 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:J;j:J:ZJJ;j:JyyyyyI<:^:vyyyyy=: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::u yyyyyo?wAwyyyqxK=j=J>j>J?j?J@j@>:W:YJ:><:y ;;\\:FBNBVB^BfBnBvB>CFCNCVC^CfCnCvC>DIFINIVI^IfInIvI>J>ELq^H_mvJ::::::::gjR<:T><::;s;cWZAYAKX:A:;R:B:F :AlqfG[maNFO=;::::::::_J;@J@Wlj^HMMufF;J:: :::::N=?R:=:AZ:vYxY:B::::::f:;J<`p:J:F::];MLtfFUMajpfF;J::::::JINZ;vyy yyy=J:B::::::N:;j:wAwAjy;::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::j:nyyYZDjysyOj;B:;JZC:bKIZ:> d::::::::::j:b:B::::::::::::::::::::: 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::;Z ::::::::::::::::::::::::::::::::::::::::::::::FZ:vc:^g:ff:J::::::::::O B:?jyyiy=J:B::::::VZ:>:cB:>::::::::::::jysy:>:<:::::::::::::::::::vYxI :;Z::::::::::::::::::::yay=J:B::::::::::3:" }{TEXT -1 11 "\n \+ " }}{PARA 0 "" 0 "" {TEXT -1 242 "We can calculate the surface area of a cone (without its base) by slicing along an edge and flattening out the surface into a planar sector of a circle. 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4011000000000000BFF1AA3587625F663FE95062CE5E9DDC4011000000000000BFF36E 6E14EC35BD3FEB9D830FAD094C4011000000000000BFF532A6A2760C143FEDEAA350FB 74BD4011000000000000BFF6F6DF2FFFE26A3FF01BE1C924F0174011000000000000BF F8BB17BD89B8C13FF14271E9CC25CF4011000000000000BFFA7F504B138F183FF26902 0A735B884011000000000000BFFC4388D89D656F3FF38F922B1A914140110000000000 00BFFE07C166273BC63FF4B6224BC1C6F94011000000000000BFFFCBF9F3B1121D3FF5 DCB26C68FCB14011000000000000C000C819409D743A3FF703428D10326A4011000000 000000C001AA3587625F653FF829D2ADB768224011000000000000C0028C51CE274A91 3FF95062CE5E9DDB4011000000000000C0036E6E14EC35BC3FFA76F2EF05D393401100 0000000000C004508A5BB120E73FFB9D830FAD094B4011000000000000C00532A6A276 0C13-%&COLORG6&%$RGBG$\"\"#!\"\"$\"\"&F3$\"\"%F3-%&STYLEG6#%%LINEG-%(S CALINGG6#%,CONSTRAINEDG-%+PROJECTIONG6%$\"#C\"\"!$\"#')FE\"\"\"" 1 2 0 1 10 0 2 1 1 1 1 1.000000 86.000000 24.000000 1 0 "Curve 1" "Curve 2 " }}}}{PARA 0 "" 0 "" {TEXT -1 50 " \+ " }{TEXT 310 51 "Figure 3: The Cone Flattened into a Ci rcular Sector" }}{PARA 0 "" 0 "" {TEXT -1 5 "\nIf " }{XPPEDIT 18 0 "t heta;" "6#%&thetaG" }{TEXT -1 86 " is the angle of the sector, then t he area of the sector, which is the surface area " }{XPPEDIT 18 0 "si gma;" "6#%&sigmaG" }{TEXT -1 84 " of the cone is given by\n\n \+ " }{XPPEDIT 19 1 "sig ma = theta*S^2/2;" "6#/%&sigmaG*(%&thetaG\"\"\"*$%\"SG\"\"#F'F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "But the arc length of the sector is given by both " } {XPPEDIT 18 0 "theta*S;" "6#*&%&thetaG\"\"\"%\"SGF%" }{TEXT -1 8 " an d " }{XPPEDIT 18 0 "2*Pi*R;" "6#*(\"\"#\"\"\"%#PiGF%%\"RGF%" }{TEXT -1 22 ", so we deduce that " }{XPPEDIT 18 0 "theta = 2*Pi*R/S;" "6#/ %&thetaG**\"\"#\"\"\"%#PiGF'%\"RGF'%\"SG!\"\"" }{TEXT -1 31 ". Substi tuting this value of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 21 "into the formula for " }{XPPEDIT 18 0 "sigma;" "6#%&sigmaG" } {TEXT -1 13 " results in \n" }}{PARA 0 "" 0 "" {TEXT -1 52 " \+ " }{XPPEDIT 18 0 "sigma = Pi *R*S;" "6#/%&sigmaG*(%#PiG\"\"\"%\"RGF'%\"SGF'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 52 "\nor, in a form that is suitable for gene ralization, " }}{PARA 0 "" 0 "" {TEXT -1 24 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 52 ". \+ " }{XPPEDIT 18 0 "sigma = 2*Pi;" "6#/%&sigmaG*&\"\"#\" \"\"%#PiGF'" }{TEXT -1 1 " " }{TEXT 312 1 "(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(0+R)/2;" "6#*&,&\"\"!\"\"\"%\"RGF&F&\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 311 1 ")" }{TEXT -1 1 " " }{XPPEDIT 18 0 "S;" "6 #%\"SG" }{TEXT -1 1 " " }{TEXT 313 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Thinking of the 0 in this formula a s the radius at the vertex of the cone, we can state the formula in wo rds as\n\n " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 314 78 "T he surface area of a cone is the average circumference times the slant height" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "6. Surface \+ Area of a Frustum of a Cone" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The frustum of a cone is the surface that lies \+ between the base and a slice parallel to the base." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ " }{OLE 1 198156 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:J j;jyyyyyI:::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::uyyyyy=V:nYnyyyYuAj=J>>:M:O:Q:S:UJ:n; v;;JB>Z:F=F=N=V=^=f=n=v=>>AFANAVA^AfAnA;jYJZB:=EFENEVE^EfEnEvE>F:::::::JEf:yyyxI N::;`:Z@[::J:Iy=DrkqK^Aj;J:@:<:=j[vGUMrvC?MoJ::::::::JCNZ;^:vYxI >:<::::::j`<:;:=ja^GEmR>@Wlj^HMMufF;J:::::::N=?R:=:AB::wAwAjy ;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::j:nyyYZDjysyOj;B:;JZC:bKIZ:>d::::::::::j:b:B::::::::::::::::::::::::::::::::::::::::::::::::::: ::=^:NJVJ^JfJnJvJ>KFKNK_>a>c>e>g>i>kN:FMNMVM^MfMnMvM> n:B:=???A?C?E?G?I?K?M?O?Q?S?U?W?Y?[_:FPNPVP^PfPnPvP>QFQNQVQ^QfQnQvQ>R< 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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 118 "The formula for the surface area of a co ne, in the verbal form stated above, remains valid for the frustum of a cone:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{TEXT 317 91 "The surface area of a frustum of a c one is the average circumference times the slant height" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "That \+ is" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 52 " " }{XPPEDIT 18 0 " sigma = 2*Pi;" "6#/%&sigmaG*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 " " } {TEXT 319 1 "(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(r+R)/2;" "6#*&,&%\"rG \"\"\"%\"RGF&F&\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 318 1 ")" }{TEXT -1 1 " " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 1 " " }{TEXT 320 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "We prove this formula, referring to the f igure below." }}{PARA 0 "" 0 "" 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