{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 } 3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 257 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 36 "Lab 2: Areas, Volumes, an d Arclength" }}{PARA 256 "" 0 "" {TEXT 277 10 "Section 4 " }{TEXT -1 37 "is most relevant to the assignment. " }}{PARA 257 "" 0 "" {TEXT -1 156 "Sections 1 and 2 illustrate the use of the \"solve\" and \"fso lve\" commands and how to use these with other Maple commands to solve area and volume problems. " }}{PARA 0 "" 0 "" {TEXT -1 133 "Section \+ 3 illustrates how to define parametric functions, how to plot them, an d how to determine arc-lengths of parametric curves. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "1. Solving Equations with \"solve\" (are a between curves)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Here we use \+ Maple's ability to solve equations to find the points of intersection \+ between two curves." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 278 15 "Sa mple Problem:" }{TEXT -1 57 " Find the area enclosed between the curve s described by " }{XPPEDIT 18 0 "y = 3-2*x;" "6#/%\"yG,&\"\"$\"\"\"*& \"\"#F'%\"xGF'!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = x^6+2*x^ 5-3*x^4+x^2;" "6#/%\"yG,**$%\"xG\"\"'\"\"\"*&\"\"#F)*$F'\"\"&F)F)*&\" \"$F)*$F'\"\"%F)!\"\"*$F'F+F)" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "First we define the two functions by:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "f1 := x -> 3 - 2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"$\"\"\"*&\"\"#F.9 $F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f2 := x - > x^6 + 2*x^5 - 3*x^4 + x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G f*6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"'\"\"\"F1*&\"\"#F1)F/\" \"&F1F1*&\"\"$F1)F/\"\"%F1!\"\"*$)F/F3F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Now if we naively \+ try to plot these two curves over an arbitrary domain:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot([f1(x),f2(x)], x=-5..5, color \+ = [red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 333 255 255 {PLOTDATA 2 " 6&-%'CURVESG6$7S7$$!\"&\"\"!$\"#8F*7$$!3YLLLe%G?y%!#<$\"3qmmm\"p0kD\"! #;7$$!3OmmT&esBf%F0$\"3GLL3f_PF0$\"3wmmT!R=00\"F37$$!3K++vo1YZNF0$\"32++vL@\\45F37$$!3;LL3 -OJNLF0$\"3?nm;/siq'*F07$$!3p***\\P*o%Q7$F0$\"3Q****\\(y$pZ#*F07$$!3Km mm\"RFj!HF0$\"3jKLL$yaE\"))F07$$!33LL$e4OZr#F0$\"3Glmm\">s%H%)F07$$!3u *****\\n\\!*\\#F0$\"3]******\\$*4)*zF07$$!3%)*****\\ixCG#F0$\"3o****** \\_&\\c(F07$$!3#******\\KqP2#F0$\"3%)******\\1aZrF07$$!39LL3-TC%)=F0$ \"3Imm;/#)[onF07$$!3[mmm\"4z)e;F0$\"3%HLLL=exJ'F07$$!3Mmmmm`'zY\"F0$\" 3lKLLL2$f$fF07$$!3#****\\(=t)eC\"F0$\"3%)****\\PYx\"\\&F07$$!3!ommmh5$ \\5F0$\"3gLLLL7i)4&F07$$!3S$***\\(=[jL)!#=$\"3o)***\\P'psm%F07$$!3)f** *\\iXg#G'F[r$\"3?****\\74_cUF07$$!3ndmmT&Q(RTF[r$\"3`JLL3x%z#QF07$$!3% \\mmTg=><#F[r$\"3*HLL3s$QMMF07$$!3vDMLLe*e$\\!#?$\"3&omm;zr)4IF07$$\"3 !=nm\"zRQb@F[r$\"3klm;/K#*oDF07$$\"3_,+](=>Y2%F[r$\"3q****\\ih2&=#F07$ $\"3summ\"zXu9'F[r$\"31lmmT3^q7$$\"3iKLL$Qx$o;F 0$!3i_mmmwanLF[r7$$\"3Y+++v.I%)=F0$!3U4+++v+'o(F[r7$$\"3?mm\"zpe*z?F0$ !3UKL$eR<*f6F07$$\"3;,++D\\'QH#F0$!3K-++])Hxe\"F07$$\"3%HL$e9S8&\\#F0$ !3!fmm\"H!o-*>F07$$\"3s++D1#=bq#F0$!3X,+]7k.6CF07$$\"3\"HLL$3s?6HF0$!3 #emmmT9C#GF07$$\"3a***\\7`Wl7$F0$!33****\\i!*3`KF07$$\"3enmmm*RRL$F0$! 3;NLLL*zym$F07$$\"3%zmmTvJga$F0$!3'eLL$3N1#4%F07$$\"3]MLe9tOcPF0$!3,pm ;HYt7XF07$$\"31,++]Qk\\RF0$!37-+++xG**[F07$$\"3![LL3dg6<%F0$!3gpmmT6KU `F07$$\"3%ymmmw(GpVF0$!3qNLLLbdQdF07$$\"3C++D\"oK0e%F0$!3[++]i`1hhF07$ $\"35,+v=5s#y%F0$!3A-+]P?WllF07$$\"\"&F*$!\"(F*-%'COLOURG6&%$RGBG$\"*+ +++\"!\")$F*F*Fc[l-F$6$7fn7$F($\"%DvF*7$$!3Gmm;HU,\"*[F0$\"3)3-/^mF*)R '!#97$F.$\"3\"[DR-/l5T&F_\\l7$$!3!***\\(=_+so%F0$\"3yKU/(RXMl%F_\\l7$F 5$\"3!=**[]qZ>)RF_\\l7$F:$\"3))>+@PK5ZFF_\\l7$F?$\"3Y:f&)zd$e#=F_\\l7$ FD$\"3A\"))Q6!z>i6F_\\l7$FI$\"3_]SR1GeLs!#:7$FN$\"35X&G29J&oSFg]l7$FS$ \"3M!=[%[->5>Fg]l7$FX$\"3$H!*Gg3;$ReF37$Fgn$!3%QgSznSgw\"F37$F\\o$!3'> 0Yu$*\\&=]F37$Fao$!3)=odD\\S@@'F37$Ffo$!3mr>=#p?9(eF37$F[p$!3)eO*[CTON [F37$F`p$!3)4fDCF37$Fjp$!3#=Tv!R4HS:F37$F_q$!3 .wc#)RzF07$Fdq$!3s8K@Z#o_u$F07$Fiq$!3m9$ymb\"\\B7F07$F_r$!3ih+fX!p% p?F[r7$Fdr$\"3OYmW*G$Q)R'Ffu7$Fir$\"3%pZ9\\B*\\jRFfu7$F^s$\"3zw(Gb6GhV #!#A7$Fds$\"3[\\>3KFF,TFfu7$Fis$\"3Ur\"pkV8P5\"F[r7$F^t$\"3y%e!ROWC!z \"F[r7$Fct$\"3SezKj#fyx$F[r7$Fht$\"3S$yZ'R`Ye7F07$F]u$\"3o5)f)*zxg'RF0 7$Fbu$\"3mDu`NR%p<\"F37$Fhu$\"375a^u$Rep#F37$F]v$\"3(*e0%zJ>,!eF37$Fbv $\"3;XT3@I1q5Fg]l7$Fgv$\"3/xr/*=K!\\>Fg]l7$F\\w$\"3#zg#HXJqYKFg]l7$Faw $\"3G.)R4X&)pG&Fg]l7$Ffw$\"3![?RQGD&*>)Fg]l7$F[x$\"39]D#Hl:ZD\"F_\\l7$ F`x$\"3J'*=I@-^P=F_\\l7$Fex$\"3aamq*>nxk#F_\\l7$Fjx$\"3_%[w/qj>s$F_\\l 7$F_y$\"3fV`C#eHS+&F_\\l7$$\"3%zm;/@-/1%F0$\"3ABFWMP\")*)eF_\\l7$Fdy$ \"30CO0Z1J,pF_\\l7$$\"3K,+voTAqUF0$\"3.Tct%y5P#zF_\\l7$Fiy$\"3aRS(=%z? o!*F_\\l7$$\"3/M$eRA5\\Z%F0$\"3YeN[!ooN/\"!#87$F^z$\"3u*3UOTUp>\"F]fl7 $$\"3m+++]oi\"o%F0$\"37SVfJfvg8F]fl7$Fcz$\"3@!=F%phpU:F]fl7$$\"3Q+D1k2 /P[F0$\"3UU7s-D[[;F]fl7$$\"3a+]P40O\"*[F0$\"3]jV\\&*H?g " 0 "" {MPLTEXT 1 0 49 "plot([f1(x),f2(x)], x=-3..1, color = [red,blue]);" }} {PARA 13 "" 1 "" {GLPLOT2D 333 240 240 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $!\"$\"\"!$\"\"*F*7$$!3cLLL$Q6G\"H!#<$\"37nmmmFiD))F07$$!3bmm;M!\\p$GF 0$\"35LLLo!)*Qn)F07$$!37LLL))Qj^FF0$\"3AmmmwxE.&)F07$$!3ALLL=KvlEF0$\" 3YmmmOk]J$)F07$$!3wmm;C2G!e#F0$\"3kKLL[9cg\")F07$$!39LL$3yO5]#F0$\"3Gm mmhN2-!)F07$$!3&*****\\nU)*=CF0$\"3!******\\`oz$yF07$$!3iLL$3WDTL#F0$ \"3Cnmm\")3DowF07$$!3))****\\d(Q&\\AF0$\"3u*****\\^x!*\\(F07$$!3gmmmc4 `i@F0$\"3BLLL8>1DtF07$$!3KLLLQW*e3#F0$\"3kmmmw))yrrF07$$!33+++q)>'**>F 0$\"3;+++S(R#**pF07$$!3.+++]5*H\">F0$\"30++++@)f#oF07$$!3,+++I\"3&H=F0 $\"3e******fi,fmF07$$!3OLL$3k(p`9F0$\"3:MLL$\\[%ReF07$$!3\"*****\\FRXL8F 0$\"3#)*****\\&y!pm&F07$$!3)*****\\#=/8D\"F0$\"3R+++l$3E]&F07$$!3immmT &*el6F0$\"3oLLL$3z6L&F07$$!3omm;Wn(o3\"F0$\"3OLLL)[`P<&F07$$!3PLLLeV(> +\"F0$\"3ummm;([R+&F07$$!3hOLL3k%y8*!#=$\"3Knmm\"Gpv#[F07$$!31-++DB:q$ )Fcs$\"3S+++l/.uYF07$$!3XNLL$o@5a(Fcs$\"34nmmOV?3XF07$$!3S,+++'[Wo'Fcs $\"3G+++?(*)oL%F07$$!3;/++]*ek%eFcs$\"3#3+++z\"HpTF07$$!39.++v3mN]Fcs$ \"3j+++v@82SF07$$!3b.++]ySNTFcs$\"3r+++q:3FQF07$$!3[pmmm/\\ELFcs$\"3!R LLL4)HlOF07$$!3y++++&)ziCFcs$\"3;++++(fD\\$F07$$!3;NLL3_;!o\"Fcs$\"3.n mmTI.OLF07$$!3q1+++ISX#)!#>$\"37+++g!3\\;$F07$$!3)p3nm;%RY>!#?$\"3F07$$\"3%*)******Rv&)z&Fcs$\"3@+++?\\GS=F07$$\"3+LLL$GUY o'Fcs$\"3SLLLV:2j;F07$$\"3=lmmm5:xuFcs$\"3'pmmmypX]\"F07$$\"3&4++]sI@K )Fcs$\"3\")*****\\&QdN8F07$$\"3)3++]2%)38*Fcs$\"3#)*****\\=BQ<\"F07$$ \"\"\"F*Fgz-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F`[l-F$6$7]oF'7$$!3iLL $e%G?yHF0$\"3G.6^ei\"y/#F07$$!3ymmm\"p0k&HF0$!3Bi50<2wNWF07$$!3&***** \\P&3Y$HF0$!3=]gF=Y+Z5!#;7$F.$!3'4],A:#Q2;Fc\\l7$$!3-n;/'zXQ*GF0$!3)[k XHag81#Fc\\l7$$!31++v3-)[(GF0$!3_K/P)H+`[#Fc\\l7$$!33L$e9i9f&GF0$!393b )>:V.)GFc\\l7$F4$!3KCv#)*>-wC$Fc\\l7$$!3(HL3xC?c\"GF0$!3')fw0*yz(GOFc \\l7$$!3#)***\\7Y\"H%z#F0$!3ki,Jo2pxRFc\\l7$$!3qm;zuE'Hx#F0$!3E4%[\"eF z&H%Fc\\l7$F9$!3m4@mB6#3&Fc\\l7$F> $!3Upr^l\\!oZ&Fc\\l7$$!3A++Drp,BEF0$!3a&ev%HcRxdFc\\l7$FC$!3#)[i^EDG&* fFc\\l7$$!3t****\\_(e1a#F0$!3%o=nBH'*48'Fc\\l7$FH$!3[m/5+`Tl#zY]hFc\\l7$FR$!3C7S5S!p\\/'Fc\\l7$FW$!3C\\asM[$)QdFc\\l7$ Ffn$!3'y.%QN/nC`Fc\\l7$F[o$!3NG%)zI;(\\!\\Fc\\l7$F`o$!31'ee-4>xR%Fc\\l 7$Feo$!3#4UztV9Y(QFc\\l7$Fjo$!3EQob_dlvLFc\\l7$F_p$!3)33R))QH&QHFc\\l7 $Fdp$!3M&o0Z?k%\\CFc\\l7$Fip$!3e?5j^Eun?Fc\\l7$F^q$!3eR[\\oSWm;Fc\\l7$ Fcq$!3]^f$>!3&>N\"Fc\\l7$Fhq$!3crvZ#H!p^5Fc\\l7$F]r$!3Aq)>R$Q&e3)F07$F br$!3_2DLnV&R(fF07$Fgr$!3'*\\T**p_0!R%F07$F\\s$!3`>\"*)p'osFIF07$Fas$! 3'oc4j*3t[>F07$Fgs$!3%G2Tch(p\\7F07$F\\t$!3)Q?.91)=`qFcs7$Fat$!3-ZGd>t @)H$Fcs7$Fft$!3%4SvHK:P0\"Fcs7$F[u$\"3]3Hm4Fq@7Ffv7$F`u$\"3_cbe(R)*)3k Ffv7$Feu$\"3ksXa/d,8nFfv7$Fju$\"3%3!H`K(>G![Ffv7$F_v$\"3C`twkPNfDFfv7$ Fdv$\"3=&y<$HLp_mF\\w7$Fjv$\"3Y>M:RrS)y$!#B7$F`w$\"3dmjKF<%)GmF\\w7$Fe w$\"3@2FQn^$>^#Ffv7$Fjw$\"3*>_J?*G%*>`Ffv7$F_x$\"3ij`+4Ohw$)Ffv7$Fdx$ \"3WV/'32`89\"Fcs7$Fix$\"3eQn?R`?99Fcs7$F^y$\"3AJN(*Q:'>m\"Fcs7$Fcy$\" 3b%f(>!H/+/#Fcs7$Fhy$\"3c;^PTHZNEFcs7$F]z$\"3=w,6X?cTQFcs7$Fbz$\"3#33S T!yHtfFcsFfz-Fjz6&F\\[lF`[lF`[lF][l-%+AXESLABELSG6$Q\"x6\"Q!Fiil-%%VIE WG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "It is clear now that f1(x) is the upper curve and the enclosed ar ea occurs between x = -3 and x = 1. How did I get these values? Answer : With Maple's \"solve\" command: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sols := solve(f1(x) = f2(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG6(\"\"\"!\"$,&*$-%%sqrtG6#\"\"#F&#F&F-*&^#F.F&F *F&F&,&F/F&*&#F&F-F&F)F&!\"\",&F)#F4F-*&^#F6F&F*F&F&,&F7F&*&F.F&F*F&F& " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 310 "\"sols\" represents a list o f solutions to the equation f1(x) = f2(x). This is alot of solutions \+ but most of them are imaginary. The symbol \"I\" represents the square root of -1 so the solutions with an \"I\" are imaginary. It may be e asier to visualize these roots by evaluating them as floating point nu mbers by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "evalf(sols,2); \+ # this gives 2 significant terms. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ($\"\"\"\"\"!$!\"$F%^$$\"#q!\"#F)^$$!#qF+F)^$F-F-^$F)F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 301 "You can see here that the roots of inter est are the first two. In general, Maple will list the real roots fir st and the imaginary roots afterwards. We will want to integrate from left to right so we set \"a\" equal the lesser of the two real roots \+ and \"b\" equal to the greater of the two real roots by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a := sols[2]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "This accesses the second term in the list \"sols\". We access the first t erm in the list with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b : = sols[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Now we integrate f1 - f2 over this interv al" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(f1(x) - f2(x), x \+ = a .. b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%_\"*\"$0\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "And this is the area enclosed by t he two curves. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 59 "2. Solving E quations with \"fsolve\" (volume using washers) " }}{PARA 0 "" 0 "" {TEXT -1 624 "In this section we look at calculating volumes by washer s and specifically doing problems that are difficult to do by hand. T his means we will once again need to use the equation solving options \+ in Maple. In this example it is impossible to find the points of inter section by the \"solve\" command so we use Maple's version of Newton's method to find the root of equations. This is called \"fsolve\". We will use this in the context of a volume problem. Suppose we wish to calculate the volume generated by rotating the region enclosed by two curves, f1(x) and f2(x), around the x-axis. The two curves are descr ibed by: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f1 := x -> x^4*(2-x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1Gf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&)9$\"\"%\"\"\",&\"\"#F0F.!\"\"F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f2 := x -> exp(.5*x)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#,$9 $$\"\"&!\"\"\"\"\"F5F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([f1(x),f2(x)],x=0..2,color = [red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 343 218 218 {PLOTDATA 2 "6&-%'CURVESG6$7[o7$$\"\"!F)F(7$$ \"39LLLL3VfV!#>$\"3c+w`8C2mq!#B7$$\"3'pmm;H[D:)F-$\"3sF\")e%*=![Z)!#A7 $$\"3LLLLe0$=C\"!#=$\"3u9B:\"3y5Y%!#@7$$\"3ILLL3RBr;F:$\"37:=$foA)H9!# ?7$$\"3Ymm;zjf)4#F:$\"3-IZ%*RU=sMFC7$$\"3=LL$e4;[\\#F:$\"3;<2(o(GU\"y' FC7$$\"3p****\\i'y]!HF:$\"3tUi5GLe<7F-7$$\"3,LL$ezs$HLF:$\"39a&pU6Q$[? F-7$$\"3_****\\7iI_PF:$\"34'*[Fqh&4A$F-7$$\"3#pmmm@Xt=%F:$\"3m[]c8KQh[ F-7$$\"3QLLL3y_qXF:$\"3$**z8#R@7LnF-7$$\"3i******\\1!>+&F:$\"3uyIg\"Ht !)Q*F-7$$\"3()******\\Z/NaF:$\"3KQQff+$4F\"F:7$$\"3'*******\\$fC&eF:$ \"3CR/IJ!>(f;F:7$$\"3ELL$ez6:B'F:$\"3!)Rf,3e:w?F:7$$\"3Smmm;=C#o'F:$\" 3kH,0,%Q`l#F:7$$\"3-mmmm#pS1(F:$\"3*>tn?U)>@KF:7$$\"3]****\\i`A3vF:$\" 3#Ros02V)pRF:7$$\"3slmmm(y8!zF:$\"3K/Cn>Mr:ZF:7$$\"3V++]i.tK$)F:$\"3+4 *)*=\\[\\i&F:7$$\"39++](3zMu)F:$\"3s'>*4?#=(ylF:7$$\"3#pmm;H_?<*F:$\"3 y0@VimBjwF:7$$\"3emm;zihl&*F:$\"3z[\"pqo?ht)F:7$$\"39LLL3#G,***F:$\"3) 3BmY8//(**F:7$$\"3Fes7$$\"3_mmmwanL8Fes$\"3A[8!*[p23@Fes7$$\"3'******\\2goP\" Fes$\"3E&oD.rn%RAFes7$$\"3CLLeR<*fT\"Fes$\"3yV#[\"oT!yM#Fes7$$\"3'**** **\\)Hxe9Fes$\"3EKY\")\\m#4X#Fes7$$\"3Ymm\"H!o-*\\\"Fes$\"3\"pF&eY8gHD Fes7$$\"3))***\\7k.6a\"Fes$\"3i62dEFes7$$\"3\"****\\i!*3`i\"Fes$\"3#Gk&*\\)3n9EFes7$$\"3 am;z\\%[gk\"Fes$\"31s(HbY\\%)f#Fes7$$\"3QLLL$*zym;Fes$\"3!QP_v.H=d#Fes 7$$\"3KLL3sr*zo\"Fes$\"3.kSP)[fI`#Fes7$$\"3GLL$3N1#4Ic@Fes7$$\"3TLL3U/37=Fes $\"3wme&='z>E?Fes7$$\"3lmm;9@BM=Fes$\"3q_BQmgOw=Fes7$$\"3'****\\P$[/a= Fes$\"3\\#o!>BxkCFes$\"3A'=K=^a38\"Fes7$$\"3/+++q `KO>Fes$\"3kK2eH$p6&*)F:7$$\"3/++v.Uac>Fes$\"3?Y[l(f[!ojF:7$$\"3/+D\"G :3u'>Fes$\"3#yLuxK,I)[F:7$$\"3-+](=5s#y>Fes$\"3cdqq&GNyK$F:7$$\"3-+v$4 0O\"*)>Fes$\"3ACbzX.x+&\\P*R$F:7$Fap$\"3?A$G01]dl$F:7$Ffp$\"3?U\"H.#p*p'RF:7$ F[q$\"3/J`&fo2iB%F:7$F`q$\"3)fYdFH**eb%F:7$Feq$\"3LpS?a_'[%[F:7$Fjq$\" 3)[;8&[1^o^F:7$F_r$\"3qBB2;RD$[&F:7$Fdr$\"388L))*o>'=eF:7$Fir$\"3'*Hg0 lX)H8'F:7$F^s$\"3/(*eh)zw!zkF:7$Fcs$\"3!H,G!GGVYoF:7$Fis$\"3I8,?/\"yG< (F:7$F^t$\"3rS>jrpbKvF:7$Fct$\"39.=v7P07zF:7$Fht$\"3sm`OA\\E\"H)F:7$F] u$\"3'*)eT_<6em)F:7$Fbu$\"3]K%)H1Jn!4*F:7$Fgu$\"3[xkl;Fn![*F:7$F\\v$\" 3c$RAPIze!**F:7$Fav$\"3hLeRd*=*H5Fes7$Ffv$\"3Mh'*f?z!Q2\"Fes7$F[w$\"3^ **33Q,(f6\"Fes7$F`w$\"35?.C'Qe4;\"Fes7$Fjw$\"3o(p,G?ne?\"Fes7$F^y$\"3E vYezG)QD\"Fes7$Fhy$\"3^8U**za6,8Fes7$Fbz$\"3+^n_#[T/N\"Fes7$Fgz$\"3wf* =(>KS+9Fes7$Fa[l$\"3='G]\"H'[sW\"Fes7$F[\\l$\"3*>O4Z;k?]\"Fes7$Fe\\l$ \"3e[Xr/78_:Fes7$F_]l$\"335')HYrh1;Fes7$Fi]l$\"3!*=xB3j&)f;Fes7$F]_l$ \"34X!f%G=G= " 0 "" {MPLTEXT 1 0 21 "solve(f1(x) = f2(x));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 220 "This results in no output. Maple couldn't find the so lutions. So we find the value of \"b\" by using Maple's version of Ne wton's method (called \"fsolve\") which finds a solution given an init ial guess. We look for \"b\" by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b := fsolve(f1(x) = f2(x), x = .7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+t!RB+)!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "Here, the initial guess is x = .7, and Maple found the s olution near this initial guess. To find the other solution we just g ive an initial guess closer to the greater solution by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "c := fsolve(f1(x) = f2(x),x = 1.8); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"+3J;u=!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Now we can plot both functions over the a ppropriate interval by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p lot([f1(x),f2(x)],x=0..c,color = [red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 342 252 252 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!F)F(7$$\"3+ :b&)>A9&3%!#>$\"3Oi*>/0tiX&!#B7$$\"3mX]@AEgRwF-$\"3!zr#pg\\P_l!#A7$$\" 3u(o2%4lpj6!#=$\"3![y_kncUX$!#@7$$\"3)R\\=zY#3m:F:$\"3xWZ:a&e)36!#?7$$ \"3At#3i&fbm>F:$\"3+:s8.w_FC7$$ \"3oTxaFcHAFF:$\"3'o-)[Zr9*[*FC7$$\"3=D[OLQ*)>JF:$\"3=\"H-*olJ*f\"F-7$ $\"3LaLpOp@;NF:$\"3Ja(o<3i(>DF-7$$\"3Tn%oF'R)Q#RF:$\"3h;?mM(f5\"QF-7$$ \"3]9KB,t&HG%F:$\"3y*R9&**>m)G&F-7$$\"3;6c0M))=(o%F:$\"3Ye;Qvf-\"R(F-7 $$\"3-U&*Q!=!3$4&F:$\"3O?EPo3-.5F:7$$\"3gI)>CqJU[&F:$\"3-bEbWj688F:7$$ \"3AJ)3uvM%ReF:$\"3ovL\"y%z]Y;F:7$$\"3'Rylda0=E'F:$\"3HoW-3c;7@F:7$$\" 3qT;(f+4'>mF:$\"3%GCf3W,#pDF:7$$\"3p8PZ!\\>e.(F:$\"3!R\")pl/-p<$F:7$$ \"3>.1T$HOUS(F:$\"3143G#o%p&y$F:7$$\"30%[ar\"zW3yF:$\"39*\\*zq=HKXF:7$ $\"3w.5oqHN$>)F:$\"3NdQk;Zv?`F:7$$\"3KiU%[,h\\f)F:$\"3GM$GdiSSA'F:7$$ \"3W*>\\yciP'*)F:$\"3@/:k6(f\\7(F:7$$\"3yOU7m[ch$*F:$\"3o3*RJ>J4<)F:7$ $\"3+TGt`'pZx*F:$\"3[')Gi`bnM$*F:7$$\"3^9OgjlW85!#<$\"3(Gg\"*)\\jpS5Fj s7$$\"3C:<6:ZH_5Fjs$\"3]B'>xwW?;\"Fjs7$$\"3'Q%R\"plGC4\"Fjs$\"3o-fBSrc #H\"Fjs7$$\"3rRo3*)=pJ6Fjs$\"3y9\\:_9DC9Fjs7$$\"3;+>4&3\"op6Fjs$\"3!RX VHDGUb\"Fjs7$$\"3C5\\F5;'=@\"Fjs$\"3]O+'\\jl)*p\"Fjs7$$\"3s!\\1#)oi(\\ 7Fjs$\"3[tLG>NCI=Fjs7$$\"3kc:sy,B!H\"Fjs$\"3?.G(\\E=p'>Fjs7$$\"3Q8*z(R (**oK\"Fjs$\"3CbGUDjc'3#Fjs7$$\"3!*=qrb#*)pO\"Fjs$\"3IG84DJS5AFjs7$$\" 3q,['fO5ZS\"Fjs$\"3\"f.h')Q(zc\"Fjs$\"3S-7 *))o#G2EFjs7$$\"3gbM;85z\"e\"Fjs$\"34CCFu87=EFjs7$$\"3WpbMVdm,;Fjs$\"3 m>e&y_69i#Fjs7$$\"3eSss+fP@;Fjs$\"3'*HCWnZj;EFjs7$$\"3t6*3\"eg3T;Fjs$ \"3\"z3&*GD\\Kg#Fjs7$$\"3C5\\sXw>f;Fjs$\"3UTDDrb#Ge#Fjs7$$\"3w34ML#4tn \"Fjs$\"3[i^!R,2Tb#Fjs7$$\"3b0+%*z]#)=H[TM$=Fjs$\"3g#p^\"4p1#)=Fjs7$ $\"3ir)faH-Q&=Fjs$\"3R+&fy&)4ms\"Fjs7$$\"33+++3J;u=Fjs$\"3o)*\\>R6__:F js-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$F+$\"3YsO$>NuN1#F-7$F 2$\"3'*\\TnkNp$*QF-7$F8$\"3C:JT8w3\"*fF-7$F?$\"3KB(fmL]^9)F-7$FE$\"3=' f$eJRCL5F:7$FJ$\"3MWcTmP)*R7F:7$FO$\"3%[*)z3/M\"e9F:7$FT$\"3)e#3*e)**> )o\"F:7$FY$\"3;49[l_7A>F:7$Fhn$\"3O\"[b`vJw;#F:7$F]o$\"3g^,]t\"e!)Q#F: 7$Fbo$\"3u8$fP!y)4k#F:7$Fgo$\"3:T(G-5_,!HF:7$F\\p$\"3uP!Qc6J\\:$F:7$Fa p$\"3y[6rA2A4B$e-jF:7$Fhs$\"3mT7V_VV)f'F:7$F^t$\"37v9AA_*R#pF:7$Fct$ \"3e!z`Z&*QqE(F:7$Fht$\"3,._+x!o$4wF:7$F]u$\"3WZSmPy/ZzF:7$Fbu$\"3&*\\ y#RTv&H$)F:7$Fgu$\"3w&*)R=#HC!o)F:7$F\\v$\"3_9>C2S1i!*F:7$Fav$\"3]#Qd. V'y9%*F:7$Ffv$\"3gT)4Jnxy!)*F:7$F[w$\"3%GG%*H0,&=5Fjs7$F`w$\"3Ib8&43!p e5Fjs7$Few$\"3_iVFxcv)4\"Fjs7$Fjw$\"3;o@Rij`T6Fjs7$Fdx$\"3Qy/CEMc$=\"F js7$F^y$\"3[vSg!H&RF7Fjs7$Fhy$\"3'>0O(>Jtr7Fjs7$Fbz$\"3j;HqqED88Fjs7$F gz$\"3g%H[*QGxh8Fjs7$F\\[l$\"3[O1399.19Fjs7$Fa[l$\"3)e'y1NU8a9Fjs7$F[ \\l$\"3xHF8b_2,:Fjs7$Fe\\l$\"3_v.IU6__:Fjs-Fj\\l6&F\\]lF(F(F]]l-%+AXES LABELSG6$Q\"x6\"Q!Fifl-%%VIEWG6$;F($\"+3J;u=!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "To find the volume generated by rotating this region about the x-axis we have to break up the interva l at x = \"b\". On this first interval f2 is the larger function so t he integration takes the form " }{XPPEDIT 18 0 "Vol1 = Pi*int(f2(x)^2- f1(x)^2,x = 0 .. b);" "6#/%%Vol1G*&%#PiG\"\"\"-%$intG6$,&*$-%#f2G6#%\" xG\"\"#F'*$-%#f1G6#F0F1!\"\"/F0;\"\"!%\"bGF'" }{TEXT -1 3 " : " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Vol1 := Pi*int((f2(x))^2 - ( f1(x))^2, x = 0..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Vol1G,$%#Pi G$\"+%4\\IP$!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "This represen ts the volume generated by rotating the first enclosed region (x = 0.. b) around the x-axis. On the next interval (x = b .. c), f1 is the lar ger function. So on the integration takes the form " }{XPPEDIT 18 0 " Vol2 = Pi*int(f1(x)^2-f2(x)^2,x = b .. c);" "6#/%%Vol2G*&%#PiG\"\"\"-% $intG6$,&*$-%#f1G6#%\"xG\"\"#F'*$-%#f2G6#F0F1!\"\"/F0;%\"bG%\"cGF'" } {TEXT -1 3 " : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Vol2 := \+ Pi*int((f1(x))^2 - (f2(x))^2, x = b..c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Vol2G,$%#PiG$\"+[ihvG!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "This represents the volume generated by rotating the second en closed region (x = b..c) around the x-axis. To find the total volume, \+ we sum Vol1 and Vol2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "to talVol := Vol1 + Vol2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)totalVolG ,$%#PiG$\"+RnM4H!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(totalVol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VB)*R\"*!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "This is the total volume generated by rotating the enclos ed region about the x-axis. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 46 "3. Parametric Curves (length of a plane curve)" }}{PARA 0 "" 0 "" {TEXT -1 119 "In this section we will study parametric curves in the c ontext of determining the length of these curves in the plane. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 276 "In order to plot a paramatric curve in Maple you \+ use the same \"plot\" command except the functions x(t), y(t), and the range of t values are all enclosed in square brackets. Outside of the square brackets you may include further plotting options such as axes labels and color. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot ([cos(t),sin(t),t = 0 .. 3*Pi/2],labels=[\"x(t)\",\"y(t)\"],color=blue );" }}{PARA 13 "" 1 "" {GLPLOT2D 334 334 334 {PLOTDATA 2 "6&-%'CURVESG 6#7S7$$\"\"\"\"\"!$F*F*7$$\"3W'\\Vaz#HZ**!#=$\"3(eeX+Uh`-\"F/7$$\"3Y+^ leR2;)*F/$\"3x;k&\\l2\"4>F/7$$\"3[e8ZbK(\\d*F/$\"3q*eq'f9U%)GF/7$$\"3X kuE,=nM#*F/$\"3)4A)3gUxOQF/7$$\"3QlX3[H?-))F/$\"3iTY1\"4`cu%F/7$$\"3@Y od7eZ@$)F/$\"3#Q-66QUba&F/7$$\"3E6lD8pRZxF/$\"3`^-IgK!GK'F/7$$\"3mft.3 LmxqF/$\"3'4#y.ehYkqF/7$$\"3nI&4VXJ(RjF/$\"3A*=?Kh]Nt(F/7$$\"3ZH$QO6D^ ^&F/$\"3gzAK*>m;M)F/7$$\"3&pCp*oabSZF/$\"3olh$[R\\\\!))F/7$$\"31Q-%z_' pAQF/$\"3?$GN3<30C*F/7$$\"3ML$G/$>QhGF/$\"3o%f%[1O)=e*F/7$$\"3&y+=s%[o 1>F/$\"3,p5;b[a;)*F/7$$\"3#[**pq+9N-\"F/$\"3A'>xB0$[Z**F/7$$!3UVFzH-!) pO!#?$\"3I]YgiE$*****F/7$$!3cn/419!*\\$*!#>$\"3O<,NAP>c**F/7$$!3GX(p&3 x!*p>F/$\"3R'eK!GM0/)*F/7$$!3OsLi9yNoGF/$\"3sx`cswzz&*F/7$$!3`LfX;<_DQ F/$\"3.T&yb\"*Q$R#*F/7$$!3Wvx)*)\\6/q%F/$\"3$RX\"yA]WE))F/7$$!3#ec%yc) [ic&F/$\"3yr#pgZRwI)F/7$$!3#*zIlv%3;J'F/$\"3kWIEQ(>lv(F/7$$!3s[-O&\\,Y 0(F/$\"3-xjUse\\(3(F/7$$!3m\\7AN&*f^xF/$\"3$3DprC]wJ'F/7$$!3#R&QRp$>0H )F/$\"3!**\\^!ogr\"f&F/7$$!3Ivs[/$Gjz)F/$\"39*3:\"HK`cZF/7$$!3>&4&R$Ge 2B*F/$\"3Y`ku>,=YQF/7$$!3S;Q.uJ!\\c*F/$\"3%\\#Q#HqSw\"HF/7$$!31:,*R9q& *z*F/$\"3Eakj314#*>F/7$$!3+8G-]$*Qb**F/$\"3P9%R)pB;N%*F[q7$$!2sZd0v'** ****!#<$!31v?\\)fl:1)!#@7$$!39A$=n!fXZ**F/$!38^&f+!yxB5F/7$$!3)=(=bUG% 4\")*F/$!32$G9Lt.`$>F/7$$!3Mq&*3\\6Rm&*F/$!3'pgY(3#eF\"HF/7$$!3]>O-\\4&[DxF/$!3-D#)ooub\\jF/7$$!3)o/t'=h/pqF/$ !3e&f*>F*)3tqF/7$$!3wi)R*4'=!GjF/$!3y,XVPw8VxF/7$$!3;02F*))G2`&F/$!3.G sv-(G8L)F/7$$!31(GlC&3/]ZF/$!3k9S(=!f$)*z)F/7$$!3'RScDtgs!QF/$!35y)z6q yoC*F/7$$!3T/,'HR&fGHF/$!3u.h')QZbh&*F/7$$!3;c>H#oXQ'>F/$!3&3)Hf3&p_!) *F/7$$!3!e5(*o%\\6A5F/$!3!*o4r/piZ**F/7$$\"3ON*fa&G5`h!#F$!\"\"F*-%'CO LOURG6&%$RGBGF+F+$\"*++++\"!\")-%+AXESLABELSG6$Q%x(t)6\"Q%y(t)Fg[l-%%V IEWG6$%(DEFAULTGF\\\\l" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Notice the circle was not completed because I only let \"t\" go from zero to " }{XPPEDIT 18 0 "3*Pi/2;" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" } {TEXT -1 12 " instead of " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#Pi GF%" }{TEXT -1 111 " . OK, let's try a trickier curve and determine \+ it's length. First define x(t) and y(t) by the following: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x := t -> (cos(t))^3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6$%)operatorG%&arrow GF(*$)-%$cosG6#9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y := t -> (sin(t))^3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*$)-%$sinG6#9$\"\"$\"\" \"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([x(t),y(t) ,t=0..2*Pi]);" }}{PARA 13 "" 1 "" {GLPLOT2D 334 334 334 {PLOTDATA 2 "6 %-%'CURVESG6$7in7$$\"\"\"\"\"!$F*F*7$$\"3J$fMKJ42L+#Rd[a#!# ?7$$\"3kF:eE2)H0*F/$\"3V2eH/;vD;!#>7$$\"3QM[NsM!4\"zF/$\"3l9Uye\\o+bF8 7$$\"3shc;7>/zkF/$\"3wkOrl)Q$f7F/7$$\"3bI%HoOb$Q\\F/$\"3#oi)eZ7\\)H#F/ 7$$\"3[#e7=a@Gb$F/$\"33TrF,WF=NF/7$$\"3C^u.\\\"H#)G#F/$\"3!eopb.8;&\\F /7$$\"3g:Ap$3'4e7F/$\"3u^d8Iy=\"['F/7$$\"3QfjhZ\"3\\d&F8$\"3o&R>nq+H*y F/7$$\"3xp@B^Ue5;F8$\"37^qi0Bye!*F/7$$\"3#)eM8Z7)QV#F2$\"3/bl%***** *!#<7$$!31%oQS-ccA$Fco$\"3'f0iH+K&H**F/7$$!3R'*Rlr_DHDF2$\"3d%o[SdUGs* F/7$$!3k')3gF3$H&=F8$\"3+rM\"3Sl#o*)F/7$$!3q=!HDJI:P&F8$\"3'[p:'pqXUzF /7$$!3[pRE+e+#G\"F/$\"3!p,v!Qq1SkF/7$$!3:VlCD_N5D)4tuB%F/$\"3csSsbhIxGF/7$$!3`7 e&Qsl#Q\\F/$\"3M\"fv!>2c)H#F/7$$!3+VD%4MfI\\'F/$\"3%*3c!3rI7D\"F/7$$!3 !*)Q^u#yplyF/$\"37P)eElF/$!3'G*=8N4$>B\"F/7$$!3ly$4mjYr ,&F/$!38\"p')41hyB#F/7$$!3=Wv)\\ClFH%F/$!3M)*)4c<\\*GGF/7$$!3E!frvl7^f $F/$!3UR\"3d&zGwMF/7$$!3[7@S[0\")pGF/$!3SXn7mw+YUF/7$$!3gTI*oS**G@#F/$ !3E%eHc98*\\]F/7$$!3?E&*yJR&=p\"F/$!3yI!*G\"GpFy&F/7$$!3!*)>]9JHIC\"F/ $!3R'y576zs]'F/7$$!3YDkvG>8q`F8$!3;?\"z'\\F/7$$\"3C?jqy?0yNF/$!3qMT X&\\&=$\\$F/7$$\"3)y&y59SsI\\F/$!3/&)Hb/%4WI#F/7$$\"3a0r,9*Hp_'F/$!3Fb %>A#\\tJ7F/7$$\"3kX\"G)*3N0&yF/$!3)HR&zU^5^dF87$$\"3OXxb#zE%***)F/$!3g \"G-6!Q8okKo?F3_#F27$F($\"3GMDF^'H?_&!#X-%'C OLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q!6\"Fb_l-%%VIEWG6$%(DEFAU LTGFg_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "This is called an \"astro id\", and the curve is traced exactly once for t = 0 .. " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 76 ". Notice if you pl ot over a larger range of \"t\", the curve is just retraced;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([x(t),y(t),t=0..3*Pi]); " }}{PARA 13 "" 1 "" {GLPLOT2D 334 334 334 {PLOTDATA 2 "6%-%'CURVESG6$ 7[q7$$\"\"\"\"\"!$F*F*7$$\"3])H)o\\=N#Q*!#=$\"37j+Y<5f)[)!#?7$$\"35Am8 **Ryg()F/$\"3O*[&)oZBHX#!#>7$$\"3!y@@Z\"prozF/$\"3+Y%o,cb\\E&F87$$\"3& [_1^r1t#pF/$\"3dY=!zt8:,\"F/7$$\"3'))*op=9k#z&F/$\"3#4VB6K1`o\"F/7$$\" 3m#GLSbS(GYF/$\"3S$eZT\\I_a#F/7$$\"3w)HuQPIF^$F/$\"3=/I)*plQeNF/7$$\"3 ?Q+'o*Rq4DF/$\"3E,[O$=uAn%F/7$$\"3?X$H-&4!*f;F/$\"3g\"[A#397JeF/7$$\"3 i<$>&>XgL5F/$\"3GO\")>[!\\b)oF/7$$\"3`twfN!fRq&F8$\"39[%=rw?='yF/7$$\" 3\\Hn$4+zGa#F8$\"3B*)))Q>WNJ()F/7$$\"3Q&Q5rO*H`!)F2$\"3YXNSUJU\\0$*!#@$\"3%[>./Ept&)*F/7$$!3G&eCfuSva(F2$\"3p/30W(H%G%*F /7$$!3'pUL6[S$zDF8$\"3ya2K'GM&>()F/7$$!3z_Rg/AL3gF8$\"3]5#Ha\\)e*y(F/7 $$!3)*Hg1VCXe5F/$\"3gK%*****H.RoF/7$$!3Gn+!Qlz'o;F/$\"3\"*)*=irJzg@HDF/$\"3uV7C$RA$[YF/7$$!3+/.Lo!Q]a$F/$\"3GXgA_#Qg_$F/7$$!3i ;5CV\"yMn%F/$\"3E73U;as3DF/7$$!3%G![g)46![eF/$\"3/mg*oC3)[;F/7$$!3_f$ \\z&yHcpF/$\"33;2taX'H'**F87$$!3%z&e:?c[tzF/$\"3'=Fbi#4pX_F87$$!3=a+EP RBk()F/$\"3MgQF;dXUCF87$$!3A!p([%eGXQ*F/$\"3k$=>****F/$!3]uDDRMu`R!#C7$$!3qh`@;mv e)*F/$!3#*\\%el2L(p\"*Fhp7$$!3)*p%o%fMf%[*F/$!3S:]e7tV`kF27$$!37cL>3Zq $y)F/$!3!Q7sU#Qo$Q#F87$$!3MG\\,*41x%yF/$!3'*ziN6r%Hw&F87$$!3Izk4#yVx(o F/$!3e@#=yFcx.\"F/7$$!3p['Q/=c7$eF/$!31zs*[H7)f;F/7$$!3MnP'Q1+)\\NpO)*F /7$$!3&eI%fM[r15Ffu$!3Y\\'*\\\"Rv'****F/7$$\"3!Q$\\V#Q,ps*Fhp$!3%=$o4r b5`)*F/7$$\"3S.&eu3V8@)F2$!3]cux+Rt&R*F/7$$\"3(=C0V3hLU#F8$!319(e>*4aq ()F/7$$\"3wb4'>qj)e_F8$!3#eI<69C-(zF/7$$\"3tnFw'H7R%**F8$!3i(Q!*Q!3%*f pF/7$$\"3'[kQx!fBT;F/$!3u4^`J=dfeF/7$$\"3%>+aU^a))[#F/$!3!HO#ee>)zp%F/ 7$$\"3)eX?AW)=\"\\$F/$!3kv\"*4cb1!e$F/7$$\"3;qMs9)4xd%F/$!3qcMzr(Hse#F /7$$\"3N#y(fJAl7dF/$!3:eRi%)*4(Q`Z*F/$!3j!4!*p'**)*HmF27$$\"3op+!**H;!p)*F/$!3E(>;R?qo=)F hp7$$\"3qVnu15'*****F/$\"35(H-BG!H\">%Fbp7$$\"3w\\P/>^tS)*F/$\"3[aQ[b_ X)4\"F27$$\"3)[:Cr)z@%Q*F/$\"3M\\\\>bnt\\%)F27$$\"3l%4BUQ$oS()F/$\"37n `oxfB9DF87$$\"3.9O^KF*o\"zF/$\"3kAsZQfP 0uDW5F/7$$\"3v9XmY'>Ws&F/$\"34z)4&>K!3t\"F/7$$\"3kYXDPpQLYF/$\"3k\"oEj [D9a#F/7$$\"3M$p`*QtG%e$F/$\"3!pOvua.q[$F/7$$\"3)o'G&4FFqe#F/$\"3o%)) \\Suazd%F/7$$\"3Z>$)\\gRINHh>sEyF/7$$!3$fvU3L\"4T5 F/$\"3Y,(GYe$[roF/7$$!3!y8jB*=O_;F/$\"34:I4_WfUeF/7$$!3au!QZWBZ`#F/$\" 3@BmK\"Q!eTYF/7$$!3UlEZJ;b!e$F/$\"3W!eh\\W12\\$F/7$$!3Y\"oB.AeDh%F/$\" 3WRY$eF2&eDF/7$$!3'*\\)3@LPio&F/$\"3)G<%fiA_c2j6m&=()F/$\"3=Z p\")z`L#e#F87$$!3/R]7aY<'Q*F/$\"3KAbLEU14%)F27$$!\"\"F*$!3#4[/t+&oj=!# W-%'COLOURG6&%$RGBG$\"#5FailF+F+-%+AXESLABELSG6$Q!6\"F^jl-%%VIEWG6$%(D EFAULTGFcjl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Therefore you must in tegrate over a range of \"t\" values exactly " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 91 " in length. Now in order to det ermine the length of the curve we must find dx/dt and dy/dt:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xprime := t -> D(x)(t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xprimeGf*6#%\"tG6\"6$%)operatorG%&a rrowGF(,$*&)-%$cosG6#9$\"\"#\"\"\"-%$sinGF1F4!\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "yprime := t -> D(y)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'yprimeGf*6#%\"tG6\"6$%)operatorG%&arrowGF (,$*&)-%$sinG6#9$\"\"#\"\"\"-%$cosGF1F4\"\"$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "You should verify that xprime is indeed the deri vative of x(t) with respect to t. Now it's just a simple matter of u sing the formula for arclenghth of a parametric curve " }{XPPEDIT 18 0 "L = int((dx/dt)^2+(dy/dt)^2,t = 0 .. 2*Pi);" "6#/%\"LG-%$intG6$,&*$ *&%#dxG\"\"\"%#dtG!\"\"\"\"#F,*$*&%#dyGF,F-F.F/F,/%\"tG;\"\"!*&F/F,%#P iGF," }{TEXT -1 45 ". We can have Maple do this quite simply by " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "L := int(sqrt( (xprime(t))^2 + (yprime(t))^2 ), t = 0 .. 2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"LG,$*$-%%sqrtG6#\"\"%\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"' " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "And this is arclength of the \+ \"astroid\" traced exactly once. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "4. Plotting Definite integrals, using \"unapply\" " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Suppose you want to define and plot a function of the for m " }{XPPEDIT 18 0 "A(x) = Int(f(t),t = a .. x);" "6#/-%\"AG6#%\"xG-%$ IntG6$-%\"fG6#%\"tG/F.;%\"aGF'" }{TEXT -1 6 " . " }}{PARA 0 "" 0 " " {TEXT -1 15 "For example if " }{XPPEDIT 18 0 "A(x) = Int(sqrt(1-t^2) ,t = 0 .. x);" "6#/-%\"AG6#%\"xG-%$IntG6$-%%sqrtG6#,&\"\"\"F/*$%\"tG\" \"#!\"\"/F1;\"\"!F'" }{TEXT -1 168 " then A(0.5) represents the area s haded in blue. Here, the curve represents the portion of the unit ci rcle that lies in the first quadrant. Ie the curve generated by " } {XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" } {TEXT -1 28 " in the first quadrant, or " }{XPPEDIT 18 0 "y = sqrt(1- x^2);" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 13 " \+ for x = 0..1" }}{PARA 0 "" 0 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'- %'CURVESG6$7Y7$$\"\"!F)$\"\"\"F)7$$\"3dmmm;arz@!#>$\"3e(=:\"QTi(***!#= 7$$\"3[LL$e9ui2%F/$\"3!e2E0a)o\"***F27$$\"3nmmm\"z_\"4iF/$\"3)f\\s^f/2 )**F27$$\"3[mmmT&phN)F/$\"3)37fk0E]'**F27$$\"3BLLe*=)H\\5F2$\"3R\\N+$H 'zW**F27$$\"3gmm\"z/3uC\"F2$\"3^l4$yi$*=#**F27$$\"3%)***\\7LRDX\"F2$\" 3%p#zm\"3WR*)*F27$$\"3^mm\"zR'ok;F2$\"3iaa\"Q\\n/')*F27$$\"3w***\\i5`h (=F2$\"3\"Rj1r%eUA)*F27$$\"3YLLL3En$4#F2$\"3_.Ayu2Py(*F27$$\"3qmm;/RE& G#F2$\"3wIBk#=x`t*F27$$\"3\")*****\\K]4]#F2$\"31q9\"H%H@#o*F27$$\"3$** ****\\PAvr#F2$\"3B_^A([sOi*F27$$\"3(******\\nHi#HF2$\"3n:#e7,zAc*F27$$ \"3jmm\"z*ev:JF2$\"3)>#\\;*p8A]*F27$$\"3?LLL347TLF2$\"3pW!GHZL`U*F27$$ \"3,LLLLY.KNF2$\"3\"=1W(>`Yb$*F27$$\"3v***\\7o7Tv$F2$\"3ztaj&Q$eo#*F27 $$\"3'GLLLQ*o]RF2$\"3=`?'yv9l=*F27$$\"3A++D\"=lj;%F2$\"3_kxU3;t!4*F27$ $\"31++vV&R5j')F2 7$$\"3&em;zRQb@&F2$\"3e0&=5t$=K&)F27$$\"3\\***\\(=>Y2aF2$\"3ld&yZ.e=T) F27$$\"39mm;zXu9cF2$\"3\\/X@$oS\\F)F27$$\"3l******\\y))GeF2$\"3k!yShX> b7)F27$$\"3'*)***\\i_QQgF2$\"34nMlUm1rzF27$$\"3@***\\7y%3TiF2$\"3bX$z* QqP8yF27$$\"36****\\P![hY'F2$\"3QXhIao;GwF27$$\"3jKLL$Qx$omF2$\"3p\"p \\+YH?X(F27$$\"3!)*****\\P+V)oF2$\"378IrHv-`sF27$$\"3?mm\"zpe*zqF2$\"3 RynnKd;iqF27$$\"3%)*****\\#\\'QH(F2$\"3G\"QTZ_=5%oF27$$\"3GKLe9S8&\\(F 2$\"3a%Rp()p\"*)>mF27$$\"3R***\\i?=bq(F2$\"3=&F27$$\"3U4Dt?F27$$\"3;+D1k2/P)*F2$\"3oLW(eL^zz\"F27$$\"37+]P40O \"*)*F2$\"31H.f%oH+Z\"F27$$\"3k]7.#Q?&=**F2$\"3F#>z^*=&RF\"F27$$\"31+v oa-oX**F2$\"3(pV\\8H')3/\"F27$$\"3Z\\PMF,%G(**F2$\"3>&=hJ*p=ltF/7$F*F( -%'COLOURG6&%$RGBGF)F)F)-%)POLYGONSG6U7&7$F(F)F'7$$\"3HLLL3x&)*3\"F/$ \"3)p=^u31%****F27$F\\]lF)7&F`]lF[]l7$$\"3umm\"H2P\"Q?F/$\"37cW*GyAz** *F27$Fc]lF)7&Fg]lFb]l7$$\"3MLL$eRwX5$F/$\"3QG\"34kz^***F27$Fj]lF)7&F^^ lFi]l7$$\"3CLL$3x%3yTF/$\"3%yVB$)F/$\"3Jc3eN+Il**F27$F]`lF)7&Fa`lF\\ `l7$$\"3z)**\\7`l2Q*F/$\"3Nk&)[$R.f&**F27$Fd`lF)7&Fh`lFc`l7$$\"3tmm;/j $o/\"F2$\"3eV30Wd0X**F27$F[alF)7&F_alFj`l7$$\"3NLL3_>jU6F2$\"3>E'=A80X $**F27$FbalF)7&FfalFaal7$$\"3!*****\\i^Z]7F2$\"3'pN&3Sv]@**F27$FialF)7 &F]blFhal7$$\"3(*****\\(=h(e8F2$\"3c*4&4^$es!**F27$F`blF)7&FdblF_bl7$$ \"3)*****\\P[6j9F2$\"3KA%4oq&Q#*)*F27$FgblF)7&F[clFfbl7$$\"3KL$e*[z(yb \"F2$\"3a*f%)4Y0z()*F27$F^clF)7&FbclF]cl7$$\"3gmm;a/cq;F2$\"3*f@m2,u%f )*F27$FeclF)7&FiclFdcl7$$\"3]mmm;t,mF2$\"3@6c!zTfH!)*F27$FjdlF)7&F^elFidl7$$\"35+]i!f#=$3#F2$\"3& \\'*\\m$4h!y*F27$FaelF)7&FeelF`el7$$\"3/+](=xpe=#F2$\"3b#[mIpu\"e(*F27 $FhelF)7&F\\flFgel7$$\"3smm\"H28IH#F2$\"3\"Gx#)G!\\bL(*F27$F_flF)7&Fcf lF^fl7$$\"3km;zpSS\"R#F2$\"3'zbbp(*\\)4(*F27$FfflF)7&FjflFefl7$$\"3GLL 3_?`(\\#F2$\"365%e9D&4$o*F27$F]glF)7&FaglF\\gl7$$\"3#HLe*)>pxg#F2$\"3$ *4=]u3*Rl*F27$FdglF)7&FhglFcgl7$$\"3u**\\Pf4t.FF2$\"3r&QanAcvi*F27$F[h lF)7&F_hlFjgl7$$\"32LLe*Gst!GF2$\"3KH#3mnYyf*F27$FbhlF)7&FfhlFahl7$$\" 3#)*****\\#RW9HF2$\"3UGX#Gsxec*F27$FihlF)7&F]ilFhhl7$$\"3[***\\7j#>>IF 2$\"3)*H7w!)\\LL&*F27$F`ilF)7&FdilF_il7$$\"3h**\\i!RU07$F2$\"3lta*G)Gk +&*F27$FgilF)7&F[jlFfil7$$\"3b***\\(=S2LKF2$\"3>^BHW*RHY*F27$F^jlF)7&F bjlF]jl7$$\"3Kmmm\"p)=MLF2$\"3OL#Ht&zyF%*F27$FejlF)7&FijlFdjl7$$\"3!** ***\\(=]@W$F2$\"3P*=v3Y3*)Q*F27$F\\[mF)7&F`[mF[[m7$$\"35L$e*[$z*RNF2$ \"33#)H'*R?Y_$*F27$Fc[mF)7&Fg[mFb[m7$$\"3#*****\\iC$pk$F2$\"3*HgayAx7J *F27$Fj[mF)7&F^\\mFi[m7$$\"39m;H2qcZPF2$\"3G*e-9$>Br#*F27$Fa\\mF)7&Fe \\mF`\\m7$$\"3p**\\7.\"fF&QF2$\"3$*3&))fE9!G#*F27$Fh\\mF)7&F\\]mFg\\m7 $$\"3Xmm;/OgbRF2$\"3a,*yR#)*R%=*F27$F_]mF)7&Fc]mF^]m7$$\"3x**\\ilAFjSF 2$\"3_N+7LL3xe,tUF2$\"3PEa4*[\"4T!*F27$Fd^mF)7&Fh^mF c^m7$$\"3em;HdO=yVF2$\"3C=Fv_Wj!**)F27$F[_mF)7&F__mFj^m7$$\"3()*****\\ #>#[Z%F2$\"3vo#*z?%GH%*)F27$Fb_mF)7&Ff_mFa_m7$$\"3immT&G!e&e%F2$\"3'Q. xumWm)))F27$Fi_mF)7&F]`mFh_m7$$\"3;LLL$)Qk%o%F2$\"3?9e=1Q#[$))F27$F``m F)7&Fd`mF_`m7$$\"37+]iSjE!z%F2$\"3%yBcum.!y()F27$Fg`mF)7&F[amFf`m7$$\" 37+]P40O\"*[F2$\"3(>5PN+v?s)F27$F^amF)7&FbamF]am7$$\"3++++++++]F2$\"3( fQWy.a-m)F27$FeamF)7\"-%&STYLEG6#%,PATCHNOGRIDG-Fc\\l6&Fe\\lF(F($\"*++ ++\"!\")-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q\"x6\"Q!F\\cm-%%FO NTG6#%(DEFAULTG-%%VIEWG6$;F(F*Facm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 5 "Note:" }{TEXT -1 200 " Shading an \+ area is done with \"filled = true\" in the plot command and combining \+ graphs with the \"display\" command found in the \"plots\" library. \+ Its a little tricky and you are not required to do so." }}{PARA 0 "" 0 "" {TEXT -1 32 "Let's first define the curve: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> sqrt(1 - x^2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,& \"\"\"F0*$)9$\"\"#F0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 36 "Tempting Techniqe that doesn't work:" }{TEXT -1 72 " Suppose you first want to find an antiderivative of f and call it F b y " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "F := x -> int(f(x),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%& arrowGF(-%$intG6$-%\"fG6#9$F2F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "This looks ok. In fact, if you type " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG \"\"\"-%%sqrtG6#,&F&F&*$)F%\"\"#F&!\"\"F&#F&F-*&F/F&-%'arcsinG6#F%F&F& " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "it still looks good. " } {TEXT 276 5 "But, " }{TEXT -1 74 "For some reason, and I don't know wh y, you cannot evaluate F at a number. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(0);" }}{PARA 8 "" 1 "" {TEXT -1 52 "Error, (in int) \+ wrong number (or type) of arguments\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "So I will present three techniques to avoid this problem , the most efficient is technique 2 using the unapply command. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT 269 12 "Technique 1:" }{TEXT -1 45 " You \+ may use the following command to define " }{XPPEDIT 18 0 "A(x) = int(f (t),t = 0 .. x);" "6#/-%\"AG6#%\"xG-%$intG6$-%\"fG6#%\"tG/F.;\"\"!F'" }{TEXT -1 2 ". " }{TEXT 268 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := x -> int(f(t),t=0..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"AGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-%\"fG6#%\"tG/F2;\" \"!9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "However, this doe s not explicitly show what A(x) looks like. This works for evaluatin g A at a number, for example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "as should be expected because thi s area is 1/4 the area of a circle with radius 1. You can also plot A (x): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(A(x),x=0..1) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6 $7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3A]TW,Gaz@F-7$$\"3[LL$e9ui2%F-$\"3 'Ri\"Q2]9vSF-7$$\"3nmmm\"z_\"4iF-$\"3)[f9-tg^?'F-7$$\"3[mmmT&phN)F-$\" 3W$\\=#fZVY$)F-7$$\"3CLLe*=)H\\5!#=$\"3ac45)[pt/\"FB7$$\"3gmm\"z/3uC\" FB$\"3;9lMaa;W7FB7$$\"3%)***\\7LRDX\"FB$\"3K#>s)f_TZ9FB7$$\"3]mm\"zR'o k;FB$\"3q_FIRb'pl\"FB7$$\"3w***\\i5`h(=FB$\"3!Rb-)Qw3l=FB7$$\"3WLLL3En $4#FB$\"31qg([_u#y?FB7$$\"3qmm;/RE&G#FB$\"3ck$4o/9_E#FB7$$\"3\")***** \\K]4]#FB$\"3wdS'zqGYZ#FB7$$\"3$******\\PAvr#FB$\"3Im_,xQp$o#FB7$$\"3) ******\\nHi#HFB$\"3#*)HAU'[\"R)GFB7$$\"3jmm\"z*ev:JFB$\"3OL5?yCekIFB7$ $\"3?LLL347TLFB$\"3iOU+tS(yF$FB7$$\"3,LLLLY.KNFB$\"3k^m$yocrX$FB7$$\"3 w***\\7o7Tv$FB$\"33V#)>3w'Rm$FB7$$\"3'GLLLQ*o]RFB$\"33]#f-yn`%QFB7$$\" 3A++D\"=lj;%FB$\"30K!=trwC/%FB7$$\"31++vV&RGUFB7$$\" 3WLL$e9Ege%FB$\"3o(felzw(>WFB7$$\"3GLLeR\"3Gy%FB$\"3c_S[#)zi$f%FB7$$\" 3cmm;/T1&*\\FB$\"3!3!yFt?yyZFB7$$\"3&em;zRQb@&FB$\"3?G9%e/_$o\\FB7$$\" 3\\***\\(=>Y2aFB$\"3p;W)fFB7$$\"3kKLL$Qx$omFB$\"3j]S:=KVMhFB7$$\"3!)*****\\P+V)oFB$\"3]<`b8B @$H'FB7$$\"3?mm\"zpe*zqFB$\"3Bre4nOFLkFB7$$\"3%)*****\\#\\'QH(FB$\"37d *\\R#o*>e'FB7$$\"3GKLe9S8&\\(FB$\"3a/QYgB[taoFB7$$\"3\"HLL$3s?6zFB$\"3E\")\\zF,o#)pFB7$$\"3a***\\7`Wl7)FB $\"3'>-*Rk#>86(FB7$$\"3#pmmm'*RRL)FB$\"3o1#3\"y35HsFB7$$\"3Qmm;a<.Y&)F B$\"38eHSZ&HGM(FB7$$\"3=LLe9tOc()FB$\"3kCBIXdH[uFB7$$\"3u******\\Qk\\* )FB$\"3L=_!\\5V\"QvFB7$$\"3CLL$3dg6<*FB$\"3twIP6s#=j(FB7$$\"3ImmmmxGp$ *FB$\"3#RfJ,Vkgq(FB7$$\"3A++D\"oK0e*FB$\"3C[.k@e\\txFB7$$\"3A++v=5s#y* FB$\"3;*4:z<%)Q#yFB7$$\"\"\"F)$\"3!G[uRj\")R&yFB-%'COLOURG6&%$RGBG$\"# 5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fa[l-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT 270 22 "Techniqe 2 (unapply): " }{TEXT -1 44 "You may use the \"unapply\" command to define " }{XPPEDIT 18 0 "A2(x) = int(f(t),t = 0 .. x);" "6#/-%#A2G6#%\"xG-%$intG6$-%\"fG6#%\"t G/F.;\"\"!F'" }{TEXT -1 52 ". Notice the lack of the \"x ->\" in this definition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A2 := unapp ly(int(f(t),t=0..x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2Gf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&*&9$\"\"\"-%%sqrtG6#,&F/F/*$)F.\"\"#F/ !\"\"F/#F/F6*&F8F/-%'arcsinG6#F.F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "and this explicitly shows you the function you are plott ing. Ie. A2(x) = F(x) - F(0), where F is an antiderivative of " } {XPPEDIT 18 0 "f(x) = sqrt(1-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\" \"F,*$F'\"\"#!\"\"" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(A2(x),x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3emmm;arz@!# >$\"3A]TW,Gaz@F-7$$\"3[LL$e9ui2%F-$\"3'Ri\"Q2]9vSF-7$$\"3nmmm\"z_\"4iF -$\"3)[f9-tg^?'F-7$$\"3[mmmT&phN)F-$\"3W$\\=#fZVY$)F-7$$\"3CLLe*=)H\\5 !#=$\"3ac45)[pt/\"FB7$$\"3gmm\"z/3uC\"FB$\"3;9lMaa;W7FB7$$\"3%)***\\7L RDX\"FB$\"3K#>s)f_TZ9FB7$$\"3]mm\"zR'ok;FB$\"3q_FIRb'pl\"FB7$$\"3w*** \\i5`h(=FB$\"3!Rb-)Qw3l=FB7$$\"3WLLL3En$4#FB$\"31qg([_u#y?FB7$$\"3qmm; /RE&G#FB$\"3ck$4o/9_E#FB7$$\"3\")*****\\K]4]#FB$\"3wdS'zqGYZ#FB7$$\"3$ ******\\PAvr#FB$\"3Im_,xQp$o#FB7$$\"3)******\\nHi#HFB$\"3#*)HAU'[\"R)G FB7$$\"3jmm\"z*ev:JFB$\"3OL5?yCekIFB7$$\"3?LLL347TLFB$\"3iOU+tS(yF$FB7 $$\"3,LLLLY.KNFB$\"3k^m$yocrX$FB7$$\"3w***\\7o7Tv$FB$\"33V#)>3w'Rm$FB7 $$\"3'GLLLQ*o]RFB$\"33]#f-yn`%QFB7$$\"3A++D\"=lj;%FB$\"30K!=trwC/%FB7$ $\"31++vV&RGUFB7$$\"3WLL$e9Ege%FB$\"3o(felzw(>WFB7$$ \"3GLLeR\"3Gy%FB$\"3c_S[#)zi$f%FB7$$\"3cmm;/T1&*\\FB$\"3!3!yFt?yyZFB7$ $\"3&em;zRQb@&FB$\"3?G9%e/_$o\\FB7$$\"3\\***\\(=>Y2aFB$\"3p;W)fFB7$$\"3kKLL$Qx$omFB$\"3j]S:= KVMhFB7$$\"3!)*****\\P+V)oFB$\"3]<`b8B@$H'FB7$$\"3?mm\"zpe*zqFB$\"3Bre 4nOFLkFB7$$\"3%)*****\\#\\'QH(FB$\"37d*\\R#o*>e'FB7$$\"3GKLe9S8&\\(FB$ \"3a/QYgB[taoFB7$$\"3\"HLL$3s?6zFB$ \"3E\")\\zF,o#)pFB7$$\"3a***\\7`Wl7)FB$\"3'>-*Rk#>86(FB7$$\"3#pmmm'*RR L)FB$\"3o1#3\"y35HsFB7$$\"3Qmm;a<.Y&)FB$\"38eHSZ&HGM(FB7$$\"3=LLe9tOc( )FB$\"3kCBIXdH[uFB7$$\"3u******\\Qk\\*)FB$\"3L=_!\\5V\"QvFB7$$\"3CLL$3 dg6<*FB$\"3twIP6s#=j(FB7$$\"3ImmmmxGp$*FB$\"3#RfJ,Vkgq(FB7$$\"3A++D\"o K0e*FB$\"3C[.k@e\\txFB7$$\"3A++v=5s#y*FB$\"3;*4:z<%)Q#yFB7$$\"\"\"F)$ \"3!G[uRj\")R&yFB-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6 \"Q!Fa[l-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "and the result is the same as for technique 1. The benefit of te chnique 2 with the \"unapply\" command is that you can define the area function with minimal effort. Make sure you use the \"unapply\" comm and exactly as shown above. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 12 "Technique 3:" }{TEXT -1 65 " We can avoid the \"unapply \" technique but it requires more work. " }}{PARA 0 "" 0 "" {TEXT 273 5 "First" }{TEXT -1 87 ", have Maple find the antiderivative of f whe re the constant of integration is zero by" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "int(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* &%\"xG\"\"\"-%%sqrtG6#,&F&F&*$)F%\"\"#F&!\"\"F&#F&F-*&F/F&-%'arcsinG6# F%F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "This gives you the ant iderivative of f (where the constant of integration equals zero), but \+ there is really no way to define this as a function without the \"unap ply\" technique. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 274 8 "Seco nd: " }{TEXT -1 118 "Enter the function F(x) . This can be done by cut ting and pasting the previous output after the \"F := x ->\" below. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "F := x -> 1/2*x*sqrt(1 -x^2)+1/2*arcsin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,&*&9$\"\"\"-%%sqrtG6#,&F/F/*$)F.\"\"#F/!\" \"F/#F/F6*&F8F/-%'arcsinG6#F.F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A3 := x -> F(x) - F(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"FG6#9$\"\"\"-F.6# \"\"!!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Again, A3(x) \+ = A2(x) = A(x) = the area under f as x goes from 0 to x. " }}{PARA 0 " " 0 "" {TEXT -1 116 "While this gives gives the correct answer, you ar e again lacking the actual function, however it is correct. Notice " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(A3(x),x=0..1);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"\"!F)F(7$$\"3emmm;arz@!#>$\"3A]TW,Gaz@F-7$$\"3[LL$e9ui2%F-$\"3'Ri\" Q2]9vSF-7$$\"3nmmm\"z_\"4iF-$\"3)[f9-tg^?'F-7$$\"3[mmmT&phN)F-$\"3W$\\ =#fZVY$)F-7$$\"3CLLe*=)H\\5!#=$\"3ac45)[pt/\"FB7$$\"3gmm\"z/3uC\"FB$\" 3;9lMaa;W7FB7$$\"3%)***\\7LRDX\"FB$\"3K#>s)f_TZ9FB7$$\"3]mm\"zR'ok;FB$ \"3q_FIRb'pl\"FB7$$\"3w***\\i5`h(=FB$\"3!Rb-)Qw3l=FB7$$\"3WLLL3En$4#FB $\"31qg([_u#y?FB7$$\"3qmm;/RE&G#FB$\"3ck$4o/9_E#FB7$$\"3\")*****\\K]4] #FB$\"3wdS'zqGYZ#FB7$$\"3$******\\PAvr#FB$\"3Im_,xQp$o#FB7$$\"3)****** \\nHi#HFB$\"3#*)HAU'[\"R)GFB7$$\"3jmm\"z*ev:JFB$\"3OL5?yCekIFB7$$\"3?L LL347TLFB$\"3iOU+tS(yF$FB7$$\"3,LLLLY.KNFB$\"3k^m$yocrX$FB7$$\"3w***\\ 7o7Tv$FB$\"33V#)>3w'Rm$FB7$$\"3'GLLLQ*o]RFB$\"33]#f-yn`%QFB7$$\"3A++D \"=lj;%FB$\"30K!=trwC/%FB7$$\"31++vV&RGUFB7$$\"3WLL$ e9Ege%FB$\"3o(felzw(>WFB7$$\"3GLLeR\"3Gy%FB$\"3c_S[#)zi$f%FB7$$\"3cmm; /T1&*\\FB$\"3!3!yFt?yyZFB7$$\"3&em;zRQb@&FB$\"3?G9%e/_$o\\FB7$$\"3\\** *\\(=>Y2aFB$\"3p;W)fFB 7$$\"3kKLL$Qx$omFB$\"3j]S:=KVMhFB7$$\"3!)*****\\P+V)oFB$\"3]<`b8B@$H'F B7$$\"3?mm\"zpe*zqFB$\"3Bre4nOFLkFB7$$\"3%)*****\\#\\'QH(FB$\"37d*\\R# o*>e'FB7$$\"3GKLe9S8&\\(FB$\"3a/QYgB[taoFB7$$\"3\"HLL$3s?6zFB$\"3E\")\\zF,o#)pFB7$$\"3a***\\7`Wl7)FB$\"3' >-*Rk#>86(FB7$$\"3#pmmm'*RRL)FB$\"3o1#3\"y35HsFB7$$\"3Qmm;a<.Y&)FB$\"3 8eHSZ&HGM(FB7$$\"3=LLe9tOc()FB$\"3kCBIXdH[uFB7$$\"3u******\\Qk\\*)FB$ \"3L=_!\\5V\"QvFB7$$\"3CLL$3dg6<*FB$\"3twIP6s#=j(FB7$$\"3ImmmmxGp$*FB$ \"3#RfJ,Vkgq(FB7$$\"3A++D\"oK0e*FB$\"3C[.k@e\\txFB7$$\"3A++v=5s#y*FB$ \"3;*4:z<%)Q#yFB7$$\"\"\"F)$\"3!G[uRj\")R&yFB-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fa[l-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "and the result is the same. You \+ will probably want to use technique 2 or 3 to answer question 1 of the assignment. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Assignment: De signing Dipsticks " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 253 "Suppose you are asked to design dip-sticks for measuring the volume of fluid in t wo different shaped tanks. In this assignment you will work on some o f the preliminary calculus involved in designing these dip-sticks for tanks of a specific size. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 17 "Cylindrical Tank:" }{TEXT 257 1 " " }{TEXT -1 312 "Suppose a cylindrical tank is positioned lenghwise on a frame. The ta nk has length 50 cm and radius 10 cm. Suppose you now fill the tank \+ with fluid from the top. By putting the tank on coordinate axes in the manner depicted below, the x-value represents the depth of the fluid, and the curve is the graph of " }{XPPEDIT 18 0 "(x-10)^2+y^2 = 10^2 ;" "6#/,&*$),&%\"xG\"\"\"\"#5!\"\"\"\"#F)F)*$)%\"yGF,F)F)*$F*F," } {TEXT -1 162 ", representing a cross-section of the tank. Now, for an y depth x you can determine the fluid volume as the product of the blu e area and the length of the tank. 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By putting one half of the tank on coordinate axes in the manner depicted below, the x-value represents the depth of the fluid in the tank, and the curve \+ is the \"top\" half of the graph of " }{XPPEDIT 18 0 "(x-10)^2+y^2 = 10^2;" "6#/,&*$),&%\"xG\"\"\"\"#5!\"\"\"\"#F)F)*$)%\"yGF,F)F)*$F*F," }{TEXT -1 182 ", representing one half of the cross-section through t he center of the tank. Now for any depth x you can determine the fluid volume by rotating the shaded region about the x-axis. 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Note: for the spherical tank, you should be able to do this without the aid of \+ Maple but you are free to do it either way. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 2 "2)" }{TEXT -1 157 " Make a plot of the fl uid volume versus fluid depth where the x-axis represents the fluid de pth (in cm) and the y-axis represents the volume (in cubic cm). " }} {PARA 0 "" 0 "" {TEXT 262 2 "3)" }{TEXT -1 129 " Suppose a dip-stick o f lenth 20 cm is inserted into the top of the tank and 3/4 of it is co vered with fluid when you retract it." }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{TEXT 263 3 "a) " }{TEXT -1 66 "How full is the tank in terms o f the percentage of total volume. " }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{TEXT 265 3 "b) " }{TEXT -1 35 "Is it more or less than 3/4 full? " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 266 3 "c) " }{TEXT -1 26 "Explain the difference? " }{TEXT 264 1 " " }}}}}{MARK "5" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }