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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 "Bulle t Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "The trapezoidal rule" }}{PARA 0 " " 0 "" {TEXT -1 58 "by Peter Stone, School of Life and Physical Scienc es, RMIT" }}{PARA 0 "" 0 "" {TEXT -1 23 "peter.stone@rmit.edu.au" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: \+ 29.3.2005" }}{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 37 "load numerical integration \+ procedures" }}{PARA 0 "" 0 "" {TEXT -1 35 "RMIT file path to read Mapl e m-file" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "read \"J:\\\\Cla ss_Notes/Peter Stone/MapleMath/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Another fil e path " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"E:\\\\Mapl eMath/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 48 "Numerical integration .. introductory examples " }}{PARA 0 "" 0 "" {TEXT -1 33 "Consider the indefinite integral " } {XPPEDIT 18 0 "Int(1/(1+exp(x)),x)" "6#-%$IntG6$*&\"\"\"F',&F'F'-%$exp G6#%\"xGF'!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Sinc e " }{XPPEDIT 18 0 "1/(1+exp(x))=(1+exp(x))/(1+exp(x))-exp(x)/(1+exp(x ))" "6#/*&\"\"\"F%,&F%F%-%$expG6#%\"xGF%!\"\",&*&,&F%F%-F(6#F*F%F%,&F% F%-F(6#F*F%F+F%*&-F(6#F*F%,&F%F%-F(6#F*F%F+F+" }{XPPEDIT 18 0 "``=1-ex p(x)/(1+exp(x))" "6#/%!G,&\"\"\"F&*&-%$expG6#%\"xGF&,&F&F&-F)6#F+F&!\" \"F/" }{TEXT -1 2 ", " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(1+exp(x)),x) = Int(``(1-exp(x)/(1+exp(x))),x);" "6#/-%$In tG6$*&\"\"\"F(,&F(F(-%$expG6#%\"xGF(!\"\"F--F%6$-%!G6#,&F(F(*&-F+6#F-F (,&F(F(-F+6#F-F(F.F.F-" }{XPPEDIT 18 0 "``=x - Int(exp(x)/(1+exp(x)),x )" "6#/%!G,&%\"xG\"\"\"-%$IntG6$*&-%$expG6#F&F',&F'F'-F-6#F&F'!\"\"F&F 2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "The remaining integ ral can be found with the aid of the substitution " }{XPPEDIT 18 0 "u= 1+exp(x)" "6#/%\"uG,&\"\"\"F&-%$expG6#%\"xGF&" }{TEXT -1 3 ". " }} {PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x)/(1+exp(x)) ,x)" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\",&F+F+-F(6#F*F+!\"\"F*" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u=1+exp(x),``],[du=exp(x)* dx,``])" "6#-%*PIECEWISEG6$7$/%\"uG,&\"\"\"F*-%$expG6#%\"xGF*%!G7$/%#d uG*&-F,6#F.F*%#dxGF*F/" }{TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Int(1/u,u)" "6#/%!G-%$IntG6$*&\"\"\"F)%\"uG! \"\"F*" }{TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(abs(u),u)+c[1];" "6#/%!G,&-%#lnG6$-%$absG6#%\"uGF,\"\"\" &%\"cG6#F-F-" }{XPPEDIT 18 0 "`` = ln(1+exp(x))+c[1];" "6#/%!G,&-%#lnG 6#,&\"\"\"F*-%$expG6#%\"xGF*F*&%\"cG6#F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "(The absolute value operation can be omitted si nce " }{XPPEDIT 18 0 "1+exp(x)" "6#,&\"\"\"F$-%$expG6#%\"xGF$" }{TEXT -1 21 " is never negative.) " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follow s that " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(1+e xp(x)),x)=x-ln(1+exp(x))+c" "6#/-%$IntG6$*&\"\"\"F(,&F(F(-%$expG6#%\"x GF(!\"\"F-,(F-F(-%#lnG6#,&F(F(-F+6#F-F(F.%\"cGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 51 "This result can be obtained with Maple as follows. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{TEXT 0 3 "Int " }{TEXT -1 23 " is the unevaluated or " }{TEXT 265 5 "inert" }{TEXT -1 35 " form of the integration procedure " }{TEXT 0 3 "int" }{TEXT -1 8 ". Using " }{TEXT 0 3 "Int" }{TEXT -1 21 " in conjunction with " }{TEXT 0 5 "value" }{TEXT -1 3 " ( " }{HYPERLNK 17 "value" 2 "value" " " }{TEXT -1 75 " ) enables the unevaluated integral to be displayed be fore it is evaluated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Int(1/(1+exp(x)),x);\nvalue(%);\nsimplify (%,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',& F'F'-%$expG6#%\"xGF'!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#ln G6#-%$expG6#%\"xG\"\"\"-F%6#,&F+F+F'F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"-%#lnG6#,&F%F%-%$expG6#F$F%!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "The foll owing slight modification of the previous code produces an output whic h \"looks like\" the standard way of presenting the evaluation of the \+ integral in print or writing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int(1/(1+exp(x)),x);\n``=sim plify(value(%),symbolic)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*&\"\"\"F',&F'F'-%$expG6#%\"xGF'!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(%\"xG\"\"\"-%#lnG6#,&F'F'-%$expG6#F&F'!\"\"%\"cGF '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Now consider the defintite integral " }{XPPEDIT 18 0 "Int (1/(1+exp(x)),x=0..1)" "6#-%$IntG6$*&\"\"\"F',&F'F'-%$expG6#%\"xGF'!\" \"/F,;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "The val ue of this integral gives the area of the region bounded by the " } {TEXT 318 1 "x" }{TEXT -1 5 " and " }{TEXT 319 1 "y" }{TEXT -1 17 " ax es, the curve " }{XPPEDIT 18 0 "y=1/(1+exp(x))" "6#/%\"yG*&\"\"\"F&,&F &F&-%$expG6#%\"xGF&!\"\"" }{TEXT -1 23 " and the vertical line " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 37 ", as shown in the f ollowing picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 209 "f := x->1/(1+exp(x)): 'f(x)'=f(x);\np1 := p lot(f(x),x=0..1,color=COLOR(RGB,.75,.75,.75),filled=true):\np2 := plot (f(x),x=-.2..1.5,color=red):\np3 := plot([[1,0],[1,f(1)]],color=black) :\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG*&\"\"\"F),&F)F)-%$expGF&F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 571 198 198 {PLOTDATA 2 "6'-%)POLYGONSG6U7&7$$\"\"!F)F)7$$\" 3emmm;arz@!#>F)7$F+$\"3EBt,s#4b%\\!#=7$F($\"3++++++++]F17&F*7$$\"3[LL$ e9ui2%F-F)7$F7$\"3UD:sas5)*[F1F.7&F67$$\"3nmmm\"z_\"4iF-F)7$F>$\"3*Hqv G.@[%[F1F97&F=7$$\"3[mmmT&phN)F-F)7$FE$\"3-i`yLs@\"z%F1F@7&FD7$$\"3CLL e*=)H\\5F1F)7$FL$\"3VB9]we\"zt%F1FG7&FK7$$\"3gmm\"z/3uC\"F1F)7$FS$\"3o m\\([t^&)o%F1FN7&FR7$$\"3%)***\\7LRDX\"F1F)7$FZ$\"3\\kw&eH-vj%F1FU7&FY 7$$\"3]mm\"zR'ok;F1F)7$F[o$\"3%*3F&p\"oy%e%F1Ffn7&Fjn7$$\"3w***\\i5`h( =F1F)7$Fbo$\"3Vu-/CFLKXF1F]o7&Fao7$$\"3WLLL3En$4#F1F)7$Fio$\"3'eI=9\\& [yWF1Fdo7&Fho7$$\"3qmm;/RE&G#F1F)7$F`p$\"3][!*>*[d6V%F1F[p7&F_p7$$\"3 \")*****\\K]4]#F1F)7$Fgp$\"3%*=@:&3,!yVF1Fbp7&Ffp7$$\"3$******\\PAvr#F 1F)7$F^q$\"3W]7fR(pZK%F1Fip7&F]q7$$\"3)******\\nHi#HF1F)7$Feq$\"3)pJm! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(1/(1+exp(x)),x=0..1);\n``=value (%);\n``=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*& \"\"\"F',&F'F'-%$expG6#%\"xGF'!\"\"/F,;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(-%#lnG6#\"\"#\"\"\"-F'6#,&F*F*-%$expG6#F*F*!\"\"F *F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+O\\&))z$!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now consider the indefinite integral " }{XPPEDIT 18 0 "Int(1/(x+exp(x)),x)" "6#-%$IntG6$*&\"\"\"F',&%\"xGF'-% $expG6#F)F'!\"\"F)" }{TEXT -1 77 ", in which we have made the seemingl y minor change of replacing a \"1\" by an \"" }{TEXT 321 1 "x" }{TEXT -1 29 "\". in the previous integral. " }}{PARA 0 "" 0 "" {TEXT -1 157 "It is not possible to obtain an analytical expression for this integr al involving basic algebraic operations and the standard so-called ele mentary functions:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s in*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*x " "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tan*x" "6 #*&%$tanG\"\"\"%\"xGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "arcsin*x" "6# *&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "arctan*x" " 6#*&%'arctanG\"\"\"%\"xGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "exp(x)" " 6#-%$expG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "ln*x" "6#*&%#lnG\" \"\"%\"xGF%" }{TEXT -1 7 ", etc. " }}{PARA 0 "" 0 "" {TEXT -1 67 "(The functions which are available on most scientific calculator). " }} {PARA 0 "" 0 "" {TEXT -1 143 "Maple 9 cannot even obtain an analytical expression for this integral in terms of special mathematical functio ns such as Bessel functions etc. " }}{PARA 0 "" 0 "" {TEXT -1 47 "For \+ a list of functions available in Maple see " }{HYPERLNK 17 "inifcn" 2 "inifcn" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "When using \+ " }{TEXT 0 3 "int" }{TEXT -1 4 " or " }{TEXT 0 9 "value/Int" }{TEXT -1 15 ", Maple simply " }{TEXT 265 31 "leaves the integral unevaluated " }{TEXT -1 158 ". However, if you select (highlight) the second integ ral, and look at the corresponding test field in the context bar you w ill notice that the inert function " }{TEXT 39 3 "Int" }{TEXT -1 22 " \+ has been replaced by " }{TEXT 281 3 "int" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(1 /(x+exp(x)),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The definite integra l " }{XPPEDIT 18 0 "Int(1/(x+exp(x)),x=0..1)" "6#-%$IntG6$*&\"\"\"F',& %\"xGF'-%$expG6#F)F'!\"\"/F);\"\"!F'" }{TEXT -1 32 " is similarly left unevaluated. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(1/(x+e xp(x)),x=0..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"/F);\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"/F);\"\"! F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The following picture sho ws that the integral " }{XPPEDIT 18 0 "Int(1/(x+exp(x)),x = 0 .. 1)" " 6#-%$IntG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"/F);\"\"!F'" }{TEXT -1 53 " does exist as the area of the region bounded by the " }{TEXT 322 1 "x" }{TEXT -1 5 " and " }{TEXT 323 1 "y" }{TEXT -1 17 " axes, th e curve " }{XPPEDIT 18 0 "y = 1/(x+exp(x));" "6#/%\"yG*&\"\"\"F&,&%\"x GF&-%$expG6#F(F&!\"\"" }{TEXT -1 23 " and the vertical line " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "f := x->1 /(x+exp(x)): 'f(x)'=f(x);\np1 := plot(f(x),x=0..1,color=COLOR(RGB,.75, .75,.75),filled=true):\np2 := plot(f(x),x=-.2..1.5,color=red):\np3 := \+ plot([[1,0],[1,f(1)]],color=black):\nplots[display]([p1,p2,p3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F)-%$expGF &F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 396 374 374 {PLOTDATA 2 "6'-%)P OLYGONSG6U7&7$$\"\"!F)F)7$$\"3emmm;arz@!#>F)7$F+$\"33M!RN)42!e*!#=7$F( $\"\"\"F)7&F*7$$\"3[LL$e9ui2%F-F)7$F7$\"3gIRBbX+R#*F1F.7&F67$$\"3nmmm \"z_\"4iF-F)7$F>$\"3%\\5Hy\"=!)z))F1F97&F=7$$\"3[mmmT&phN)F-F)7$FE$\"3 e7XH+gzT&)F1F@7&FD7$$\"3CLLe*=)H\\5F1F)7$FL$\"3E:G[qTkE#)F1FG7&FK7$$\" 3gmm\"z/3uC\"F1F)7$FS$\"31>zjK=o^zF1FN7&FR7$$\"3%)***\\7LRDX\"F1F)7$FZ $\"3%Gb![M!GHo(F1FU7&FY7$$\"3]mm\"zR'ok;F1F)7$F[o$\"3HMqTvji?uF1Ffn7&F jn7$$\"3w***\\i5`h(=F1F)7$Fbo$\"3i0&)4$4!otrF1F]o7&Fao7$$\"3WLLL3En$4# F1F)7$Fio$\"3w**R?/(QN$pF1Fdo7&Fho7$$\"3qmm;/RE&G#F1F)7$F`p$\"3wSrkC'o Ft'F1F[p7&F_p7$$\"3\")*****\\K]4]#F1F)7$Fgp$\"3x2jhWX(y^'F1Fbp7&Ffp7$$ \"3$******\\PAvr#F1F)7$F^q$\"3b<%=(R_28jF1Fip7&F]q7$$\"3)******\\nHi#H F1F)7$Feq$\"3u4Z')3uMDhF1F`q7&Fdq7$$\"3jmm\"z*ev:JF1F)7$F\\r$\"3))\\nZ #y$\\ifF1Fgq7&F[r7$$\"3?LLL347TLF1F)7$Fcr$\"3]%*G\"3tNwx&F1F^r7&Fbr7$$ \"3,LLLLY.KNF1F)7$Fjr$\"3KA,!yp=!GcF1Fer7&Fir7$$\"3w***\\7o7Tv$F1F)7$F as$\"3uk%H%yG\\haF1F\\s7&F`s7$$\"3'GLLLQ*o]RF1F)7$Fhs$\"3())\\'RvpS?`F 1Fcs7&Fgs7$$\"3A++D\"=lj;%F1F)7$F_t$\"3sg?i[6+s^F1Fjs7&F^t7$$\"31++vV& R3R;l.&F1Fat7&Fet7$$\"3WLL$e9Ege%F1F)7$F]u$\"3]D&G /oW3!\\F1Fht7&F\\u7$$\"3GLLeR\"3Gy%F1F)7$Fdu$\"3gF\"\\>Ew5y%F1F_u7&Fcu 7$$\"3cmm;/T1&*\\F1F)7$F[v$\"3!p,Hsqjnl%F1Ffu7&Fju7$$\"3&em;zRQb@&F1F) 7$Fbv$\"3T,h,RloKXF1F]v7&Fav7$$\"3\\***\\(=>Y2aF1F)7$Fiv$\"3g$peQFJ'GW F1Fdv7&Fhv7$$\"39mm;zXu9cF1F)7$F`w$\"3]PB;,!e,K%F1F[w7&F_w7$$\"3l***** *\\y))GeF1F)7$Fgw$\"3&R&*pk&G87UF1Fbw7&Ffw7$$\"3'*)***\\i_QQgF1F)7$F^x $\"3%[#=(><6-6%F1Fiw7&F]x7$$\"3@***\\7y%3TiF1F)7$Fex$\"3,,O'=L_\\,%F1F `x7&Fdx7$$\"35****\\P![hY'F1F)7$F\\y$\"3ueQ8-(\\G\"RF1Fgx7&F[y7$$\"3kK LL$Qx$omF1F)7$Fcy$\"3Z8pc$pIU#QF1F^y7&Fby7$$\"3!)*****\\P+V)oF1F)7$Fjy $\"3+>o^g5rKPF1Fey7&Fiy7$$\"3?mm\"zpe*zqF1F)7$Faz$\"3u$Q@xc9Cl$F1F\\z7 &F`z7$$\"3%)*****\\#\\'QH(F1F)7$Fhz$\"3g=c23(etc$F1Fcz7&Fgz7$$\"3GKLe9 S8&\\(F1F)7$F_[l$\"3_$4!4/C\")*[$F1Fjz7&F^[l7$$\"3R***\\i?=bq(F1F)7$Ff [l$\"3aR**HY5@6MF1Fa[l7&Fe[l7$$\"3\"HLL$3s?6zF1F)7$F]\\l$\"3m?3X[MoOLF 1Fh[l7&F\\\\l7$$\"3a***\\7`Wl7)F1F)7$Fd\\l$\"3!QC5`425E$F1F_\\l7&Fc\\l 7$$\"3#pmmm'*RRL)F1F)7$F[]l$\"3M%e^\"Q=H!>$F1Ff\\l7&Fj\\l7$$\"3Qmm;a<. Y&)F1F)7$Fb]l$\"3kt:HI83?JF1F]]l7&Fa]l7$$\"3=LLe9tOc()F1F)7$Fi]l$\"3Ku $\\t#eY_IF1Fd]l7&Fh]l7$$\"3u******\\Qk\\*)F1F)7$F`^l$\"39@1')>&H?*HF1F [^l7&F_^l7$$\"3CLL$3dg6<*F1F)7$Fg^l$\"3Cy%>qlxY#HF1Fb^l7&Ff^l7$$\"3Imm mmxGp$*F1F)7$F^_l$\"3am4v'[,h'GF1Fi^l7&F]_l7$$\"3A++D\"oK0e*F1F)7$Fe_l $\"3,!3o,W3`!GF1F`_l7&Fd_l7$$\"3A++v=5s#y*F1F)7$F\\`l$\"3KK9CPgm[FF1Fg _l7&F[`l7$F3F)7$F3$\"3/^**p8UT*o#F1F^`l7\"-%&STYLEG6#%,PATCHNOGRIDG-%& COLORG6&%$RGBG$\"#v!\"#F_alF_al-%'CURVESG6$7U7$$!35+++++++?F1$\"3dT%p- *=@;;!#<7$$!3aLLe*=CZ\"=F1$\"3%\\\"\\;-aSK:F[bl7$$!3'pmm\"z$[%H;F1$\"3 ;0ltsVDc9F[bl7$$!3A+v=()3Co9F1$\"3#Q`fdEOaR\"F[bl7$$!3]L$3_RLqI\"F1$\" 3c6XqZ>4R8F[bl7$$!30om;a-WW%*F-$\"3w2ulV4ME7F[bl7$$!3smm;z<^%z&F-$\"3c HMq&Hy*G6F[bl7$$!3gML3x2$>;#F-$\"3<@;ly#R\\/\"F[bl7$$\"3YILe9o$f?\"F-$ \"3Ad>RKnzj(*F17$$\"31(**\\7joJp%F-$\"3cpTw3%pD8*F17$$\"3[MLek(o'*H)F- $\"3/%Q]s`2/b)F17$$\"3')**\\i!Gg%*=\"F1$\"3nx[e7F[I!)F17$$\"3mmm;MMCf: F1$\"3+LlSc[4\\vF17$$\"3%HL$3P'[\\)=F1$\"3%Q>A*G[qjrF17$$\"3M++]_bh^AF 1$\"3-x1@+[LnnF17$$\"3*)****\\P!)y>EF1$\"3^C'\\km*=/kF17$$\"37++]Z/fuH F1$\"3c9)=GmBJ3'F17$$\"3gK$ek-&y'H$F1$\"3&>R%Q!=zK\"eF17$$\"3qmm;Wb!*z OF1$\"38Rnxz1F;bF17$$\"36mmmw)eW+%F1$\"3Qm1=3!*z#G&F17$$\"3i**\\7e:*>Q %F1$\"3N^D$Q#f*)H]F17$$\"39mmm^><;ZF1$\"3)Gb$)4dL6#[F17$$\"3E**\\733#G 3&F1$\"3;llS]sx1YF17$$\"3<**\\PCs&>V&F1$\"3r,^Bp^g:WF17$$\"3[nm\"zWWiz &F1$\"36u_'RnV$GUF17$$\"3en;HPQxIhF1$\"3!ecb()4%RmSF17$$\"3%=L$3x*3;\\ 'F1$\"37+`n1I`,RF17$$\"3/L$ekF:k'oF1$\"3kYCuER(F1 $\"3A'\\Kj'RC2OF17$$\"3aLLe%yl]a(F1$\"3m3u[>W$4Z$F17$$\"3H)****\\M4\"4 zF1$\"3z[&p)RBVPLF17$$\"3C++DY\\Dl#)F1$\"3g1z(eG\"[8KF17$$\"3L**\\7GT% )4')F1$\"3%[9(4&)*f$*4$F17$$\"3s++vj;X#**)F1$\"3A-[M>p&)yHF17$$\"3Ommm ^:CO$*F1$\"3q(Re)=^wvGF17$$\"3+++]P1J.(*F1$\"3Q\"o7HEO2x#F17$$\"3ILeky Hf.5F[bl$\"3_F/S&=t(zEF17$$\"3A++DPq&*R5F[bl$\"3'R!Q%=77Ye#F17$$\"3[m \"zCysT2\"F[bl$\"3!Rb6Y86*)\\#F17$$\"3'**\\i]4Q*46F[bl$\"3`5&GQ9kIT#F1 7$$\"3^mmTD_!\\9\"F[bl$\"3QM&=a+2EL#F17$$\"3++DJqD^\"=\"F[bl$\"3gl_\\Q *R=D#F17$$\"3aLLL%zpn@\"F[bl$\"3Ig#p(f,@x@F17$$\"3GL$3#)RDGD\"F[bl$\"3 #*zK#4#\\\"R5#F17$$\"3%o;zMW#e)G\"F[bl$\"3jQ&y'o=3M?F17$$\"36++]a%R9K \"F[bl$\"3iq$*Q\"4)Gs>F17$$\"3#omTqH(4f8F[bl$\"3ok&pA5=T!>F17$$\"3QLLL ?*yFR\"F[bl$\"3cOW*>aEa%=F17$$\"3/+D\"eb!pG9F[bl$\"37+#[]v,^y\"F17$$\" 3++v=tD1j9F[bl$\"3b#H^pnO%H " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "More formally, " }{XPPEDIT 18 0 "Int(1/(x+exp(x)),x = 0 .. 1)" "6#-%$IntG6$*&\"\"\"F',&%\"xGF'-%$expG6 #F)F'!\"\"/F);\"\"!F'" }{TEXT -1 26 " is the limit of the form " } {XPPEDIT 18 0 "Limit(``,delta*x=0)" "6#-%&LimitG6$%!G/*&%&deltaG\"\"\" %\"xGF*\"\"!" }{XPPEDIT 18 0 "Sum(``(1/(x+exp(x))),x = 0 .. 1);" "6#-% $SumG6$-%!G6#*&\"\"\"F*,&%\"xGF*-%$expG6#F,F*!\"\"/F,;\"\"!F*" }{TEXT -1 64 ", where the summation extends over an ever increasing number of " }{TEXT 324 1 "x" }{TEXT -1 71 " values between 0 and 1 lying in a c orresponding subintervals of width " }{XPPEDIT 18 0 "delta" "6#%&delta G" }{TEXT 326 1 "x" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "delta" "6#%& deltaG" }{XPPEDIT 18 0 "x -> 0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6 \"\"\"!F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "Maple can obtain a numerical value for th is integral by a suitable algorithm for " }{TEXT 265 21 "numerical int egration" }{TEXT -1 21 " based on evaluating " }{XPPEDIT 18 0 "f(x)=1/ (x+exp(x))" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F)-%$expG6#F'F)!\"\"" } {TEXT -1 22 " at various values of " }{TEXT 325 1 "x" }{TEXT -1 30 " t hroughout the interval from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 35 " . This can be achieved as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(1/(x+exp(x)),x=0..1);\n evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&%\"x GF'-%$expG6#F)F'!\"\"/F);\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+Mw+j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The following command produces the same result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(I nt(1/(x+exp(x)),x=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mw+j^ !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "A more accurate value can be obtained by increasing precision. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf[20](Int(1/(x+exp(x)),x=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5$>n;!pLw+j^!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 65 "As a rough check on these values consider the foll owing picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 426 "f := x->1/(x+exp(x)):\np1 := plot(f(x),x=0..1 ,color=COLOR(RGB,.75,.75,.75),filled=true):\np2 := plot(f(x),x=-.2..1. 5,color=red):\np3 := plot([[0,0],[0,1],[1,f(1)],[1,0],[0,0]],\n \+ color=COLOR(RGB,.4,0,.9),thickness=2):\np4 := plot([[0,0],[0,1],[1,1 ],[1,0],[0,0]],\n color=COLOR(RGB,.4,0,.9),linestyle=2):\np5 : = plot([[0,f(1)],[1,f(1)]],\n color=COLOR(RGB,0,.9,0),thicknes s=2):\nplots[display]([p1,p2,p3,p4,p5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 370 336 336 {PLOTDATA 2 "6)-%)POLYGONSG6U7&7$$\"\"!F)F)7$$\" 3emmm;arz@!#>F)7$F+$\"33M!RN)42!e*!#=7$F($\"\"\"F)7&F*7$$\"3[LL$e9ui2% F-F)7$F7$\"3gIRBbX+R#*F1F.7&F67$$\"3nmmm\"z_\"4iF-F)7$F>$\"3%\\5Hy\"=! )z))F1F97&F=7$$\"3[mmmT&phN)F-F)7$FE$\"3e7XH+gzT&)F1F@7&FD7$$\"3CLLe*= )H\\5F1F)7$FL$\"3E:G[qTkE#)F1FG7&FK7$$\"3gmm\"z/3uC\"F1F)7$FS$\"31>zjK =o^zF1FN7&FR7$$\"3%)***\\7LRDX\"F1F)7$FZ$\"3%Gb![M!GHo(F1FU7&FY7$$\"3] mm\"zR'ok;F1F)7$F[o$\"3HMqTvji?uF1Ffn7&Fjn7$$\"3w***\\i5`h(=F1F)7$Fbo$ \"3i0&)4$4!otrF1F]o7&Fao7$$\"3WLLL3En$4#F1F)7$Fio$\"3w**R?/(QN$pF1Fdo7 &Fho7$$\"3qmm;/RE&G#F1F)7$F`p$\"3wSrkC'oFt'F1F[p7&F_p7$$\"3\")*****\\K ]4]#F1F)7$Fgp$\"3x2jhWX(y^'F1Fbp7&Ffp7$$\"3$******\\PAvr#F1F)7$F^q$\"3 b<%=(R_28jF1Fip7&F]q7$$\"3)******\\nHi#HF1F)7$Feq$\"3u4Z')3uMDhF1F`q7& Fdq7$$\"3jmm\"z*ev:JF1F)7$F\\r$\"3))\\nZ#y$\\ifF1Fgq7&F[r7$$\"3?LLL347 TLF1F)7$Fcr$\"3]%*G\"3tNwx&F1F^r7&Fbr7$$\"3,LLLLY.KNF1F)7$Fjr$\"3KA,!y p=!GcF1Fer7&Fir7$$\"3w***\\7o7Tv$F1F)7$Fas$\"3uk%H%yG\\haF1F\\s7&F`s7$ $\"3'GLLLQ*o]RF1F)7$Fhs$\"3())\\'RvpS?`F1Fcs7&Fgs7$$\"3A++D\"=lj;%F1F) 7$F_t$\"3sg?i[6+s^F1Fjs7&F^t7$$\"31++vV&R3R;l.&F1F at7&Fet7$$\"3WLL$e9Ege%F1F)7$F]u$\"3]D&G/oW3!\\F1Fht7&F\\u7$$\"3GLLeR \"3Gy%F1F)7$Fdu$\"3gF\"\\>Ew5y%F1F_u7&Fcu7$$\"3cmm;/T1&*\\F1F)7$F[v$\" 3!p,Hsqjnl%F1Ffu7&Fju7$$\"3&em;zRQb@&F1F)7$Fbv$\"3T,h,RloKXF1F]v7&Fav7 $$\"3\\***\\(=>Y2aF1F)7$Fiv$\"3g$peQFJ'GWF1Fdv7&Fhv7$$\"39mm;zXu9cF1F) 7$F`w$\"3]PB;,!e,K%F1F[w7&F_w7$$\"3l******\\y))GeF1F)7$Fgw$\"3&R&*pk&G 87UF1Fbw7&Ffw7$$\"3'*)***\\i_QQgF1F)7$F^x$\"3%[#=(><6-6%F1Fiw7&F]x7$$ \"3@***\\7y%3TiF1F)7$Fex$\"3,,O'=L_\\,%F1F`x7&Fdx7$$\"35****\\P![hY'F1 F)7$F\\y$\"3ueQ8-(\\G\"RF1Fgx7&F[y7$$\"3kKLL$Qx$omF1F)7$Fcy$\"3Z8pc$pI U#QF1F^y7&Fby7$$\"3!)*****\\P+V)oF1F)7$Fjy$\"3+>o^g5rKPF1Fey7&Fiy7$$\" 3?mm\"zpe*zqF1F)7$Faz$\"3u$Q@xc9Cl$F1F\\z7&F`z7$$\"3%)*****\\#\\'QH(F1 F)7$Fhz$\"3g=c23(etc$F1Fcz7&Fgz7$$\"3GKLe9S8&\\(F1F)7$F_[l$\"3_$4!4/C \")*[$F1Fjz7&F^[l7$$\"3R***\\i?=bq(F1F)7$Ff[l$\"3aR**HY5@6MF1Fa[l7&Fe[ l7$$\"3\"HLL$3s?6zF1F)7$F]\\l$\"3m?3X[MoOLF1Fh[l7&F\\\\l7$$\"3a***\\7` Wl7)F1F)7$Fd\\l$\"3!QC5`425E$F1F_\\l7&Fc\\l7$$\"3#pmmm'*RRL)F1F)7$F[]l $\"3M%e^\"Q=H!>$F1Ff\\l7&Fj\\l7$$\"3Qmm;a<.Y&)F1F)7$Fb]l$\"3kt:HI83?JF 1F]]l7&Fa]l7$$\"3=LLe9tOc()F1F)7$Fi]l$\"3Ku$\\t#eY_IF1Fd]l7&Fh]l7$$\"3 u******\\Qk\\*)F1F)7$F`^l$\"39@1')>&H?*HF1F[^l7&F_^l7$$\"3CLL$3dg6<*F1 F)7$Fg^l$\"3Cy%>qlxY#HF1Fb^l7&Ff^l7$$\"3ImmmmxGp$*F1F)7$F^_l$\"3am4v'[ ,h'GF1Fi^l7&F]_l7$$\"3A++D\"oK0e*F1F)7$Fe_l$\"3,!3o,W3`!GF1F`_l7&Fd_l7 $$\"3A++v=5s#y*F1F)7$F\\`l$\"3KK9CPgm[FF1Fg_l7&F[`l7$F3F)7$F3$\"3/^**p 8UT*o#F1F^`l7\"-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$\"#v!\"#F_al F_al-%'CURVESG6$7U7$$!35+++++++?F1$\"3dT%p-*=@;;!#<7$$!3aLLe*=CZ\"=F1$ \"3%\\\"\\;-aSK:F[bl7$$!3'pmm\"z$[%H;F1$\"3;0ltsVDc9F[bl7$$!3A+v=()3Co 9F1$\"3#Q`fdEOaR\"F[bl7$$!3]L$3_RLqI\"F1$\"3c6XqZ>4R8F[bl7$$!30om;a-WW %*F-$\"3w2ulV4ME7F[bl7$$!3smm;z<^%z&F-$\"3cHMq&Hy*G6F[bl7$$!3gML3x2$>; #F-$\"3<@;ly#R\\/\"F[bl7$$\"3YILe9o$f?\"F-$\"3Ad>RKnzj(*F17$$\"31(**\\ 7joJp%F-$\"3cpTw3%pD8*F17$$\"3[MLek(o'*H)F-$\"3/%Q]s`2/b)F17$$\"3')** \\i!Gg%*=\"F1$\"3nx[e7F[I!)F17$$\"3mmm;MMCf:F1$\"3+LlSc[4\\vF17$$\"3%H L$3P'[\\)=F1$\"3%Q>A*G[qjrF17$$\"3M++]_bh^AF1$\"3-x1@+[LnnF17$$\"3*)** **\\P!)y>EF1$\"3^C'\\km*=/kF17$$\"37++]Z/fuHF1$\"3c9)=GmBJ3'F17$$\"3gK $ek-&y'H$F1$\"3&>R%Q!=zK\"eF17$$\"3qmm;Wb!*zOF1$\"38Rnxz1F;bF17$$\"36m mmw)eW+%F1$\"3Qm1=3!*z#G&F17$$\"3i**\\7e:*>Q%F1$\"3N^D$Q#f*)H]F17$$\"3 9mmm^><;ZF1$\"3)Gb$)4dL6#[F17$$\"3E**\\733#G3&F1$\"3;llS]sx1YF17$$\"3< **\\PCs&>V&F1$\"3r,^Bp^g:WF17$$\"3[nm\"zWWiz&F1$\"36u_'RnV$GUF17$$\"3e n;HPQxIhF1$\"3!ecb()4%RmSF17$$\"3%=L$3x*3;\\'F1$\"37+`n1I`,RF17$$\"3/L $ekF:k'oF1$\"3kYCuER(F1$\"3A'\\Kj'RC2OF17$$\"3aLL e%yl]a(F1$\"3m3u[>W$4Z$F17$$\"3H)****\\M4\"4zF1$\"3z[&p)RBVPLF17$$\"3C ++DY\\Dl#)F1$\"3g1z(eG\"[8KF17$$\"3L**\\7GT%)4')F1$\"3%[9(4&)*f$*4$F17 $$\"3s++vj;X#**)F1$\"3A-[M>p&)yHF17$$\"3Ommm^:CO$*F1$\"3q(Re)=^wvGF17$ $\"3+++]P1J.(*F1$\"3Q\"o7HEO2x#F17$$\"3ILekyHf.5F[bl$\"3_F/S&=t(zEF17$ $\"3A++DPq&*R5F[bl$\"3'R!Q%=77Ye#F17$$\"3[m\"zCysT2\"F[bl$\"3!Rb6Y86*) \\#F17$$\"3'**\\i]4Q*46F[bl$\"3`5&GQ9kIT#F17$$\"3^mmTD_!\\9\"F[bl$\"3Q M&=a+2EL#F17$$\"3++DJqD^\"=\"F[bl$\"3gl_\\Q*R=D#F17$$\"3aLLL%zpn@\"F[b l$\"3Ig#p(f,@x@F17$$\"3GL$3#)RDGD\"F[bl$\"3#*zK#4#\\\"R5#F17$$\"3%o;zM W#e)G\"F[bl$\"3jQ&y'o=3M?F17$$\"36++]a%R9K\"F[bl$\"3iq$*Q\"4)Gs>F17$$ \"3#omTqH(4f8F[bl$\"3ok&pA5=T!>F17$$\"3QLLL?*yFR\"F[bl$\"3cOW*>aEa%=F1 7$$\"3/+D\"eb!pG9F[bl$\"37+#[]v,^y\"F17$$\"3++v=tD1j9F[bl$\"3b#H^pnO%H " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The area of the shaded region which represents " }{XPPEDIT 18 0 "Int(1/(x +exp(x)),x=0..1)" "6#-%$IntG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"/F) ;\"\"!F'" }{TEXT -1 52 " lies between the area of the trapezoid outlin ed in " }{TEXT 265 6 "purple" }{TEXT -1 77 " and the area of the recta ngle with base given by the line segment along the " }{TEXT 327 1 "x" }{TEXT -1 34 " axis from the origin to the point" }{XPPEDIT 18 0 " ``( 1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 26 ", and having the parallel " }{TEXT 270 5 "green" }{TEXT -1 56 " line segment as another side. In t erms of the function " }{XPPEDIT 18 0 "f(x)=1/(x+exp(x))" "6#/-%\"fG6# %\"xG*&\"\"\"F),&F'F)-%$expG6#F'F)!\"\"" }{TEXT -1 37 ", the area of t his rectangle is f(1) " }{TEXT 329 1 "~" }{TEXT -1 44 " 0.2689414214, \+ while the trapezoid has area " }{XPPEDIT 18 0 "1- (1 - f(1))/2 = (1+f( 1))/2" "6#/,&\"\"\"F%*&,&F%F%-%\"fG6#F%!\"\"F%\"\"#F+F+*&,&F%F%-F)6#F% F%F%F,F+" }{TEXT -1 1 " " }{TEXT 328 1 "~" }{TEXT -1 22 " 0 .634470710 7, where " }{XPPEDIT 18 0 "(1-f(1))/2" "6#*&,&\"\"\"F%-%\"fG6#F%!\"\"F %\"\"#F)" }{TEXT -1 35 " is the area of the upper triangle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "f \+ := x->1/(x+exp(x)):\nrectangle_area := evalf(f(1));\ntrapezoid_area := evalf((1+f(1))/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/rectangle_ar eaG$\"+9UT*o#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/trapezoid_areaG $\"+2rqWj!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "To obtain information concerning Maple's numerical integr ation see the relevant help page .. " }{HYPERLNK 17 "int[numerical]" 2 "int[numerical]" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 " This help page can also be accessed by executing the following command . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?evalf/Int" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We now return t o the first integral " }{XPPEDIT 18 0 "Int(1/(1+exp(x)),x=0..1)" "6#-% $IntG6$*&\"\"\"F',&F'F'-%$expG6#%\"xGF'!\"\"/F,;\"\"!F'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 68 "Distinguish carefully between the f ollowing two Maple computations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(int(1/(1+exp(x)),x=0.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+O\\&))z$!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf (Int(1/(1+exp(x)),x=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I\\ &))z$!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "In the first case \+ where " }{TEXT 0 3 "int" }{TEXT -1 39 " is used, Maple evaluates the i ntegral " }{TEXT 265 12 "symbolically" }{TEXT -1 41 " (analytically) t o obtain the expression " }{XPPEDIT 18 0 "ln(2)-ln(1+exp(1))+1" "6#,(- %#lnG6#\"\"#\"\"\"-F%6#,&F(F(-%$expG6#F(F(!\"\"F(F(" }{TEXT -1 82 ", a nd then evaluates this expression numerically using Maple's built-in f unctions " }{TEXT 281 2 "ln" }{TEXT -1 5 " and " }{TEXT 281 3 "exp" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "In the second case Mapl e uses " }{TEXT 265 21 "numerical integration" }{TEXT -1 135 ". The se cond result is accurate to 10 digits while the first result exhibits r ounding error. This can be compensated for by evaluating " }{XPPEDIT 18 0 "ln(2)-ln(1+exp(1))+1" "6#,(-%#lnG6#\"\"#\"\"\"-F%6#,&F(F(-%$expG 6#F(F(!\"\"F(F(" }{TEXT -1 55 " with a higher precision and then round ing the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "int(1/(1+exp(x)),x=0..1);\nevalf[14](%);\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%#lnG6#\"\"#\"\"\"-F%6#,&F (F(-%$expG6#F(F(!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/vTI\\ &))z$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I\\&))z$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Examples to compare symbolic \+ and numerical integration " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 151 " When learning integral calculus, it rapidly becomes clear that it is i s not always possible to find an analytical formula for the value of a n integral." }}{PARA 0 "" 0 "" {TEXT -1 64 "Students may often attempt to give a value an integral such as " }{XPPEDIT 18 0 "Int(cos(x^2),x )" "6#-%$IntG6$-%$cosG6#*$%\"xG\"\"#F*" }{TEXT -1 55 " by perhaps sug gesting that some expression involving " }{XPPEDIT 18 0 "sin(x^2)" "6# -%$sinG6#*$%\"xG\"\"#" }{TEXT -1 14 " would \"work\"." }}{PARA 0 "" 0 "" {TEXT -1 96 "Maple can in fact give an analytical expression for th is integral, but it is certainly not just " }{XPPEDIT 18 0 "sin(x^2)" "6#-%$sinG6#*$%\"xG\"\"#" }{TEXT -1 9 ", because" }}{PARA 259 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[sin(x^2)] = 2*x*cos(x^2)" "6#/7#-%$sin G6#*$%\"xG\"\"#*(F*\"\"\"F)F,-%$cosG6#*$F)F*F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "As mentio ned in the first section, there are many " }{TEXT 265 30 "special math ematical functions" }{TEXT -1 21 " \"built-into\" Maple. " }}{PARA 0 " " 0 "" {TEXT -1 47 "For a list of functions available in Maple see " } {HYPERLNK 17 "inifcn" 2 "inifcn" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 125 "This enables Maple to evaluate many more integrals symbo lically than would be possible just using the elementary functions: \+ " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "ln*x" "6#*&%#lnG\"\"\"%\"xGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "tan*x" "6#*&%$tanG\"\"\"%\"xGF%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "arcsin*x" "6#*&%'arcsinG\"\"\"%\"xGF%" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "arctan*x" "6#*&%'arctanG\"\"\"%\"xGF%" }{TEXT -1 7 " , etc. " }}{PARA 0 "" 0 "" {TEXT -1 65 "For example, Maple gives a sym bolic expression for the value of " }{XPPEDIT 18 0 "Int(cos(x^2),x)" "6#-%$IntG6$-%$cosG6#*$%\"xG\"\"#F*" }{TEXT -1 65 " which involves th e Fresnel cosine function denoted in Maple by " }{TEXT 281 11 "Fresnel C(x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 93 "This special fun ction is defined by an integral very similar to the one we are conside ring: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "FresnelC(x ) = int(cos(Pi/2*t^2), t=0..x)" "6#/-%)FresnelCG6#%\"xG-%$intG6$-%$co sG6#*(%#PiG\"\"\"\"\"#!\"\"%\"tGF1/F3;\"\"!F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "See " }{HYPERLNK 17 "Fresnel" 2 "Fresnel" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "Int(cos(x^2),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#*$)%\"xG\"\"#\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%%sqrtG6#\"\"#\"\"\"-F&6#%#PiGF)-%)Fres nelCG6#*&*&F%F)%\"xGF)F)*$-F&6#F,F)!\"\"F)#F)F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Now consider the definite integral " }{XPPEDIT 18 0 "Int(cos(x^2),x=0..sqrt(Pi/2)) " "6#-%$IntG6$-%$cosG6#*$%\"xG\"\"#/F*;\"\"!-%%sqrtG6#*&%#PiG\"\"\"F+! \"\"" }{TEXT -1 85 ". The value of this integral gives the area of the region enclosed between the curve " }{XPPEDIT 18 0 "y=cos(x^2)" "6#/% \"yG-%$cosG6#*$%\"xG\"\"#" }{TEXT -1 9 " and the " }{TEXT 330 1 "x" } {TEXT -1 11 " axis from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 17 " up to the first " }{TEXT 333 1 "x" }{TEXT -1 33 " intercept of the curve with the " }{TEXT 331 1 "x" }{TEXT -1 37 " axis which lies \+ to the right of the " }{TEXT 332 1 "y" }{TEXT -1 59 " axis. This is th e shaded region in the following picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f := x->cos(x^2): 'f(x)'=f(x);\np1 := plot(f(x),x=0..sqrt(Pi/2),color=COLOR(RGB,.75,.75 ,.75),filled=true):\np2 := plot(f(x),x=-.1..2.3,color=red,thickness=2) :\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%\"xG-%$cosG6#*$)F'\"\"#\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 434 313 313 {PLOTDATA 2 "6&-%)POLYGONSG6&7^q7$$\"\"!F)F)7$$\"3;-;dT\"o=t#! #>F)7$$\"3ZwML/?&)3^F-F)7$$\"3503**e*=?y(F-F)7$$\"3YjR%*R0HZ5!#=F)7$$ \"3vM`j_-5:8F7F)7$$\"393(H$QTRj:F7F)7$$\"3\")*3B_2)[?=F7F)7$$\"3g-\\`# *\\P'3#F7F)7$$\"3#H\"382#49N#F7F)7$$\"3A)o)ot%HSi#F7F)7$$\"3)*oJ'Gb`T' GF7F)7$$\"3)Q;!y#RwW8$F7F)7$$\"3qT*\\U?4fS$F7F)7$$\"3%H(zP8][nOF7F)7$$ \"3P#ewu!4-0RF7F)7$$\"3W$RB01uu=%F7F)7$$\"3)H7&fI*[nU%F7F)7$$\"3O$4!3( [#30ZF7F)7$$\"3IpaPY[X^\\F7F)7$$\"3c`3WsVw@_F7F)7$$\"3e'p*)Q'H;zaF7F)7 $$\"3_W6=\"RJxu&F7F)7$$\"3_7TUX5O%*fF7F)7$$\"3-MT.YWQgiF7F)7$$\"3OhKv% *zqOlF7F)7$$\"3Wzh@c%[sx'F7F)7$$\"3%)4uTW(Qq.(F7F)7$$\"32rsiKvU0tF7F)7 $$\"3YU'>4g$*zc(F7F)7$$\"3'H?))GxR?#yF7F)7$$\"3g#psJt9T5)F7F)7$$\"3C)) *y2jrvN)F7F)7$$\"3k%Gl#o4>G')F7F)7$$\"3Kh*p*4BTt))F7F)7$$\"3$>Q(*y+/:9 *F7F)7$$\"3O7Dx-uv$R*F7F)7$$\"3oB*GR)[Vd'*F7F)7$$\"3xF\\h%GqT2?\"F[sF)7$$\"3*fAqHD#3E7F[sF)7$$\"35++\"[89LD \"F[sF)7$Fet$\"3Y*em&GwN!G'!#E7$Fbt$\"3e?*ffrMnu'F-7$F_t$\"3c]\"Hav!e' G\"F77$F\\t$\"3@&R`#e%*>2>F77$Fis$\"3WL\\()e48qCF77$Ffs$\"3=JM')*>1e2$ F77$Fcs$\"3MY\"HAv*)y1\\3&F77$Ffr$\"3D:rr&H3Va&F77$Fcr$\"3kf%y)3<*p&fF77$F`r$\"3 7OOm]&)y_jF77$F]r$\"3.+qC[:!oq'F77$Fjq$\"3?;#eJ[zq0(F77$Fgq$\"3#3;*oB2 catF77$Fdq$\"3e%3#[219ewF77$Faq$\"30$3=9b/(>zF77$F^q$\"3%3c_-'z!f=)F77 $F[q$\"3qLN n>ld0j*F77$F]o$\"3P(3;t1j4q*F77$Fjn$\"3'fW**y*z&fv*F77$Fgn$\"3du]i]*4' 3)*F77$FZ$\"3)H^j9Zcm%)*F77$FW$\"3[Y3Ahi&R))*F77$FT$\"3kH6))R!z'4**F77 $FQ$\"3t1!>?*GzK**F77$FN$\"35#[%4nSx^**F77$FK$\"3i^h!)z6Pm**F77$FH$\"3 'Rl;y4/j(**F77$FE$\"3M\\Bq!G=Z)**F77$FB$\"3)oIeGOF0***F77$F?$\"3g`Wi=' 3X***F77$F<$\"31M^())38q***F77$F9$\"3Q_1urW])***F77$F5$\"370Lu,&)R**** F77$F2$\"3)y4wfi;)****F77$F/$\"3IcnbQf'*****F77$F+$\"2OS/^@(******F[s7 $F($\"\"\"F)7\"-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$\"#v!\"#Fd^l Fd^l-%'CURVESG6%7jn7$$!3/+++++++5F7$\"3i_m;/+]****F77$$!3M,+++IooZF-$ \"35QN'Q9u*****F77$$!3)\\+++]?%p@!#?$\"2'\\#*))**********F[s7$$\"3!*)* *****pm>!\\F-$\"3Ql\\qH6(*****F77$$\"3!********o![05F7$\"3Q(GHD&*)[*** *F77$$\"3g*****\\l:$=:F7$\"3E\"HOT&HM(***F77$$\"3m*****\\JzP*>F7$\"3Q) \\;z5+@***F77$$\"3Y*****\\R%4'[#F7$\"3OQLcnd!4)**F77$$\"3;*****\\NZ_*H F7$\"3S@RJ>Iyf**F77$$\"3')*****\\XnF]$F7$\"3%y\"G1Nb#[#**F77$$\"3J+++g U\"[-%F7$\"3a([^?w!3p)*F77$$\"3i******pLj%[%F7$\"3Qzik/eV)z*F77$$\"3?+ ++!y!G-]F7$\"3!=k$Gi&f&)o*F77$$\"3<*******p`?_&F7$\"3\"[M7QRx'Q&*F77$$ \"3;+++?7&H-'F7$\"3atpP8>@\\$*F77$$\"3-*****\\:9yZ'F7$\"3#*Gb(z7NC8*F7 7$$\"36+++!=!p=qF7$\"3)>%oVet'4\"))F77$$\"3m******>J)oZ(F7$\"3o?Xv(>ew Z)F77$$\"3S*****\\Vq)4!)F7$\"39;\"fOU5:,)F77$$\"3?)*****>Xl\"[)F7$\"3, 9P+(\\8@_(F77$$\"3')*****\\Vw#***)F7$\"3/L!\\qbFf*oF77$$\"3Q+++0\\<#\\ *F7$\"3N[rOuZ:3iF77$$\"3\")*****\\FY1+\"F[s$\"335Ci(fQ@R&F77$$\"3!**** *\\`R(y/\"F[s$\"3#H`[ct@Mb%F77$$\"3u*****\\Q:))4\"F[s$\"3q*\\EQY_Xb$F7 7$$\"3')****\\:#H<:\"F[s$\"3n2#>%4$G*=CF77$$\"3#)****\\g3z(>\"F[s$\"3' fmXKjOnN\"F77$$\"3#*******)pQvC\"F[s$\"3%3Oe2\">IW9F-7$$\"3u*****R3L*) H\"F[s$!3!HyVU\\z;;\"F77$$\"3u*****HY7#\\8F[s$!3/&G\\Y,]*pCF77$$\"3w** **\\Z.'yR\"F[s$!3r^<(4ni!RPF77$$\"3%)******Gb(=X\"F[s$!3Yo/TP>'o6&F77$ $\"3h*****>d5/]\"F[s$!3Y^G1AkJ\"H'F77$$\"3u*******3KAb\"F[s$!3<$GRF]ns V(F77$$\"3e****\\(3!>*f\"F[s$!3:*R%)peD;M)F77$$\"3#)*****>eF0l\"F[s$!3 z5(R2\\d;9*F77$$\"3i***\\F()zYn\"F[s$!3wT3QLmPP%*F77$$\"3k****\\j@$))p \"F[s$!3)>]Oo(R@v'*F77$$\"3&)****\\m#ySs\"F[s$!3Qc5TrvGd)*F77$$\"3#)** **\\pVK\\fX,!>F[s$!3mDrM(G&Q?*)F77$$\"3%)*****4i Z5&>F[s$!3Q$3\"\\^Q?pyF77$$\"3'*****\\b\"G:+#F[s$!3SxdzSh.!\\'F77$$\"3 8++vR8sC?F[s$!3M:9-K(eBv&F77$$\"3%)*****R_9z/#F[s$!31*[Szk9_&\\F77$$\" 3#)****\\Il\\u?F[s$!3_%3EUVsb(RF77$$\"3!)*****p`y55#F[s$!3$pEj%H;tMHF7 7$$\"3]****\\+Q&[7#F[s$!3gb.p731h>F77$$\"3k*****R1H'[@F[s$!3Y\"Gcs&Qdj &*F-7$$\"3w***\\PXyR<#F[s$\"3iuc3h**Gz8F-7$$\"3')****\\VyK*>#F[s$\"3o. \\z%39LC\"F77$$\"30+++W/fBAF[s$\"3?fs!)f!3*)H#F77$$\"3!)****\\WI&yC#F[ s$\"35S9f(y`\"RLF77$$\"3-++DAl#RF#F[s$\"3f2e:4drCWF77$$\"3#)********** ***H#F[s$\"3Ovk\")>3CgaF7-%'COLOURG6&Fc^l$\"*++++\"!\")F(F(-%*THICKNES SG6#\"\"#-%+AXESLABELSG6%Q\"x6\"Q!F`cm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$ !\"\"F[dm$\"#BF[dmFecm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 328 "A numerical value for the area of this r egion can be obtained by evaluating the definite integral symbolically and then evaluating the symbolic expression numerically. Often such a result exhibits rounding errors, but these can be compensated for by \+ performing the evaluation with higher precision and then rounding the \+ result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Int(cos(x^2),x=0..sqrt(Pi/2));\nvalue(%);\nevalf[13]( %);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#*$ )%\"xG\"\"#\"\"\"/F+;\"\"!,$*(F,!\"\"F,#F-F,%#PiGF4F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*(-%)FresnelCG6#F&F&F'F%%#PiGF%F &F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".:HC9Xx*!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VU^u(*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "Alternatively we can get Maple to perform numerical integration via the " }{TEXT 0 9 "evalf/Int" }{TEXT -1 9 " \+ scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(cos(x^2),x=0..sqrt(Pi/2));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#*$)%\"xG\"\"#\"\"\"/F+;\" \"!,$*(F,!\"\"F,#F-F,%#PiGF4F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ VU^u(*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "Consider the two indefinite integrals " }{XPPEDIT 18 0 "I nt(x^2*cos(x^3),x)" "6#-%$IntG6$*&%\"xG\"\"#-%$cosG6#*$F'\"\"$\"\"\"F' " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int(x*cos(x^3),x)" "6#-%$IntG6$* &%\"xG\"\"\"-%$cosG6#*$F'\"\"$F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 60 "The first integral can be found by means of the subsitu tion " }{XPPEDIT 18 0 "u=x^3" "6#/%\"uG*$%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x^2*cos(x^3),x)" "6#-%$IntG6$*&%\"xG\"\"#-%$cosG6#* $F'\"\"$\"\"\"F'" }{TEXT -1 9 " ... " }{XPPEDIT 18 0 "PIECEWISE([u = x^3, ``],[du = 3*x^2*dx, ``(1/3)*`.`*du = x^2*dx]);" "6#-%*PIECEWIS EG6$7$/%\"uG*$%\"xG\"\"$%!G7$/%#duG*(F+\"\"\"*$F*\"\"#F1%#dxGF1/*(-F,6 #*&F1F1F+!\"\"F1%\".GF1F/F1*&F*F3F4F1" }{TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*u,u)" "6#-%$IntG6$*&%$cosG \"\"\"%\"uGF(F)" }{TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sin*u+c = 1/3" "6#/,&*&%$sinG\"\"\"%\"uGF'F'%\"cGF'*&F' F'\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x^3) +c" "6#,&-%$sin G6#*$%\"xG\"\"$\"\"\"%\"cGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(x^2*cos( x^3),x);\n``=value(%)+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$* &)%\"xG\"\"#\"\"\"-%$cosG6#*$)F(\"\"$F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"$F(-%$sinG6#*$)%\"xGF)F(F(F(%\"cGF( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Cont rast this with Maple's (release 9) analytical expression for " } {XPPEDIT 18 0 "Int(x*cos(x^3),x)" "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#*$ F'\"\"$F(F'" }{TEXT -1 50 ", which involves a fairly exotic special fu nction " }{TEXT 281 11 "LommelS1(x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "See " }{HYPERLNK 17 "LommelS1" 2 "LommelS1" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(x*cos(x^3),x);\n``=value(%)+c;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#*$)F'\"\"$F(F(F'" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"'F(*()\"\"##F,\"\"$ F(%#PiG#F(F,,**&#F.F,F(**F/#!\"\"F,%\"xGF6F,#F(F.-%$sinG6#*$)F7F.F(F(F (F(*&F3F(**F/F5F7F6F,F8,&*&-%$cosGF;F(F=F(F(F9F6F(F(F(*&#F(F,F(*.F/F5F 7\"\"&F,F8F<#!\"(F)F9F(-%)LommelS1G6%F'F3F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The definite integral " }{XPPEDIT 18 0 "Int(x*c os(x^3),x = 0 .. (Pi/2)^(1/3));" "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#*$F '\"\"$F(/F';\"\"!)*&%#PiGF(\"\"#!\"\"*&F(F(F-F5" }{TEXT -1 57 " gives \+ the area of the region enclosed between the curve " }{XPPEDIT 18 0 "y \+ = x*cos(x^3);" "6#/%\"yG*&%\"xG\"\"\"-%$cosG6#*$F&\"\"$F'" }{TEXT -1 9 " and the " }{TEXT 334 1 "x" }{TEXT -1 11 " axis from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 17 " up to the first " }{TEXT 337 1 "x" }{TEXT -1 33 " intercept of the curve with the " }{TEXT 335 1 "x" }{TEXT -1 37 " axis which lies to the right of the " }{TEXT 336 1 "y" }{TEXT -1 59 " axis. This is the shaded region in the following \+ picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "f := x->x*cos(x^3): 'f(x)'=f(x);\np1 := plot(f(x),x= 0..(Pi/2)^(1/3),color=COLOR(RGB,.75,.75,.75),filled=true):\np2 := plot (f(x),x=-.1..2,color=red,thickness=2):\nplots[display]([p1,p2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\"-%$cosG6#*$)F' \"\"$F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 419 305 305 {PLOTDATA 2 "6&-%) POLYGONSG6&7jq7$$\"\"!F)F)7$$\"3rX423W!Q`#!#>F)7$$\"3WF\\NvSXQZF-F)7$$ \"3F8lK.K\"y@(F-F)7$$\"3pt!4I82Or*F-F)7$$\"3')=0)*)*Qv>7!#=F)7$$\"3oFl kyh/]9F:F)7$$\"3W_7>&\\+&)o\"F:F)7$$\"30KUZ!H5^$>F:F)7$$\"3_Nh,0#H4=#F :F)7$$\"3)*fygtTyLCF:F)7$$\"3Ez_dn*)\\cEF:F)7$$\"3.5rsvIA2HF:F)7$$\"3k (3c6ow*eJF:F)7$$\"3,.czGze,MF:F)7$$\"3tz^A%=->i$F:F)7$$\"3Wd>=Vr(Q)QF: F)7$$\"3d'GInH/e5%F:F)7$$\"3i%3U[LeRO%F:F)7$$\"3y,%*3,%oCf%F:F)7$$\"3' G\"*f1;!=V[F:F)7$$\"3[qv$R/<>3&F:F)7$$\"38JT\\OR,J`F:F)7$$\"37u^\\VEwf bF:F)7$$\"3Y8F(o-*\\1eF:F)7$$\"3_LZ?&y)yigF:F)7$$\"3MMFCt(*)eG'F:F)7$$ \"37n]P_\\%o_'F:F)7$$\"3e4+!*H_xvnF:F)7$$\"3=.2sT\\I>qF:F)7$$\"3W-<(*f C$\\D(F:F)7$$\"3FMtfXmb;vF:F)7$$\"3M(p(e7wj^xF:F)7$$\"3kIw(=sOE+)F:F)7 $$\"3&H6(p?#z+B)F:F)7$$\"3%*HKMZRty%)F:F)7$$\"3HY7+n')p7()F:F)7$$\"3)3 a8H@fs&*)F:F)7$$\"3g\"zS\"[=O'>*F:F)7$$\"3?)yW$[,oY%*F:F)7$$\"3cDpZAkw (o*F:F)7$$\"32Z\"R'e>JM**F:F)7$$\"3Cz$Gtf\")y,\"!#U^!>['o86FdsF)7$$\"3]TKvY\"Qa7\"FdsF)7$$\"3\"3M,;\")*=P6FdsF)7$$ \"3Uq1R!e=)\\6FdsF)7$$\"3/++=\\tWi6FdsF)7$Fgu$\"3ah?1*e5y4\"!#D7$Fdu$ \"3v)G./J2.#eF-7$Fau$\"3MONj69NP6F:7$F^u$\"3I8fE!*ybH;F:7$F[u$\"39mw;a $ex4#F:7$Fht$\"3Y$\\OCO%4hDF:7$Fet$\"3#>SoSd&)z*HF:7$Fbt$\"3o8()*H9/QQ $F:7$F_t$\"3i)4sUtRmu$F:7$F\\t$\"3=<0yyXWDTF:7$Fis$\"3C'yr'>@QwWF:7$Ff s$\"3c)35qb&HgZF:7$Fbs$\"3[.\"o-GHR-&F:7$F_s$\"3hNC*RPZ,`&F:7$F\\s$\"3 =*e7Vad<&fF:7$Fir$\"3YGi!obQSG'F:7$Ffr$\"3#)4KT7(fAb'F:7$Fcr$\"3\"3Zk0 43?u'F:7$F`r$\"3)[6O.kGb(oF:7$F]r$\"3[IRQMO(=&pF:7$Fjq$\"3U>]([gnS)pF: 7$Fgq$\"3g=)zQ[VW(pF:7$Fdq$\"32*R)\\g\"pe#pF:7$Faq$\"3a6EX-3))[oF:7$F^ q$\"3A^wT7>RKnF:7$F[q$\"3+/gT'p&o.mF:7$Fhp$\"3!)>3*z#[b]kF:7$Fep$\"3i# *3:1,=wiF:7$Fbp$\"3))>.,zC+$4'F:7$F_p$\"3UUZ1$4iG\"fF:7$F\\p$\"3![;65) ee&p&F:7$Fio$\"3]<(fxigyZ&F:7$Ffo$\"3'\\FZ0wY*p_F:7$Fco$\"3UJA\")*>6#Q ]F:7$F`o$\"3kW*)\\&Gh>\"[F:7$F]o$\"3/f**\\zG%4d%F:7$Fjn$\"39K=B\"G'*)[ VF:7$Fgn$\"3o*z(>*ftf4%F:7$FZ$\"3OFV8;O@xQF:7$FW$\"3hf#G8![\"yh$F:7$FT $\"3'\\rGP]`*)R$F:7$FQ$\"3U%H:'*=2u:$F:7$FN$\"3rl#\\cZXj!HF:7$FK$\"3!> bESa!GJNV#F:7$FE$\"35WJ/i=\"3=#F:7$FB$\"38_ZF&[f]$>F :7$F?$\"3A!o'HI4[)o\"F:7$F<$\"3K3l\"*Q%R+X\"F:7$F8$\"3R\"3k/*=v>7F:7$F 5$\"3$H`dK0.Or*F-7$F2$\"3*>VXIp7y@(F-7$F/$\"3-1p`[SXQZF-7$F+$\"3w&oNxS /Q`#F-7$F(F(7\"-%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$\"#v!\"#Ff`l Ff`l-%'CURVESG6%7[p7$$!3/+++++++5F:$!2mT+++&******F:7$$!3?+++DwfAaF-$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(x*cos(x^ 3),x=0..(Pi/2)^(1/3));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*&%\"xG\"\"\"-%$cosG6#*$)F'\"\"$F(F(/F';\"\"!,$*(\"\"#!\"\"F4# F4F.%#PiG#F(F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+XKy.\\!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The trapezoidal rule " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 265 16 "trapezoidal rule" }{TEXT -1 118 " for numerical integrat ion is based on the idea of estimating the area of the region enclosed between a curve and the " }{TEXT 339 1 "x" }{TEXT -1 163 " axis by ad ding together the area of trapezoids formed by various ordinates of po ints on the curve (vertical lines which join points on the curve to po ints on the " }{TEXT 338 1 "x" }{TEXT -1 8 " axis). " }}{PARA 0 "" 0 " " {TEXT -1 27 "First consider four points " }{XPPEDIT 18 0 "``(x[0],y[ 0]),``(x[1],y[1]),``(x[2],y[2]),``(x[3],y[3])" "6&-%!G6$&%\"xG6#\"\"!& %\"yG6#F)-F$6$&F'6#\"\"\"&F+6#F1-F$6$&F'6#\"\"#&F+6#F8-F$6$&F'6#\"\"$& F+6#F?" }{TEXT -1 15 " along a curve " }{XPPEDIT 18 0 "y = f(x)" "6#/% \"yG-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 259 "" 0 "" {TEXT -1 1 " \+ " }{GLPLOT2D 431 442 442 {PLOTDATA 2 "6A-%'CURVESG6%7S7$$!3++++++++:!# <$\"3++++++]iIF*7$$!3MLLL$Q6GT\"F*$\"310w%H:)yZJF*7$$!3bmm;M!\\pL\"F*$ \"3lP&o9c/*=KF*7$$!3MLLL))Qj^7F*$\"3#>c5.oWaH$F*7$$!3ALLL=Kvl6F*$\"3qN I!Rb;)oLF*7$$!3wmm;C2G!3\"F*$\"3j=o?3#y\"QMF*7$$!3OLL$3yO5+\"F*$\"3!fP Yc9A#*\\$F*7$$!3i+++vE%)*=*!#=$\"3)=(o9747fNF*7$$!3)RLL$3WDT$)FN$\"35; ..Ft_<$3xw$F*7$$!3w++++()>'*\\FN$\"3&eHv)G+p 7QF*7$$!3E++++0\"*HTFN$\"3=u\\lN=6aQF*7$$!35++++83&H$FN$\"3yvfcl!z/*QF *7$$!3\\LLL3k(p`#FN$\"3xm0bm\\[?RF*7$$!3Anmmmj^N;FN$\"33Z#pj![U_RF*7$$ !3tzmmmYh=()!#>$\"3/)o!H2JIwRF*7$$\"3-+*****\\s]k\"!#?$\"3r<\"f:f5/+%F *7$$\"3U9LLL`dF!)F[q$\"3XC+>*)yX=SF*7$$\"3'3++]sgam\"FN$\"3G1(e&>@qMSF *7$$\"3G+++v\"ep[#FN$\"3%=]k>b6n/%F*7$$\"3#QLLLe/TM$FN$\"3=cE#f-Xc0%F* 7$$\"39LLLeDBJTFN$\"3q\\2$y5811%F*7$$\"3Immm;kD!)\\FN$\"3-$Hxa-*\\iSF* 7$$\"3Qjmm\"f`@'eFN$\"3!3%4'zsT11%F*7$$\"3%z****\\nZ)HmFN$\"3eU2R\"**e e0%F*7$$\"3ckmm;$y*euFN$\"3o2a4kNQZSF*7$$\"3f)******R^bJ)FN$\"3&[*[G(z <].%F*7$$\"3'e*****\\5a`\"*FN$\"3V7\\'=Cq$>SF*7$$\"3'o****\\7RV'**FN$ \"3\\&)*HEM))3+%F*7$$\"3k*****\\@fk3\"F*$\"3'eR&G(R;l(RF*7$$\"3/LLL`4N n6F*$\"3)HFring6&RF*7$$\"3#*******\\,s`7F*$\"3B%*4(Q[w/#RF*7$$\"3[mm;z M)>L\"F*$\"3CK%)fs3X*)QF*7$$\"3$*******pfa<9F*$\"3e*=M#[t-_QF*7$$\"3#H LLeg`!)\\\"F*$\"3Gq2(*\\AZ8QF*7$$\"3w****\\#G2Ae\"F*$\"3a\"*fU\\oqpPF* 7$$\"3;LLL$)G[k;F*$\"3s@fg.\\\\BPF*7$$\"3#)****\\7yh]x$*yh$F*7$$\"3bmmm,FT=>F*$\"3)pmP>NE&fNF *7$$\"3FLL$e#pa-?F*$\"3y*>>c5$F*7$$!3FLL3L$F*7$$!3!om;/siq;\"F*$\"3]X#\\:ZDxO$F*7$$ !3%****\\(y$pZ7\"F*$\"3z%3pk6JDS$F*7$$!3ILLLyaE\"3\"F*$\"3e8yvr)*RPMF* 7$$!3mmm;>s%H/\"F*$\"3Oikrp%GtY$F*7$$!3Q+++]$*4)***!#=$\"3sB)=(RD9+NF* 7$$!39+++]_&\\c*Fao$\"3VVPT'>b@`$F*7$$!31+++]1aZ\"*Fao$\"3+%\\\"*Qt<@c $F*7$$!3umm;/#)[o()Fao$\"3Y3$H9$=d)e$F*7$$!3hLLL$=exJ)Fao$\"3$\\kDCH$4 >OF*7$$!3*RLLLtIf$zFao$\"3?0g:SU:WOF*7$$!3]++]PYx\"\\(Fao$\"3MzZ,;*)Qs OF*7$$!3EMLLL7i)4(Fao$\"3cR3j5%elp$F*7$$!3c****\\P'psm'Fao$\"3fwY,[q=A PF*7$$!3')****\\74_ciFao$\"3kDhOUosXPF*7$$!34LLL3x%z#eFao$\"37(*R1r))Q pPF*7$$!3VLL$3s$QMaFao$\"3k?g\"444.z$F*7$$!3&omm;zr)4]Fao$\"3p*)fymh+7 QF*7$$!3Iom;/K#*oXFao$\"3y,>yr#*eLQF*7$$!3-,+]ih2&=%Fao$\"3[&oA(Rfe^QF *7$$!3snmmT3^qPFao$\"3_=0%*[`>qQF*7$$!3q++++VAULFao$\"3oB7KM#=&))QF*7$ $!33-++v%HK#HFao$\"3yFir'3cb!RF*7$$!3d,+]P/$y^#Fao$\"3H'\\#yjb?@RF*7$$ !3y,++DRqn?Fao$\"3()[82.*=w$RF*7$$!3uMLLL_Cj;Fao$\"3i^6S2H]^RF*7$$!3R+ ++]#*RJ7Fao$\"3w[xYeTUlRF*7$$!3zvmmTg#3S)!#>$\"3![FTzeLs(RF*7$$!3N.+++ :qATF`v$\"3gZ&e&H$o#*)RF*7$$!3*[VNL$3(>t*!#@$\"3oD5&RYc(**RF*7$$\"3#y) ***\\7k.6%F`v$\"3r(\\\")H``)4SF*7$$\"3:emmmT9C#)F`v$\"309[tY%p)=SF*7$$ \"33****\\i!*3`7Fao$\"3^2h1Y;SFSF*7$$\"3%QLLL$*zym\"Fao$\"3$HH`'RCuMSF *7$$\"3wKLL3N1#4#Fao$\"3g$3,Mwf8/%F*7$$\"3Nmm;HYt7DFao$\"3ELwVxP.ZSF*7 $$\"3Y*******p(G**GFao$\"39s:e>vY^SF*7$$\"3]mmmT6KULFao$\"3+(e1?DIc0%F *7$$\"3fKLLLbdQPFao$\"3yOsy??_eSF*7$$\"3[++]i`1hTFao$\"3i1]or/ugSF*7$$ \"3W++]P?WlXFao$\"3ypIW)*y-iSF*7$$\"3++++++++]Fao$\"3++++++]iSF*-%&COL ORG6&%$RGBG$\"\"\"\"\"!$\"\"&!\"\"Fa[l-%*THICKNESSG6#F_[l-F$6$7&7$$Fc[ lF`[l$F`[lF`[l7$F[\\l$\"3++++++++NF*7$F\\\\l$\"\"%F`[l7$F\\\\lF\\\\l-% 'COLOURG6&F][lF`[lF`[lF`[l-F$6$7$7$F(F\\\\l7$FfzF\\\\lFd\\l-F$6$7%F]\\ l7$F[\\l$\"3+++++++]PF*7$F\\\\lF`]l-Fe\\l6&F][lF\\\\lF\\\\l$\"*++++\"! \")-%)POLYGONSG6%7'Fj[l7$F[\\l$\"#NFc[l7$$!\"&Fc[l$\"$v$!\"#7$F\\\\lFb ^lFc\\l-F[[l6&F][l$\"\"*Fc[lFh^lF^[l-%&STYLEG6#%,PATCHNOGRIDG-Fi]l6%7& F\\^l7$F[\\lFb^lFe^lF`\\l-F[[l6&F][l$\"\"(Fc[lFd_lF^[lFj^l-F$6&7$F]\\l F`\\l-%'SYMBOLG6#%'CIRCLEGFd\\l-F[_l6#%&POINTG-F$6&Fh_l-Fj_l6#%(DIAMON DGFd\\lF]`l-F$6&Fh_l-Fj_l6#%&CROSSGFd\\lF]`l-F$6%7$7$$!\"%Fc[l$\"#AFc[ l7$F^alFb^l7%7$$!+++++V!#5$\"++++dO!\"*Fbal7$$!+++++PFgalFhalFj^l-F$6% 7$7$F^al$\"$b\"Fd^l7$F^alF\\\\l7%7$F\\bl$\")+++$*FjalFdbl7$FealFgblFj^ l-%%TEXTG6%7$Fa[l$Fc[lFc[lQ\"x6\"-%%FONTG6$%*HELVETICAG\"#5-F[cl6%7$$! $D\"Fd^l$\"#PFc[lQ*(x~~,y~~)F`clFacl-F[cl6%7$$Fa^lFd^l$\"$D%Fd^lF]dlFa cl-F[cl6%7$F[\\l$!#8Fd^lF_clFacl-F[cl6%7$F\\\\lFgdlF_clFacl-F[cl6%7$F^ al$\"$0#Fd^lQ'y~~+~yF`clFacl-F[cl6%7$$!#QFd^l$\"$)>Fd^lQ&_____F`clFacl -F[cl6%7$Feel$\"$v\"Fd^lQ\"2F`clFacl-F[cl6%7$$!$@\"Fd^l$\"$j$Fd^lQ'0~~ ~~0F`cl-Fbcl6$Fdcl\"\")-F[cl6%7$$Fd^lFd^l$\"$=%Fd^lQ'1~~~~1F`clFhfl-F[ cl6%7$$!#$*Fd^l$Fd^lFc[lQ\"0F`clFhfl-F[cl6%7$$Fe_lFd^lFgglQ\"1F`clFhfl -F[cl6%7$$!#NFd^lFgelQ*0~~~~~~~1F`clFhfl-%+AXESLABELSG6%F_clQ!F`cl-Fbc l6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#:Fc[lFa[l;Fggl$\"#XFc [l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curv e 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21 " "Curve 22" "Curve 23" "Curve 24" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "The area of the first trapezoid is therefore" }{XPPEDIT 18 0 " ``((y[0]+y[1])/2)*h" "6#*&-%!G6#*&,&&%\"yG6#\"\"!\"\"\"&F*6#F-F -F-\"\"#!\"\"F-%\"hGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 120 "The area of the other three trapezoids is calculated in a similar way so that the total area of the three trapezoids is " }}{PARA 259 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``((y[0]+y[1])/2)*h+``((y[1]+y[2 ])/2)*h+``((y[2]+y[3])/2)*h;" "6#,(*&-%!G6#*&,&&%\"yG6#\"\"!\"\"\"&F+6 #F.F.F.\"\"#!\"\"F.%\"hGF.F.*&-F&6#*&,&&F+6#F.F.&F+6#F1F.F.F1F2F.F3F.F .*&-F&6#*&,&&F+6#F1F.&F+6#\"\"$F.F.F1F2F.F3F.F." }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 256 3 " = " } {XPPEDIT 18 0 "``(y[0]/2+y[1]+y[2]+y[3]/2)*h;" "6#*&-%!G6#,**&&%\"yG6# \"\"!\"\"\"\"\"#!\"\"F-&F*6#F-F-&F*6#F.F-*&&F*6#\"\"$F-F.F/F-F-%\"hGF- " }{TEXT 257 5 " = " }{XPPEDIT 18 0 "[y[0]+2*y[1]+2*y[2]+y[3]];" "6# 7#,*&%\"yG6#\"\"!\"\"\"*&\"\"#F)&F&6#F)F)F)*&F+F)&F&6#F+F)F)&F&6#\"\"$ F)" }{TEXT 272 1 " " }{XPPEDIT 18 0 "h/2" "6#*&%\"hG\"\"\"\"\"#!\"\"" }{TEXT 258 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Generalising from t his, the definite integral " }{XPPEDIT 18 0 "Int(f(x),x=a .. b)" "6#-% $IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 48 " can be approximated by the sum of the areas of " }{TEXT 265 10 "trapezoids" }{TEXT -1 26 " formed by joining points " }}{PARA 259 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``(x[0],f(x[0])),``(x[1],f(x[1])),` . . . `,``(x[n],f(x [n]));" "6&-%!G6$&%\"xG6#\"\"!-%\"fG6#&F'6#F)-F$6$&F'6#\"\"\"-F+6#&F'6 #F3%(~.~.~.~G-F$6$&F'6#%\"nG-F+6#&F'6#F=" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 16 "along the curve " }{XPPEDIT 18 0 "y = f(x)" "6#/% \"yG-%\"fG6#%\"xG" }{TEXT -1 30 " with straight line segments. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "It is usu al to take the " }{TEXT 312 1 "x" }{TEXT -1 13 " coordinates " } {XPPEDIT 18 0 "a = x[0],x[1],` . . . `,x[n] = b;" "6&/%\"aG&%\"xG6#\" \"!&F&6#\"\"\"%(~.~.~.~G/&F&6#%\"nG%\"bG" }{TEXT -1 7 " to be " } {TEXT 265 14 "equally spaced" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 258 "" 0 "" {TEXT 273 2 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b)" "6#-%$IntG6$-%\"fG6#% \"xG/F);%\"aG%\"bG" }{TEXT 276 2 " " }{TEXT 259 1 "~" }{TEXT 277 2 " \+ " }{XPPEDIT 18 0 "[f(x[0])+2*f(x[1])+2*f(x[2])*`+ . . . +`*2*f(x[n-1] )+f(x[n])];" "6#7#,*-%\"fG6#&%\"xG6#\"\"!\"\"\"*&\"\"#F,-F&6#&F)6#F,F, F,*,F.F,-F&6#&F)6#F.F,%*+~.~.~.~+GF,F.F,-F&6#&F)6#,&%\"nGF,F,!\"\"F,F, -F&6#&F)6#F>F," }{TEXT 274 1 " " }{XPPEDIT 18 0 "h/2;" "6#*&%\"hG\"\" \"\"\"#!\"\"" }{TEXT 275 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "h=(b-a)/n" "6#/%\"hG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF* " }{TEXT -1 36 " is the distance between successive " }{TEXT 280 1 "x " }{TEXT -1 13 " coordinates." }}{PARA 0 "" 0 "" {TEXT -1 12 "This is \+ the " }{TEXT 265 16 "trapezoidal rule" }{TEXT -1 61 " for estimating n umerically the value of a definite integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "x[i]=a+i*h " "6#/&%\"xG6#%\"iG,&%\"aG\"\"\"*&F'F*%\"hGF*F*" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "a+i*``((b-a)/n);" "6#,&%\"aG\"\"\"*&%\"iGF%-%!G6#*&,&% \"bGF%F$!\"\"F%%\"nGF.F%F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 259 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(f(x ),x = a .. b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 " \+ " }{TEXT 260 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h/2" "6#*&%\"hG\" \"\"\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``(f(a)+2*Sum(f(a+i*h ),i = 1 .. n-1)+f(b));" "6#-%!G6#,(-%\"fG6#%\"aG\"\"\"*&\"\"#F+-%$SumG 6$-F(6#,&F*F+*&%\"iGF+%\"hGF+F+/F5;F+,&%\"nGF+F+!\"\"F+F+-F(6#%\"bGF+ " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The trapezoidal rule can be applied by means of the proce dure " }{TEXT 0 9 "trapezoid" }{TEXT -1 8 " in the " }{TEXT 0 7 "stude nt" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can obta in essentially the formula above (with \"=\" instead of \"" }{TEXT 279 1 "~" }{TEXT -1 5 "\" ). " }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" } {TEXT -1 16 ": The procedure " }{TEXT 0 9 "trapezoid" }{TEXT -1 37 " c an be accessed without loading the " }{TEXT 0 7 "student" }{TEXT -1 27 " package by using the name " }{TEXT 0 18 "student[trapezoid]" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "f := 'f': x := 'x': a := 'a': b := 'b': n := 'n ': h := 'h':\nInt(f(x),x=a .. b)=student[trapezoid](f(x),x=a.. a+n*h,n );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG %\"bG,$*&#\"\"\"\"\"#F2*&%\"hGF2,(-F(6#F-F2*&F3F2-%$SumG6$-F(6#,&F-F2* &%\"jGF2F5F2F2/FA;F2,&%\"nGF2F2!\"\"F2F2-F(6#,&F-F2*&FEF2F5F2F2F2F2F2F 2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following code will generate specific instances of the formula for th e trapezoidal rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "n := 6:\nstudent[trapezoid](f(x),x=a..a+n* h,n);\nvalue(%);\nfor i from 0 to n do subs(f(a+i*h)=y[i],%) end do:\n %; n:= 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&%\" hGF&,(-%\"fG6#%\"aGF&*&F'F&-%$SumG6$-F,6#,&F.F&*&%\"jGF&F)F&F&/F7;F&\" \"&F&F&-F,6#,&F.F&*&\"\"'F&F)F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&%\"hGF&,0-%\"fG6#%\"aGF&*&F'F&-F,6# ,&F.F&F)F&F&F&*&F'F&-F,6#,&F.F&*&F'F&F)F&F&F&F&*&F'F&-F,6#,&F.F&*&\"\" $F&F)F&F&F&F&*&F'F&-F,6#,&F.F&*&\"\"%F&F)F&F&F&F&*&F'F&-F,6#,&F.F&*&\" \"&F&F)F&F&F&F&-F,6#,&F.F&*&\"\"'F&F)F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&,0&%\"yG6#\"\"!F&*&F'F&&F+6#F&F&F&* &F'F&&F+6#F'F&F&*&F'F&&F+6#\"\"$F&F&*&F'F&&F+6#\"\"%F&F&*&F'F&&F+6#\" \"&F&F&&F+6#\"\"'F&F&%\"hGF&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 51 "A procedure for illustrating the trapezoidal rule: " } {TEXT 0 8 "drawtrap" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "d rawtrap: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 262 4 " \+ " }{TEXT -1 52 "drawtrap(fx, xrngs ) or drawtrap(fx, xrngs, yrng ) \n " }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 177 " fx - an expres sion involving a single variable, say x.\n\n xrngs - an equ ation of the form x=a..b, where a..b is the range for the area to be s hown and also the" }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ horizontal plotting range.\n" }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ " }{TEXT 265 2 "OR" }{TEXT -1 102 ": an equation of the form \+ x=[a..b,c..d], where the range a..b is the range for the area to be sh own" }}{PARA 0 "" 0 "" {TEXT -1 64 " and c..d is \+ the horizontal plotting range." }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 7 "yrng - " } {TEXT -1 91 " vertical range (optional), which can be given in the form s..t, or in the form y=s..t." }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The procedure " }{TEXT 0 8 "drawtrap" }{TEXT -1 140 " plots the graph of a function, and show s the trapezoids used by the trapezoidal rule to calculate an approxim ation for a definite integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 66 "color/colour=clr\nThis option determines the colour \+ for the curve. " }}{PARA 0 "" 0 "" {TEXT -1 27 "The default colour is \+ red.\n" }}{PARA 0 "" 0 "" {TEXT -1 31 "scheme=normal or 1/stripes or 2 " }}{PARA 0 "" 0 "" {TEXT -1 64 "This option determines the colouring \+ scheme for the trapezoids." }}{PARA 0 "" 0 "" {TEXT -1 204 "The \"nor mal\" scheme (which is the default) is to colour the trapezoids above \+ and below the horizontal axis differently, while the \"stripes\" schem e is to use two different colours for alternate trapezoids." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "shading=clr or \+ shading=[clr1,clr2] " }}{PARA 0 "" 0 "" {TEXT -1 83 "This option can b e used to specify one or two different colours for the trapezoids." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "intervals =n" }}{PARA 0 "" 0 "" {TEXT -1 62 "The number subintervals used in app lying the trapezoidal rule." }}{PARA 0 "" 0 "" {TEXT -1 108 "The defau lt number of intervals is 4 which coincides with the default number of intervals for the procedure " }{TEXT 0 9 "trapezoid" }{TEXT -1 8 " in the " }{TEXT 0 7 "student" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 40 "Standard plot options ar e also available" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 16 "How to activate:" } {TEXT -1 156 "\nTo make the procedures active open the subsection, pla ce the cursor anywhere after the prompt [ > and press [Enter].\nYou c an then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "drawtrap: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6229 "drawtrap := proc(fx,eqn)\n local a,b,h,div,traps_minus,traps_plus,i,j,ai,aip,\n fai,faip,x,r s,xmin,xmax,xrange,startoptions,\n yrange,Options,n,thck,tp,p1,pp,am ,plts,shdclr,typ,traps;\n\n if nargs<2 then\n error \"at least \+ 2 arguments are required; the basic syntax is: 'drawtrap(f(x),x=a..b)' or 'drawtrap(f(x),x=[a..b,c..d])'.\"\n end if;\n\n if not type(eq n,`=`) then \n error \"the 2nd argument, %1, is invalid ..it shou ld be an equation of the form 'x=a..b' or 'x=[a..b,c..d]' to give the \+ required range or ranges\",eqn;\n end if; \n x := op(1,eqn);\n \+ if not type(x,name) then\n error \"the 2nd argument equation lef t side, %1, should be the independent variable\",x;\n end if;\n if not type(indets(fx,name) minus \{x\},set(realcons)) then\n error \"the 1st argument, %1, must depend only on the variable %2\",fx,x;\n end if; \n rs := op(2,eqn);\n\n if not (type(rs,[realcons..rea lcons,realcons..realcons]) or \n type(rs,realcons..realcons)) then\n error \"the 2nd argument,%1, is invalid .. the right side of \+ the equation, %2, should be a range of real values or a list of two ra nges\",eqn,rs;\n end if;\n if type(rs,range) then\n a := eval f(op(1,rs));\n b := evalf(op(2,rs));\n if a>=b then\n \+ error \"parameter range is invalid\"\n end if;\n xmin := a ;\n xmax := b; # plotting interval same as area interval\n els e # have two ranges\n a := evalf(op(1,rs[1]));\n b := evalf( op(2,rs[1]));\n if a>=b then\n error \"1st parameter rang e is invalid\"\n end if; \n xmin := evalf(op(1,rs[2]));\n \+ xmax := evalf(op(2,rs[2]));\n if xmin>=xmax or a>=b then\n \+ error \"2nd parameter range is invalid\"\n end if;\n if not (xmin<=a and xmax>=b) then\n error \"range for area must \+ be a subrange of the plotting range\"\n end if; \n end if;\n \+ xrange := xmin..xmax;\n\n startoptions := 3; \n yrange := NULL;\n if nargs>2 then \n if type(args[3],range) or type(args[3],name =range) then\n yrange := args[3];\n startoptions := 4; \n end if;\n end if;\n \n # Get the allowed options.\n Op tions := [];\n n := 4;\n thck := 2;\n shdclr := [COLOR(RGB,.83,. 83,.83),COLOR(RGB,.73,.73,.87)];\n typ := 1;\n if nargs>=startopti ons then\n Options:=[args[startoptions..nargs]];\n if not ty pe(Options,list(equation)) then\n error \"each optional argume nt must be an equation\"\n end if;\n if hasoption(Options,'i ntervals','n','Options') then\n if not type(n,posint) then\n \+ error \"\\\"intervals\\\" must be a positive integer\"\n \+ end if;\n end if;\n if hasoption(Options,'thickness','t hck','Options') then\n if not type(thck,nonnegint) then\n \+ error \"\\\"thickness\\\" must be a non-negative integer\"\n \+ end if;\n end if;\n if hasoption(Options,'scheme','typ ','Options') then\n if not member(typ,\{'normal','stripes',1,2 \}) then\n error \"\\\"scheme\\\" must be 'normal', 'stripe s', 1 or 2\"\n end if;\n if typ='normal' then typ := 1 \n elif typ='stripes' then typ := 2 end if;\n end if;\n \+ if typ=2 then\n shdclr := [COLOR(RGB,.8,.75,1),COLOR(RGB,1 ,.75,.8)];\n end if;\n if hasoption(Options,'shading','tp',' Options') then\n if type(tp,list) then\n for j to m in(nops(shdclr),nops(tp))\n do shdclr[j] := tp[j] end do ;\n else \n shdclr[1] := tp;\n shdclr[2] := tp;\n end if;\n end if;\n end if;\n \n if n>256 then\n error \"too many subdivisions\"\n end if;\n h := eval f((b-a)/n);\n p1 := plot(fx,x=xrange,yrange,\n thi ckness=thck,op(Options));\n\n if typ=1 then\n ai := a;\n f ai := traperror(evalf(subs(x=ai,fx)));\n if fai=lasterror or not \+ type(fai,numeric) then\n error \"evaluation failed at %1\",ai; \n end if;\n\n traps_plus := [];\n traps_minus := [];\n for i to n do\n aip := a+i*h;\n faip := traperro r(evalf(subs(x=aip,fx)));\n if faip=lasterror or not type(faip ,numeric) then\n error \"evaluation failed at %1\",aip;\n \+ end if;\n\n if fai>=0 and faip>=0 then\n tra ps_plus := [op(traps_plus),\n [[ai,0],[aip,0],[aip,faip] ,[ai,fai]]];\n elif fai<=0 and faip<=0 then\n traps _minus := [op(traps_minus),\n [[ai,0],[aip,0],[aip,faip] ,[ai,fai]]];\n else\n am := ai-fai*h/(faip-fai);\n \+ if fai>=0 and faip<=0 then\n traps_plus := [op (traps_plus),[[ai,0],[ai,fai],[am,0]]];\n traps_minus := \+ \n [op(traps_minus),[[am,0],[aip,faip],[aip,0]]];\n else \n traps_minus := [op(traps_minus),[[a i,0],[ai,fai],[am,0]]];\n traps_plus := [op(traps_plus),[ [am,0],[aip,faip],[aip,0]]];\n end if;\n end if;\n \+ ai := aip;\n fai := faip; \n end do;\n plts := [p1];\n if traps_plus<>[] then\n pp := plots[polygonp lot](traps_plus,color=shdclr[1]);\n plts := [op(plts),pp];\n \+ end if;\n if traps_minus<>[] then\n pp := plots[polyg onplot](traps_minus,color=shdclr[2]);\n plts := [op(plts),pp]; \n end if;\n plots[display](plts);\n else \n ai := a ;\n fai := traperror(evalf(subs(x=ai,fx)));\n if fai=lasterr or or not type(fai,numeric) then\n error \"evaluation failed a t %1\",ai;\n end if;\n\n traps := [];\n for i to n do\n aip := a+i*h;\n faip := traperror(evalf(subs(x=aip,fx )));\n if faip=lasterror or not type(faip,numeric) then\n \+ error \"evaluation failed at %1\",aip;\n end if; \n \+ if (fai>=0 and faip>=0) or (fai<0 and faip<0) then\n t raps := [op(traps),[[ai,0],[aip,0],[aip,faip],[ai,fai]]];\n el se\n am := ai-fai*h/(faip-fai);\n traps := [op(t raps),[[ai,0],[ai,fai],[am,0]],\n [[am,0],[aip,faip], [aip,0]]];\n end if;\n ai := aip;\n fai := fai p; \n end do; \n plts := [p1,seq(plots[polygonplot](tr aps[i],\n color=`if`(irem(i,2)=0,shdclr[2],shdclr[ 1])),i=1..n)];\n plots[display](plts);\n end if; \nend proc:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "A procedur e for numerical integration using the trapezoidal rule: " }{TEXT 0 4 " trap" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "trap: usage " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 282 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 283 2 " " }{TEXT -1 19 " trap( gx, rng ) " }{TEXT 284 1 "\n" }{TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " gx - " }{TEXT -1 55 " \+ an expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 83 " where gx evaluates t o a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 285 61 "the range x=a..b for the definite integral to be aproximated." }} {PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 4 "trap" }{TEXT -1 43 " calculates a numerical approximation f or " }{XPPEDIT 18 0 "Int(gx,x = a .. b);" "6#-%$IntG6$%#gxG/%\"xG;%\" aG%\"bG" }{TEXT -1 336 " by using the trapezoidal rule with a specifi ed number of sub-intervals of the interval from a to b obtained using \+ equally spaced x values. The trapezoidal rule can alse be applied iter atively whereby the number of intervals is successively doubled until \+ the desired accuracy is achieved or the maximum number of iterations i s reached." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "intervals=n" }}{PARA 0 "" 0 "" {TEXT -1 53 "This option d etermines the number of intervals used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "iterate=true/false" }}{PARA 0 "" 0 "" {TEXT -1 124 "This option controls whether the trapezoidal rule i s to be applied iteratively rather than with a fixed number of interva ls." }}{PARA 0 "" 0 "" {TEXT -1 31 "The default is \"iterate=false\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "maxit erations=n\nThis option controls the maximum number of iterations of t he integration procedure." }}{PARA 0 "" 0 "" {TEXT -1 186 "The default is \"maxiterations=20\" when the computation can be performed using h ardware floating point arithmetic and \"maxiterations=16\" when softw are floating point arithmetic is used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "info=true/false\nWhen the trapezoid al rule is applied iteratively, option \"info=true\" allows the progre ss of the procedure to be monitored by printing each approximation to \+ the integral immediately after it has been calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 16 "How to activate:" }{TEXT -1 156 "\nTo make the procedure s active open the subsection, place the cursor anywhere after the prom pt [ > and press [Enter].\nYou can then close up the subsection." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "trap: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6997 "tr ap := proc(fx,eqn)\n local a,b,h,i,j,x,rs,Options,n,saveDigits,f,fa, fb,dist,\n dotrap,oddsum,it,sm,oddsm,val,lastval,eps,maxit,\n \+ prntflg,ends,t,sign,maxit_opt,usehf,vals,ff;\n\n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: \+ 'trap(f(x),x=a..b)'.\"\n end if;\n\n if not type(fx,algebraic) the n \n error \"the 1st argument, %1, is invalid ..it should be an a lgebraic expression in a single variable\",fx;\n end if; \n if not type(eqn,`=`) then \n error \"the 2nd argument, %1, is invalid . .it should be an equation of the form 'x=a..b' to give the required in terval for the integral to be estimated\",eqn;\n end if; \n x := op(1,eqn);\n if not type(x,name) then\n error \"the 2nd argume nt equation left side, %1, should be the independent variable\",x;\n \+ end if;\n if not type(indets(fx,name) minus \{x\},set(realcons)) th en\n error \"the 1st argument, %1, must depend only on the variab le %2\",fx,x;\n end if; \n rs := op(2,eqn);\n if not type(rs,re alcons..realcons) then\n error \"the 2nd argument,%1, is inval id .. the right side of the equation, %2, should be a range of real va lues\",eqn,rs;\n end if;\n \n # Get the allowed options.\n Opt ions := [];\n n := 4;\n it := false;\n prntflg := false;\n max it_opt := false;\n if nargs>=3 then\n Options:=[args[3..nargs]] ;\n if not type(Options,list(equation)) then\n error \"ea ch optional argument must be an equation\"\n end if;\n if ha soption(Options,'intervals','n','Options') then\n if not type( n,posint) then\n error \"\\\"intervals\\\" must be a positi ve integer\"\n end if;\n end if;\n if hasoption(Opti ons,'iterate','it','Options') then\n if not it=true then it := false end if;\n end if;\n if hasoption(Options,'maxiteratio ns','maxit','Options') then\n if not type(maxit,posint) then\n error \"\\\"maxiterations\\\" must be a positive integer\" \n end if;\n maxit_opt := true;\n elif hasoption( Options,'maxiter','maxit','Options') then\n if not type(maxit, posint) then\n error \"\\\"maxiter\\\" must be a positive i nteger\"\n end if;\n maxit_opt := true;\n end if; \n if hasoption(Options,'info','prntflg','Options') then\n \+ if prntflg<>true then prntflg := false end if;\n end if;\n \+ if nops(Options)>0 then\n error \"%1 is not a valid option fo r %2 .. the recognised options are \\\"intervals\\\", \\\"iterate\\\", \\\"maxiterations\\\",(or \\\"maxiter\\\") and \\\"info\\\"\",op(1,Op tions),procname;\n end if;\n end if;\n \n saveDigits := Dig its;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n a := evalf (op(1,rs));\n b := evalf(op(2,rs));\n if a=b then return 0 end if; \n sign := 1;\n if b `,lastval)\n end if ;\n if usehf then\n if not maxit_opt then maxit := 22 end if; \n for j to maxit do\n n := 2*n;\n \+ h := dist/n; \n oddsm := traperror(evalhf(oddsum(f,a ,h,n)));\n if oddsm=lasterror or not type(oddsm,numeric) th en\n error \"computation of trapezoid estimate failed\" \n end if;\n sm := sm+oddsm;\n val := (ends+sm)*h;\n if prntflg then\n print(`appr oximation with `||n||` intervals ---> `,val)\n end if;\n \+ if abs(val-lastval)0 then\n \+ return evalf(val);\n else\n return eva lf(-val);\n end if;\n end if;\n la stval := val;\n end do;\n else\n if not maxit_opt then maxit := 16 end if;\n for j to maxit do\n n : = 2*n;\n h := dist/n; \n oddsm := traperror(odds um(f,a,h,n));\n if oddsm=lasterror then \n er ror \"computation of trapezoid estimate failed\"\n end if; \n sm := sm+oddsm;\n val := (ends+sm)*h;\n \+ if prntflg then\n print(`approximation with `||n|| ` intervals ---> `,val)\n end if;\n if abs(val -lastval)0 then\n return evalf(val); \n else\n return evalf(-val);\n \+ end if;\n end if;\n lastval := val;\n \+ end do;\n end if;\n WARNING(\"reached max %1 iterations without convergence\",j-1);\n Digits := saveDigits;\n if si gn>0 then\n return evalf(val);\n else\n return ev alf(-val);\n end if;\n else\n dotrap := proc(f,a,b,n)\n \+ local h,s,i;\n h := (b-a)/n;\n s := (f(a)+f(b))/ 2;\n for i to n-1 do s := s+f(a+i*h) end do;\n s*h;\n \+ end proc:\n if usehf then \n sm := traperror(ev alhf(dotrap(f,a,b,n)));\n if sm=lasterror or not type(sm,numer ic) then\n sm := traperror(dotrap(f,a,b,n));\n end \+ if;\n if sm=lasterror then \n error \"computation o f trapezoid estimate failed\"\n end if;\n else\n \+ sm := traperror(evalf(dotrap(f,a,b,n)));\n if sm=lasterror or \+ not type(sm,numeric) then\n error \"computation of trapezoi d estimate failed\"\n end if;\n end if;\n Digits := \+ saveDigits;\n if sign>0 then\n return evalf(sm);\n e lse\n return evalf(-sm);\n end if;\n end if; \nend pro c:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 " Examples using: " }{TEXT 0 18 "student[trapezoid]" }{TEXT 266 1 "," } {TEXT -1 1 " " }{TEXT 0 8 "drawtrap" }{TEXT -1 5 " and " }{TEXT 0 4 "t rap" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }{TEXT 290 30 ".. the trapezoidal rule using " }{TEXT 266 18 "student[trapezoid]" }{TEXT 291 1 " " }}{PARA 0 "" 0 "" {TEXT -1 97 "The following picture illustr ates the region corresponding to the approximation for the integral " }{XPPEDIT 18 0 "Int(``(2*x^2-x^3/3),x = 1 .. 5);" "6#-%$IntG6$-%!G6#,& *&\"\"#\"\"\"*$%\"xGF+F,F,*&F.\"\"$F0!\"\"F1/F.;F,\"\"&" }{TEXT -1 107 " given by using the trapezoidal rule with 4 intervals. The defaul t number of intervals is 4 so the option \"" }{TEXT 281 11 "intervals= 4" }{TEXT -1 20 "\" could be omitted. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "drawtrap(2*x^2-x^3/3,x=[1..5,0..6],intervals=4);" }}{PARA 13 " " 1 "" {GLPLOT2D 494 306 306 {PLOTDATA 2 "6&-%'CURVESG6%7W7$$\"\"!F)F( 7$$\"3%*******\\#HyI\"!#=$\"3aV05L-FYL!#>7$$\"33++]([kdW#F-$\"3c!Goj;' eZ6F-7$$\"3++++v;\\DPF-$\"3L/wp,/].EF-7$$\"3W+++D!o&3!HFO7$$\"36+++&> q0]\"FO$\"3Iv$\\PRQrP$FO7$$\"3'******\\U80j\"FO$\"3w()HFw6?sQFO7$$\"35 +++0ytb@FO$\"3+()pR,;l4eFO7$$\"3'* ***\\(3wY_AFO$\"3E!Rr@oRyL'FO7$$\"3#)******HOTqBFO$\"3)H>E2$o/)z'FO7$$ \"37++v3\">)*\\#FO$\"31>.73$))4H(FO7$$\"3:++DEP/BEFO$\"30'y\\+\")*)[u( FO7$$\"3=++](o:;v#FO$\"3=BT()*Hg#)>)FO7$$\"3=++v$)[opGFO$\"3+6RW$H5Gf) FO7$$\"3%*****\\i%Qq*HFO$\"3g\"*QXDm5\"**)FO7$$\"3&****\\(QIKHJFO$\"39 ?xt:P_q$*FO7$$\"3#****\\7:xWC$FO$\"3MI!o;l\"zo'*FO7$$\"37++]Zn%)oLFO$ \"3!y12ZPlP&**FO7$$\"3y******4FL(\\$FO$\"3)R[]5\\l.-\"!#;7$$\"3#)**** \\d6.BOFO$\"3\\T$))*47.S5Fct7$$\"3(****\\(o3lWPFO$\"3Suit;5=a5Fct7$$\" 3!*****\\A))ozQFO$\"3)4'*Q+wHQ1\"Fct7$$\"3e******Hk-,SFO$\"3%3\"*=fXmm 1\"Fct7$$\"36+++D-eITFO$\"3eoxL4A=j5Fct7$$\"3u***\\(=_(zC%FO$\"3))[lw7&FO$\"3+b/Pv`oXwFO7$$\"3O++v)Q?QD&F O$\"3c,$)GXi]loFO7$$\"3G+++5jyp`FO$\"3WZ&y?%4KdgFO7$$\"3<++]Ujp-bFO$\" 39Kse)Q'R>]FO7$$\"33++D,X8ibFO$\"3M6FoLYY:XFO7$$\"3++++gEd@cFO$\"3wOoY 'ejj)RFO7$$\"31+]PMh%\\o&FO$\"3#3l\\_gMSR$FO7$$\"39++v3'>$[dFO$\"3%*)y PHo1@x#FO7$$\"3p******4h(*3eFO$\"3!o%Q5p'\\'[@FO7$$\"37++D6EjpeFO$\"39 R%o0gkr\\\"FO7$$\"3^+]i0j\"[$fFO$\"3qEI@k*4Il(F-7$$\"\"'F)F(-%'COLOURG 6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG6'7&7$$\"\" \"F)F(7$$\"+++++?!\"*F(7$F]]l$\"+LLLL`F_]l7$Fj\\l$\"+nmmm;F_]l7&F\\]l7 $$\"+++++IF_]lF(7$Fh]l$\"+++++!*F_]lF`]l7&Fg]l7$$\"+++++SF_]lF(7$F_^l$ \"+nmmm5F`\\lFj]l7&F^^l7$$\"+++++]F_]lF(7$Ff^l$\"*LLLL)F`\\lFa^l-%&COL ORG6&F]\\l$\"#$)!\"#F^_lF^_l-%+AXESLABELSG6$Q\"x6\"Q!Fe_l-%%VIEWG6$;F( Fh[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "student [trapezoid](2*x^2-x^3/3,x=1..5,4);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"&\"\"\"-%$SumG6$,&*&\"\"#F%),&F%F%%\"iGF%F+F%F%*& \"\"$!\"\"F-F0F1/F.;F%F0F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" } {TEXT -1 52 ": The default number of intervals for the procedure " } {TEXT 0 9 "trapezoid" }{TEXT -1 59 " is also 4, so the third argument \+ could have been omitted. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(2*x^2-x^3/3,x=1..5);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"#\"\"\")%\"xGF(F) F)*&#F)\"\"$F)*$)F+F.F)F)!\"\"/F+;F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"##*\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The trapezpoidal rule estimate for the integra l with 4 intervals is 30, while the exact value is 30 " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 92 "The next picture illustrates the region corresponding t o the approximation for the integral " }{XPPEDIT 18 0 "Int(``(2*x^2-x^ 3/3),x = 1 .. 5);" "6#-%$IntG6$-%!G6#,&*&\"\"#\"\"\"*$%\"xGF+F,F,*&F. \"\"$F0!\"\"F1/F.;F,\"\"&" }{TEXT -1 55 " given by using the trapezoid al rule with 8 intervals. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "draw trap(2*x^2-x^3/3,x=[1..5,0..6],intervals=8);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 306 306 {PLOTDATA 2 "6&-%'CURVESG6%7W7$$\"\"!F)F(7$$\"3% *******\\#HyI\"!#=$\"3aV05L-FYL!#>7$$\"33++]([kdW#F-$\"3c!Goj;'eZ6F-7$ $\"3++++v;\\DPF-$\"3L/wp,/].EF-7$$\"3W+++D!o&3!HFO7$$\"36+++&>q0]\"F O$\"3Iv$\\PRQrP$FO7$$\"3'******\\U80j\"FO$\"3w()HFw6?sQFO7$$\"35+++0yt b@FO$\"3+()pR,;l4eFO7$$\"3'****\\( 3wY_AFO$\"3E!Rr@oRyL'FO7$$\"3#)******HOTqBFO$\"3)H>E2$o/)z'FO7$$\"37++ v3\">)*\\#FO$\"31>.73$))4H(FO7$$\"3:++DEP/BEFO$\"30'y\\+\")*)[u(FO7$$ \"3=++](o:;v#FO$\"3=BT()*Hg#)>)FO7$$\"3=++v$)[opGFO$\"3+6RW$H5Gf)FO7$$ \"3%*****\\i%Qq*HFO$\"3g\"*QXDm5\"**)FO7$$\"3&****\\(QIKHJFO$\"39?xt:P _q$*FO7$$\"3#****\\7:xWC$FO$\"3MI!o;l\"zo'*FO7$$\"37++]Zn%)oLFO$\"3!y1 2ZPlP&**FO7$$\"3y******4FL(\\$FO$\"3)R[]5\\l.-\"!#;7$$\"3#)****\\d6.BO FO$\"3\\T$))*47.S5Fct7$$\"3(****\\(o3lWPFO$\"3Suit;5=a5Fct7$$\"3!***** \\A))ozQFO$\"3)4'*Q+wHQ1\"Fct7$$\"3e******Hk-,SFO$\"3%3\"*=fXmm1\"Fct7 $$\"36+++D-eITFO$\"3eoxL4A=j5Fct7$$\"3u***\\(=_(zC%FO$\"3))[lw7&FO$\"3+b/Pv`oXwFO7$$\"3O++v)Q?QD&FO$\"3c, $)GXi]loFO7$$\"3G+++5jyp`FO$\"3WZ&y?%4KdgFO7$$\"3<++]Ujp-bFO$\"39Kse)Q 'R>]FO7$$\"33++D,X8ibFO$\"3M6FoLYY:XFO7$$\"3++++gEd@cFO$\"3wOoY'ejj)RF O7$$\"31+]PMh%\\o&FO$\"3#3l\\_gMSR$FO7$$\"39++v3'>$[dFO$\"3%*)yPHo1@x# FO7$$\"3p******4h(*3eFO$\"3!o%Q5p'\\'[@FO7$$\"37++D6EjpeFO$\"39R%o0gkr \\\"FO7$$\"3^+]i0j\"[$fFO$\"3qEI@k*4Il(F-7$$\"\"'F)F(-%'COLOURG6&%$RGB G$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG6+7&7$$\"\"\"F)F(7$ $\"+++++:!\"*F(7$F]]l$\"++++vLF_]l7$Fj\\l$\"+nmmm;F_]l7&F\\]l7$$\"++++ +?F_]lF(7$Fh]l$\"+LLLL`F_]lF`]l7&Fg]l7$$\"+++++DF_]lF(7$F_^l$\"+nmm\"H (F_]lFj]l7&F^^l7$$\"+++++IF_]lF(7$Ff^l$\"+++++!*F_]lFa^l7&Fe^l7$$\"+++ ++NF_]lF(7$F]_l$\"+LL$3-\"F`\\lFh^l7&F\\_l7$$\"+++++SF_]lF(7$Fd_l$\"+n mmm5F`\\lF__l7&Fc_l7$$\"+++++XF_]lF(7$F[`l$\"+++]75F`\\lFf_l7&Fj_l7$$ \"+++++]F_]lF(7$Fb`l$\"*LLLL)F`\\lF]`l-%&COLORG6&F]\\l$\"#$)!\"#Fj`lFj `l-%+AXESLABELSG6$Q\"x6\"Q!Faal-%%VIEWG6$;F(Fh[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "student[trapezoid](2*x^2-x^3 /3,x=1..5,8);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"&\" \"#\"\"\"*&#F'F&F'-%$SumG6$,&*&F&F'),&F'F'*&F&!\"\"%\"iGF'F'F&F'F'*&\" \"$F2F0F5F2/F3;F'\"\"(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#h\" \"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The trapezpoidal rule estimate for the \+ integral with 8 intervals is 30 " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\" \"#!\"\"" }{TEXT -1 30 ", while the exact value is 30 " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 123 "As the number of intervals used increases, the closer \+ the trapezoidal estimate becomes to the exact value of the integral. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The f ollowing picture illustrates the region corresponding to the approxima tion for the integral " }{XPPEDIT 18 0 "Int(``(2*x^2-x^3/3),x = 1 .. 5 );" "6#-%$IntG6$-%!G6#,&*&\"\"#\"\"\"*$%\"xGF+F,F,*&F.\"\"$F0!\"\"F1/F .;F,\"\"&" }{TEXT -1 56 " given by using the trapezoidal rule with 16 \+ intervals. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "drawtrap(2*x^2-x^3 /3,x=[1..5,0..6],intervals=16);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 306 306 {PLOTDATA 2 "6&-%'CURVESG6%7W7$$\"\"!F)F(7$$\"3%*******\\#HyI\"!#= $\"3aV05L-FYL!#>7$$\"33++]([kdW#F-$\"3c!Goj;'eZ6F-7$$\"3++++v;\\DPF-$ \"3L/wp,/].EF-7$$\"3W+++D!o&3!HFO7$$\"36+++&>q0]\"FO$\"3Iv$\\PRQrP$FO 7$$\"3'******\\U80j\"FO$\"3w()HFw6?sQFO7$$\"35+++0ytb@FO$\"3+()pR,;l4eFO7$$\"3'****\\(3wY_AFO$\"3E!Rr@oR yL'FO7$$\"3#)******HOTqBFO$\"3)H>E2$o/)z'FO7$$\"37++v3\">)*\\#FO$\"31> .73$))4H(FO7$$\"3:++DEP/BEFO$\"30'y\\+\")*)[u(FO7$$\"3=++](o:;v#FO$\"3 =BT()*Hg#)>)FO7$$\"3=++v$)[opGFO$\"3+6RW$H5Gf)FO7$$\"3%*****\\i%Qq*HFO $\"3g\"*QXDm5\"**)FO7$$\"3&****\\(QIKHJFO$\"39?xt:P_q$*FO7$$\"3#****\\ 7:xWC$FO$\"3MI!o;l\"zo'*FO7$$\"37++]Zn%)oLFO$\"3!y12ZPlP&**FO7$$\"3y** ****4FL(\\$FO$\"3)R[]5\\l.-\"!#;7$$\"3#)****\\d6.BOFO$\"3\\T$))*47.S5F ct7$$\"3(****\\(o3lWPFO$\"3Suit;5=a5Fct7$$\"3!*****\\A))ozQFO$\"3)4'*Q +wHQ1\"Fct7$$\"3e******Hk-,SFO$\"3%3\"*=fXmm1\"Fct7$$\"36+++D-eITFO$\" 3eoxL4A=j5Fct7$$\"3u***\\(=_(zC%FO$\"3))[lw7&FO$\"3+b/Pv`oXwFO7$$\"3O++v)Q?QD&FO$\"3c,$)GXi]loFO7$$\"3G++ +5jyp`FO$\"3WZ&y?%4KdgFO7$$\"3<++]Ujp-bFO$\"39Kse)Q'R>]FO7$$\"33++D,X8 ibFO$\"3M6FoLYY:XFO7$$\"3++++gEd@cFO$\"3wOoY'ejj)RFO7$$\"31+]PMh%\\o&F O$\"3#3l\\_gMSR$FO7$$\"39++v3'>$[dFO$\"3%*)yPHo1@x#FO7$$\"3p******4h(* 3eFO$\"3!o%Q5p'\\'[@FO7$$\"37++D6EjpeFO$\"39R%o0gkr\\\"FO7$$\"3^+]i0j \"[$fFO$\"3qEI@k*4Il(F-7$$\"\"'F)F(-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F (-%*THICKNESSG6#\"\"#-%)POLYGONSG637&7$$\"\"\"F)F(7$$\"++++]7!\"*F(7$F ]]l$\"+L$eRZ#F_]l7$Fj\\l$\"+nmmm;F_]l7&F\\]l7$$\"+++++:F_]lF(7$Fh]l$\" ++++vLF_]lF`]l7&Fg]l7$$\"++++])F_]lFf_l7&Fj_l7$$\"+++++IF_]lF(7$Fb`l$ \"+++++!*F_]lF]`l7&Fa`l7$$\"++++]KF_]lF(7$Fi`l$\"*n\"H#o*F`\\lFd`l7&Fh `l7$$\"+++++NF_]lF(7$F`al$\"+LL$3-\"F`\\lF[al7&F_al7$$\"++++]PF_]lF(7$ Fgal$\"++voa5F`\\lFbal7&Ffal7$$\"+++++SF_]lF(7$F^bl$\"+nmmm5F`\\lFial7 &F]bl7$$\"++++]UF_]lF(7$Febl$\"+Lek`5F`\\lF`bl7&Fdbl7$$\"+++++XF_]lF(7 $F\\cl$\"+++]75F`\\lFgbl7&F[cl7$$\"++++]ZF_]lF(7$Fccl$\"*nT5S*F`\\lF^c l7&Fbcl7$$\"+++++]F_]lF(7$Fjcl$\"*LLLL)F`\\lFecl-%&COLORG6&F]\\l$\"#$) !\"#FbdlFbdl-%+AXESLABELSG6$Q\"x6\"Q!Fidl-%%VIEWG6$;F(Fh[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "student[trapezoid](2*x^2-x^3/3,x=1..5,16);\nvalue(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"&\"\"%\"\"\"*&#F'F&F'-%$Sum G6$,&*&\"\"#F'),&F'F'*&F&!\"\"%\"iGF'F'F/F'F'*&\"\"$F3F1F6F3/F4;F'\"#: F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$X#\"\")" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The trapezpoidal rule e stimate for the integral with 16 intervals is 30 " }{XPPEDIT 18 0 "5/8 ;" "6#*&\"\"&\"\"\"\"\")!\"\"" }{TEXT -1 39 " = 30.625, while the exac t value is 30 " }{XPPEDIT 18 0 "2/3 = 30.666666*` . . . `;" "6#/*&\"\" #\"\"\"\"\"$!\"\"*&-%&FloatG6$\")mmmI!\"'F&%(~.~.~.~GF&" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }{TEXT 292 70 ".. absolute and relative error in a numerical estimat e for an integral" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " We \+ find an approximation for " }{XPPEDIT 18 0 "Int(sin(x),x = Pi/4 .. 3*P i/4);" "6#-%$IntG6$-%$sinG6#%\"xG/F);*&%#PiG\"\"\"\"\"%!\"\"*(\"\"$F.F -F.F/F0" }{TEXT -1 45 " using the trapezoidal rule with 4 intervals." }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" }{TEXT -1 52 ": The default numbe r of intervals for the procedure " }{TEXT 0 8 "drawtrap" }{TEXT -1 6 " is 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "drawtrap(sin(x),x=[Pi/4..3*Pi/4,0..Pi]);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 237 237 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$\"\"!F)F (7$$\"3k\\0\"3%zxZo!#>$\"31q+d@uUUoF-7$$\"3W7SA\"H*f!G\"!#=$\"3uZ%)4&* >5x7F37$$\"3GfjyzGm]>F3$\"3a)>O?m:$Q>F37$$\"3!\\&yw%3o^i#F3$\"35)>yf`> ^f#F37$$\"3G*\\]U[nkH$F3$\"3#3seB<)3PKF37$$\"3'*za()fz%)=RF3$\"3/,zuO+ J>QF37$$\"3!G2,F*oGjXF3$\"3!=9zKNalS%F37$$\"3m@(3*elwH_F3$\"3)y(p0=8h% *\\F37$$\"3i\"Guj\")3T*eF3$\"3/VQ$)RSrebF37$$\"3q>5A'[mud'F3$\"3ftV=o; N8hF37$$\"3yg8y$Ho$zrF3$\"3w=@'=m@$ylF37$$\"378*Q!pr'p&yF3$\"3]qmy)eyJ 2(F37$$\"3,L1QI$[t`)F3$\"3(*Q\"oZ'3SPvF37$$\"3y3o*3l@I>*F3$\"3/!Gv*Qmy ^zF37$$\"3Ku@bSeV)y*F3$\"3#z\\(pbm_)H)F37$$\"3s!eu,4W'\\5!#<$\"39]$RB! 3Ys')F37$$\"3Re`x09i46Fdp$\"3e%e+@v-`&*)F37$$\"3YC.<#G*Qz6Fdp$\"3$[ICT -KPC*F37$$\"3)*y:\\uc9T7Fdp$\"3[IP#[*[bh%*F37$$\"3-Q'HZA-*38Fdp$\"3;Ij +\"38!f'*F37$$\"3OJYe$[AMP\"Fdp$\"3OB7VW$\\e!)*F37$$\"3Z$>!30EuS9Fdp$ \"3)pIb6G\\b\"**F37$$\"3uS1o\"\\jD]\"Fdp$\"3+nE@J/tw**F37$$\"3;44fpcCp :Fdp$\"2!)e8t(z)*****Fdp7$$\"3%H*\\v6(4&Q;Fdp$\"3[4'*zXK3x**F37$$\"3cu H2kU!))p\"Fdp$\"3()pAta;==**F37$$\"3OK'*RKS#Rw\"Fdp$\"3Ew$4\"yt39)*F37 $$\"3G&)\\0D\"*>J=Fdp$\"3-OkT\"[jGm*F37$$\"37'=p#yY,(*=Fdp$\"3cK5WG*4E Z*F37$$\"3%)>m<5Ypg>Fdp$\"3e6Y_lsZ\\#*F37$$\"3Y^'GuJ+9.#Fdp$\"3W\"y7g$ >%y&*)F37$$\"3#R(z:SD$\\4#Fdp$\"3!y!)Q%)flvl)F37$$\"3[WHg[nwi@Fdp$\"3= @)z#3!G%)H)F37$$\"3DS)*fBYBCAFdp$\"3CexilG)*RzF37$$\"3CE[wY_V\"H#Fdp$ \"3]i=%o\"G%Q^(F37$$\"3'G0'4'zlYN#Fdp$\"354j1X!p=3(F37$$\"3(f#=?R*f2U# Fdp$\"3cErLF[5+mF37$$\"3m%=(*\\/z`[#Fdp$\"3yjGuUC@,hF37$$\"3A()yS%eM3QF37$$\"35 Gw]PNh6GFdp$\"3.bj`7FBSKF37$$\"3kW%)zn]?\")GFdp$\"3#)3`(f\"4buDF37$$\" 3i1s4i&[M%HFdp$\"3)>EF37$$\"3[#>#G4J\")4IFdp$\"3O)f2vk%)RJ\"F37 $$\"3wRO')[CLtIFdp$\"34(R(G%y?2#oF-7$$\"35+++aEfTJFdp$!30BmIq&o?5%!#F- %'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG6'7 &7$$\"+N;)R&y!#5F(7$$\"+Xs4y6!\"*F(7$F\\\\l$\"+D`zQ#*Fj[l7$Fh[l$\"+8y1 rqFj[l7&F[\\l7$$\"+Ejzq:F^\\lF(7$Fg\\l$\"\"\"F)F_\\l7&Ff\\l7$$\"+3a\\j >F^\\lF(7$F^]l$\"+F`zQ#*Fj[lFi\\l7&F]]l7$$\"+!\\%>cBF^\\lF(7$Fe]lFc\\l F`]l-%&COLORG6&F[[l$\"#$)!\"#F[^lF[^l-%+AXESLABELSG6$Q\"x6\"Q!Fb^l-%%V IEWG6$;F($\"+aEfTJF^\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" }{TEXT -1 52 ": The defau lt number of intervals for the procedure " }{TEXT 0 18 "student[trapez oid]" }{TEXT -1 11 " is also 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "student[trapezoid](sin(x),x= Pi/4..3*Pi/4);\ntrap4 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&%#PiG\"\"\",&*$-%%sqrtG6#\"\"#F&F&*&F,F&-%$SumG6$-%$sinG6#,&F%#F& \"\"%*(#F&\"\")F&%\"iGF&F%F&F&/F:;F&\"\"$F&F&F&#F&\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&trap4G$\"+aD*fR\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "We only have a rather ro ugh approximation for the area corresponding to the integral, as you c an see from the picture." }}{PARA 0 "" 0 "" {TEXT -1 109 "It is also c learly an under-estimate for the required area, as there are \"gaps\" \+ indicating the \"missed area\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 133 "We can obtain a numerical value for this integral which is accurate to about 10 digits by first evaluating the integral analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(sin(x),x=Pi/4..3*Pi/4);\nvalue( %);\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$ sinG6#%\"xG/F);,$%#PiG#\"\"\"\"\"%,$F-#\"\"$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 265 14 "absolute error" }{TEXT -1 96 " in the value for the integral obtained by using the trapezoidal r ule with 4 intervals is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "abserr := abs(trap4-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\")35A=!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 15 " . . . and the " }{TEXT 265 14 "relative error" } {TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "relerr := abserr/abs(area);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+K)>%)G\"!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " . . . which is a bout 1.3%." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example \+ 3 " }{TEXT 293 23 ".. using the procedure " }{TEXT 266 4 "trap" } {TEXT 294 12 " instead of " }{TEXT 266 18 "student[trapezoid]" }{TEXT 295 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " We find an approximation for " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. 2);" "6#-%$IntG6$*&\"\"\"F'%\"xG! \"\"/F(;F'\"\"#" }{TEXT -1 45 " using the trapezoidal rule with 8 inte rvals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "drawtrap(1/x,x=[1..2,.5..2.5],y=0..1.3,intervals=8,xt ickmarks=4);" }}{PARA 13 "" 1 "" {GLPLOT2D 392 295 295 {PLOTDATA 2 "6' -%'CURVESG6%7U7$$\"3++++++++]!#=$\"\"#\"\"!7$$\"35mmmT:(z@&F*$\"3r![VP ``k\">!#<7$$\"3JLLL$3VfV&F*$\"3lt:e$>2'R=F37$$\"3t***\\i&*)fDcF*$\"3Y$ Rz\")\\)ex+5F3$\"3M&Ry6r*4)***F*7$$\"3 *******\\Z/N/\"F3$\"3!GAz!)e#4$e*F*7$$\"35+++NfC&3\"F3$\"3!*y&=tA,X@*F *7$$\"3LLLez6:B6F3$\"3/t/'))*3_.*)F*7$$\"3_mmm\"=C#o6F3$\"3U?IAg3+g&)F *7$$\"3gmmmEpS17F3$\"3#G9ne2x!*G)F*7$$\"3%)***\\i`A3D\"F3$\"3'[RR(H#RZ *zF*7$$\"3Ymmmwy8!H\"F3$\"37ph4P&46v(F*7$$\"3/++DOIFL8F3$\"3-9FV'=R.]( F*7$$\"3!****\\(3zMu8F3$\"3)*\\\"3p)zq 0rmF*7$$\"3F3$\"3iWV:#=H#>_F*7$$\"3'******\\)Hxe>F3$\"3%)>i%y$oB0^F*7$$\"3Ymm \"H!o-**>F3$\"3t(Rvr0ZF*7$ $\"3;LLL$*zym@F3$\"3#Gy=D=E^h%F*7$$\"30LL$3N1#4AF3$\"3#GD8mU7l_%F*7$$ \"3kmm\"HYt7D#F3$\"3&f4c]QI>W%F*7$$\"3%*******p(G**G#F3$\"3i7b8g![pO%F *7$$\"3Umm;9@BMBF3$\"3e=PT*4kSG%F*7$$\"3/LLL`v&QP#F3$\"3\"o`W2o_D@%F*7 $$\"30++DOl5;CF3$\"3yFejQ.*)QTF*7$$\"3/++v.UacCF3$\"3'e5MNBf22%F*7$$\" 3++++++++DF3$\"3A+++++++SF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F-F-Fj[l-% *THICKNESSG6#F,-%)POLYGONSG6+7&7$$\"\"\"F-Fj[l7$$\"++++D6!\"*Fj[l7$Ff \\l$\"+*)))))))))!#57$Fc\\lFc\\l7&Fe\\l7$$\"++++]7Fh\\lFj[l7$F`]l$\"++ +++!)F\\]lFi\\l7&F_]l7$$\"++++v8Fh\\lFj[l7$Fg]l$\"+tssssF\\]lFb]l7&Ff] l7$$\"+++++:Fh\\lFj[l7$F^^l$\"+nmmmmF\\]lFi]l7&F]^l7$$\"++++D;Fh\\lFj[ l7$Fe^l$\"+ah%Q:'F\\]lF`^l7&Fd^l7$$\"++++] " 0 "" {MPLTEXT 1 0 43 "student[t rapezoid](1/x,x=1..2,8);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&#\"\"$\"#K\"\"\"*&#F'\"\")F'-%$SumG6$*&F'F',&F'F'*&F*!\"\"%\"iGF'F' F1/F2;F'\"\"(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/&=7%p!#5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The same result can be obtained with the procedure " }{TEXT 0 4 "trap" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "trap8 := trap(1/x,x=1..2,intervals=8);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&trap8G$\"+/&=7%p!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "We can obtain a numerica l value for this integral which is accurate to about 10 digits by firs t evaluating the integral analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(1/x,x=1..2);\nvalue (%);\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*& \"\"\"F'%\"xG!\"\"/F(;F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#ln G6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+1=ZJp!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 265 14 "absolute error" }{TEXT -1 96 " in the value for the inte gral obtained by using the trapezoidal rule with 8 intervals is . . . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "abserr := abs(trap8-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'abserrG$\"()pY(*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 15 " . . . and the " }{TEXT 265 14 "relative error" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rele rr := abserr/abs(area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$ \"+(G^hS\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 " . . . which is about 0.14%." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }{TEXT 296 110 ".. empirical demons tration that the absolute error is proportional to the square of the w idth of the intervals" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 " First we find an approximation for " }{XPPEDIT 18 0 "Int(exp(-x^2),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"#!\"\"/F+;\"\"!\"\"\"" }{TEXT -1 46 " using the trapezoidal rule with 32 intervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "drawtrap(exp(-x^2),x=[0..1,0 ..1.2],intervals=32,shading=wheat);" }}{PARA 13 "" 1 "" {GLPLOT2D 376 316 316 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$\"\"!F)$\"\"\"F)7$$\"3h******* \\ech#!#>$\"3AA(*\\q1;$***!#=7$$\"3-+++v*G:*[F/$\"3Q#epma,h(**F27$$\"3 u******\\L)4X(F/$\"3OE#R+nOY%**F27$$\"3)******\\MSF+\"F2$\"3q'Q-L*\\&* **)*F27$$\"3#)****\\Fy:f7F2$\"3+52D6CqU)*F27$$\"3')****\\d'*)o\\\"F2$ \"3/&3\")3\"QUy(*F27$$\"3w****\\(>ZIu\"F2$\"3Vf)Q(Gwu+(*F27$$\"3u**** \\xOi(*>F2$\"3egFb]r!)3'*F27$$\"3#)****\\FPQ^AF2$\"3;La/;)ed]*F27$$\"3 /+++IrS7DF2$\"3)*H6Mt-H)Q*F27$$\"3p*****\\o;Bu#F2$\"3)4\"4(3m^bF*F27$$ \"3*********QS6+$F2$\"3(fcnCT'oQ\"*F27$$\"3[******\\o-hKF2$\"3u(z;CLi6 **)F27$$\"3(*******4cZ6NF2$\"3]Y^$Q[V*R))F27$$\"3S****\\xq!*QPF2$\"3go ='R9r`p)F27$$\"3&********3X$4SF2$\"3j6r'e9i]^)F27$$\"3s******f:WQUF2$ \"3!31kNIxcN)F27$$\"3f****\\<_$\\]%F2$\"3yf6KqhBj\")F27$$\"3**)*****fs #3u%F2$\"3W>V%fLBr)zF27$$\"3!)****\\<#Q'**\\F2$\"3;:86#e*G)y(F27$$\"33 ++]_u3Y_F2$\"3orqHtL4%f(F27$$\"3[*****\\PJK]&F2$\"3Mo=H0z0(Q(F27$$\"3% *****\\n(p$RdF2$\"3?kC-\"*e_$>(F27$$\"3A*****\\#p2%*fF2$\"3%pV$4T9s\") pF27$$\"3o****\\xgkeiF2$\"3;utMtX-fnF27$$\"3g****\\-V&*)['F2$\"3WL;cSD ZjlF27$$\"3.+++&\\$pPnF2$\"3Q;\\\\\\([5N'F27$$\"37******>am%*pF2$\"3P4 KT_#R38'F27$$\"3?*****\\JigC(F2$\"3e&p\">%3L_\"fF27$$\"3G****\\Pr_F27$$\"35******\\/;h#)F2$\"3O'z*o[Mo`]F27$$\"3M )***\\P/&f\\)F2$\"3+WR)f'=re[F27$$\"3e******4zj_()F2$\"3_OVpa_G[YF27$$ \"3u)***\\<3;%**)F2$\"3'oC*G&HdKX%F27$$\"3]****\\Z=iY#*F2$\"35Y!yxCJGD %F27$$\"39)*****\\'[M\\*F2$\"3mTO0Ed R2\"F^x$\"3)4K1==qc:$F27$$\"3&*****\\o#R05\"F^x$\"3-w^6TiVyHF27$$\"3() *****>`9V7\"F^x$\"3<@(y8\"=)\\#GF27$$\"3)****\\<#Rm\\6F^x$\"3%eu)y>Kum EF27$$\"3%****\\A_ER<\"F^x$\"3,a(4jHh0_#F27$$\"3%**************>\"F^x$ \"3#z@@oex#pBF2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\" #-%)POLYGONSG6C7&7$F(F(7$$\"++++DJ!#6F(7$Fj[l$\"+U\"R-***!#5F'7&Fi[l7$ $\"++++]iF\\\\lF(7$Fc\\l$\"+&p85'**F`\\lF]\\l7&Fb\\l7$$\"++++v$*F\\\\l F(7$Fj\\l$\"+%[%\\7**F`\\lFe\\l7&Fi\\l7$$\"++++]7F`\\lF(7$Fa]l$\"+qV' \\%)*F`\\lF\\]l7&F`]l7$$\"+++]i:F`\\lF(7$Fh]l$\"+,b\")e(*F`\\lFc]l7&Fg ]l7$$\"++++v=F`\\lF(7$F_^l$\"+Abaa'*F`\\lFj]l7&F^^l7$$\"+++](=#F`\\lF( 7$Ff^l$\"+%y_F`*F`\\lFa^l7&Fe^l7$$\"+++++DF`\\lF(7$F]_l$\"+G18%R*F`\\l Fh^l7&F\\_l7$$\"+++]7GF`\\lF(7$Fd_l$\"+93YR#*F`\\lF__l7&Fc_l7$$F[\\lF` \\lF(7$F[`l$\"+zhgp!*F`\\lFf_l7&Fj_l7$$\"+++]PMF`\\lF(7$Fa`l$\"+HE]&)) )F`\\lF\\`l7&F``l7$$\"++++]PF`\\lF(7$Fh`l$\"+j0:)o)F`\\lFc`l7&Fg`l7$$ \"+++]iSF`\\lF(7$F_al$\"+Regy%)F`\\lFj`l7&F^al7$$\"++++vVF`\\lF(7$Ffal $\"++/(zD)F`\\lFaal7&Feal7$$\"+++](o%F`\\lF(7$F]bl$\"+nEQF!)F`\\lFhal7 &F\\bl7$$\"+++++]F`\\lF(7$Fdbl$\"+Jy+)y(F`\\lF_bl7&Fcbl7$$\"+++]7`F`\\ lF(7$F[cl$\"+Q\"G5a(F`\\lFfbl7&Fjbl7$$\"++++DcF`\\lF(7$Fbcl$\"+*HLwG(F `\\lF]cl7&Facl7$$\"+++]PfF`\\lF(7$Ficl$\"+A6,HqF`\\lFdcl7&Fhcl7$$Fd\\l F`\\lF(7$F`dl$\"+i%Qjw'F`\\lF[dl7&F_dl7$$\"+++]ilF`\\lF(7$Ffdl$\"+%fs2 ]'F`\\lFadl7&Fedl7$$\"++++voF`\\lF(7$F]el$\"+!4VMB'F`\\lFhdl7&F\\el7$$ \"+++](=(F`\\lF(7$Fdel$\"+gUWlfF`\\lF_el7&Fcel7$$\"+++++vF`\\lF(7$F[fl $\"+Z#Gyp&F`\\lFfel7&Fjel7$$\"+++]7yF`\\lF(7$Fbfl$\"+4))fJaF`\\lF]fl7& Fafl7$$\"++++D\")F`\\lF(7$Fifl$\"+Geqn^F`\\lFdfl7&Fhfl7$$\"+++]P%)F`\\ lF(7$F`gl$\"+)fSq!\\F`\\lF[gl7&F_gl7$$\"++++]()F`\\lF(7$Fggl$\"+\")=V] YF`\\lFbgl7&Ffgl7$$\"+++]i!*F`\\lF(7$F^hl$\"+jFk)R%F`\\lFigl7&F]hl7$$F []lF`\\lF(7$Fehl$\"+(GoB:%F`\\lF`hl7&Fdhl7$$\"+++](o*F`\\lF(7$F[il$\"+ gQB7RF`\\lFfhl7&Fjhl7$$\"+++++5!\"*F(7$Fbil$\"+7WzyOF`\\lF]il-Fjz6&F\\ [l$\")#)eq%)F_[lFjil$\")h>!\\(F_[l-%+AXESLABELSG6$Q\"x6\"Q!Fbjl-%%VIEW G6$;F($\"#7!\"\"%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "student[trapezoid](exp(-x^2) ,x=0..1,32);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\" #kF%*&#F%\"#KF%-%$SumG6$-%$expG6#,$*&\"%C5!\"\"%\"iG\"\"#F3/F4;F%\"#JF %F%*&F$F%-F.6#F3F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZDknu!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The sa me result can be obtained with the procedure " }{TEXT 0 4 "trap" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "trap32 := trap(exp(-x^2),x=0..1,intervals=32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trap32G$\"+ZDknu!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "A better estimate \+ is obtained by doubling the number of sub-intervals used to 64. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "trap64 := trap(exp(-x^2),x=0..1,intervals=64);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'trap64G$\"+O;4ou!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "We can obtain a numerical value f or this integral which is accurate to about 10 digits by first evaluat ing the integral analytically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(exp(-x^2),x=0..1);\nvalu e(%);\narea := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$- %$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"/F,;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&-%$erfG6#F&F&%#PiGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+I8Cou!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 265 15 "absol ute errors" }{TEXT -1 119 " in the values for the integral obtained by using the trapezoidal rule with 32 and 64 intervals respectively are \+ . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "abserr32 := abs(trap32-area);\nabserr64 := abs(trap64 -area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr32G$\"'$y)f!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr64G$\"'%p\\\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The ratio of thes e absolute errors is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(abserr32/abserr64,6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'0+S!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "When the number of intervals is doubled from 32 to 64 , the absolute error decreases by a factor of 4 approximately. " }} {PARA 0 "" 0 "" {TEXT -1 140 "It can be shown theoretically that in ma ny situations the absolute error in using the trapezoidal rule is appr oximately proportional to the " }{TEXT 265 6 "square" }{TEXT -1 118 " \+ of the width of the sub-intervals (on which the trapezoids are based ) . This last result seems to support this idea. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "Similarly, when the numb er of intervals is tripled from 32 to 96, the absolute error decreases by a factor of 9 approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "trap96 := trap(exp(-x^2),x= 0..1,intervals=96);\nabserr32 := abs(trap32-area);\nabserr96 := abs(tr ap96-area);\nevalf(abserr32/abserr96,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trap96G$\"+*zu\"ou!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)a bserr32G$\"'$y)f!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr96G$\" &Jl'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'1+!*!\"&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }{TEXT 301 41 ".. us ing the trapezoidal rule iteratively" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "We start by finding an approximation for \+ " }{XPPEDIT 18 0 "Int(ln(2*sin(x)+cos(x)),x = 0 .. 1);" "6#-%$IntG6$-% #lnG6#,&*&\"\"#\"\"\"-%$sinG6#%\"xGF,F,-%$cosG6#F0F,/F0;\"\"!F," } {TEXT -1 46 " using the trapezoidal rule with 32 intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "dr awtrap(ln(2*sin(x)+cos(x)),x=[0..1,0..1.1],intervals=32,scheme=stripes );" }}{PARA 13 "" 1 "" {GLPLOT2D 490 262 262 {PLOTDATA 2 "6E-%'CURVESG 6%7S7$$\"\"!F)F(7$$\"3WLLLepo(R#!#>$\"3wx'3#3T2cYF-7$$\"35nmTg:!R[%F-$ \"3Cf&e(yf=$\\)F-7$$\"3!RLL32o+$oF-$\"3[6.1yX&*e7!#=7$$\"3_ML$e\\'y\"> *F-$\"3YE!z'35o\\;F:7$$\"3um;a3!GU:\"F:$\"3\\co%zYB'=?F:7$$\"3OL$3F&)[ @P\"F:$\"3'RU%ej&fTM#F:7$$\"33+]PkKz(f\"F:$\"3;G$R2`ybm#F:7$$\"3bL$3x. b6$=F:$\"3Kx+n]$zA)HF:7$$\"31+](oToP1#F:$\"3-V!o:bIIG$F:7$$\"3.nm;p)RI I#F:$\"3ss[PZ;vxNF:7$$\"3ZLLe%H!z8DF:$\"3*)G'*R#p-T)=]F:7$$\"3?nmm'4Q_)QF:$\"3MPEpk+Y1_F:7$$\"3B+]P\\R_HTF:$\" 3*yAiPsG\\T&F:7$$\"3wmmm@$edM%F:$\"3_2+R1V,\"f&F:7$$\"3c+]P*p,Ie%F:$\" 3qn%=mDCax&F:7$$\"3\"4+D\")\\8*3[F:$\"3lXX`3)yF%fF:7$$\"3InmTg(GY/&F:$ \"3myYY?+64hF:7$$\"3;n;a`*)3h_F:$\"3))eUcV^jaiF:7$$\"3WLLe90d%\\&F:$\" 3G&Q,1@!4/kF:7$$\"3KL$3xB#4PdF:$\"3)G:A5s:8b'F:7$$\"3m+]i5\"3#[fF:$\"3 7,:vY$3In'F:7$$\"3kLL3P!>i<'F:$\"3#[34u+`yz'F:7$$\"3]+++Nmx6kF:$\"3]I6 'H\"3#)>pF:7$$\"3>++v)yBAk'F:$\"3hw?$4z7C.(F:7$$\"3!4+v$fK>loF:$\"3M$[ o!=m:NrF:7$$\"3\"****\\7%Gw7rF:$\"3*RNNt9]AC(F:7$$\"37mmm@^@NtF:$\"3OJ 9V\\ALKtF:7$$\"3c++]7/tsvF:$\"3[ueDeP@AuF:7$$\"3%RL3xcazy(F:$\"3'y9UbS N\")\\(F:7$$\"3P,+]<9DB!)F:$\"3?(H+*RVAvvF:7$$\"32m;/;ukW#)F:$\"3gEt?? 3?UwF:7$$\"35,](o-qgZ)F:$\"3`$**GFNHlq(F:7$$\"3kmm;HzK-()F:$\"3i(p-ZSy Qw(F:7$$\"3r**\\P%)*)>R*)F:$\"3!ya1^18\"=yF:7$$\"31NLLjRLn\"*F:$\"3k(G -LM$zkyF:7$$\"3pMLeH\\j+%*F:$\"3:$)>4Oy%p!zF:7$$\"3#ymTg//?j*F:$\"3)=k -V=CK%zF:7$$\"3#3++]B3Y%)*F:$\"3'eFQ&eF_]lFjclF]\\l-Fc[l6$7&F`dl7$$\"+++++]F_]lF(7$Fjdl$\"+6XD ygF_]lFcdlF`]l-Fc[l6$7&Fidl7$$\"+++]7`F_]lF(7$Fcel$\"+$y0#)G'F_]lF\\el F]\\l-Fc[l6$7&Fbel7$$\"++++DcF_]lF(7$F\\fl$\"+B*oU['F_]lFeelF`]l-Fc[l6 $7&F[fl7$$\"+++]PfF_]lF(7$Fefl$\"+_$ypm'F_]lF^flF]\\l-Fc[l6$7&Fdfl7$$F []lF_]lF(7$F^gl$\"+ON\"o$oF_]lFgflF`]l-Fc[l6$7&F]gl7$$\"+++]ilF_]lF(7$ Ffgl$\"+$p/U*pF_]lF_glF]\\l-Fc[l6$7&Fegl7$$\"++++voF_]lF(7$F_hl$\"+uw` RrF_]lFhglF`]l-Fc[l6$7&F^hl7$$\"+++](=(F_]lF(7$Fhhl$\"+I\"eJF(F_]lFahl F]\\l-Fc[l6$7&Fghl7$$\"+++++vF_]lF(7$Fail$\"+)4v`R(F_]lFjhlF`]l-Fc[l6$ 7&F`il7$$\"+++]7yF_]lF(7$Fjil$\"+\"4kk](F_]lFcilF]\\l-Fc[l6$7&Fiil7$$ \"++++D\")F_]lF(7$Fcjl$\"+](pmg(F_]lF\\jlF`]l-Fc[l6$7&Fbjl7$$\"+++]P%) F_]lF(7$F\\[m$\"+S\"3ip(F_]lFejlF]\\l-Fc[l6$7&F[[m7$$\"++++]()F_]lF(7$ Fe[m$\"+y'o_x(F_]lF^[mF`]l-Fc[l6$7&Fd[m7$$\"+++]i!*F_]lF(7$F^\\m$\"+Oe ,WyF_]lFg[mF]\\l-Fc[l6$7&F]\\m7$$Fg]lF_]lF(7$Fg\\m$\"+40f-zF_]lF`\\mF` ]l-Fc[l6$7&Ff\\m7$$\"+++](o*F_]lF(7$F_]m$\"+E76^zF_]lFh\\mF]\\l-Fc[l6$ 7&F^]m7$$\"+++++5!\"*F(7$Fh]m$\"+Z^n*)zF_]lFa]mF`]l-%+AXESLABELSG6$Q\" x6\"Q!Fb^m-%%VIEWG6$;F($\"#6F][l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "trap32 := trap(ln(2*sin( x)+cos(x)),x=0..1,intervals=32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'trap32G$\"+Suf:a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "A value which is accurate to 10 digits can be obtaine d from a (very complicated) analytical expression. " }}{PARA 0 "" 0 " " {TEXT -1 113 "Evaluating this expression as a floating point number \+ yields a small imaginary part which should be removed with " }{TEXT 0 2 "Re" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Int(ln(2*sin(x)+cos(x)),x=0..1);\nv alue(%);\nevalf(evalf(%,15));\nRe(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%#lnG6#,&*&\"\"#\"\"\"-%$sinG6#%\"xGF,F,-%$cosGF/F,/F0;\" \"!F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*(-%#lnG6#\"\"#\"\"\"-F&6#- %$expG6#^#F)F)F/F)F)*(^#!\"\"F)F*F)-F&6#*&^$#F)\"\"&#!\"#F8F),(*&\"\"$ F)-F-6#F1F)F2*&^#\"\"%F)F>F)F)*&F8F)F,F)F)F)F)F)*&^##F2F(F))F*F(F)F)*( F1F)F*F)-F&6#F8F)F)*(F*F)-F&6#,(*$F8#F)F(F)*&F(F)F,F)F2*&F1F)F,F)F)F)F /F)F)*(F*F)-F&6#,(FOF)*&F(F)F,F)F)*&F,F)F/F)F)F)F/F)F)*&-%&dilogG6#,(F )F)*&#F(F8F)*&F8FPF,F)F)F)*(^#F7F)F8FPF,F)F)F)F/F)F)*&-Fen6#,(F)F)*&#F (F8F)FjnF)F2*(^##F2F8F)F8FPF,F)F)F)F/F)F)*&F1F)-Fen6#,(F)F)*(F(F)F8F2F 8FPF)*&F\\oF)F8FPF)F)F)*&F1F)-Fen6#,(F)F)*(F(F)F8F2F8FPF2*&FdoF)F8FPF) F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+\\s8 " 0 "" {MPLTEXT 1 0 60 "Int(ln(2*sin(x)+cos(x)), x=0..1);\narea := evalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%#lnG6#,&*&\"\"#\"\"\"-%$sinG6#%\"xGF,F,-%$cosGF/F,/F0;\" \"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+\\s8 " 0 "" {MPLTEXT 1 0 29 "abserr32 := abs(trap32-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr32G$\"(4)R:!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "If we assume that the absolute error is proportional to \+ the square of the width of the intervals used, we can estimate how man y intervals will be needed to obtain a value which is correct to 10 di gits. " }}{PARA 0 "" 0 "" {TEXT -1 40 "If we aim for an error no great er than " }{XPPEDIT 18 0 "epsilon = 2*`.`*10^(-11);" "6#/%(epsilonG*( \"\"#\"\"\"%\".GF')\"#5,$\"#6!\"\"F'" }{TEXT -1 95 ", this is unlikely to affect the last decimal place, since a unit in the last place (1 u lp) is " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" }{TEXT -1 73 ": A bound for an \+ error aimed for in a numerical computation is called a " }{TEXT 265 9 "tolerance" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eps := 2e-11;\nabserr32/eps; \nevalf(log[4](%),7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"# !#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++]/*p(!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(7Q9\"!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "log[4](7699045)" "6#-&%$logG6#\"\"%6# \"(X!*p(" }{TEXT -1 1 " " }{TEXT 304 1 "~" }{TEXT -1 29 " 11.43812 mea ns that 7699045 " }{TEXT 303 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4^1 1.43812" "6#)\"\"%-%&FloatG6$\"(7Q9\"!\"&" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The desired tolera nce is reached when the absolute error is divided by 4 about 11 times. " }}{PARA 0 "" 0 "" {TEXT -1 70 "Thus we should double the number of \+ intervals 11 times to reach about " }{XPPEDIT 18 0 "32*`.`*2^11 = 2^16 ;" "6#/*(\"#K\"\"\"%\".GF&\"\"#\"#6*$F(\"#;" }{XPPEDIT 18 0 "``=65536 " "6#/%!G\"&Ob'" }{TEXT -1 12 " intervals. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "The following calculations giv e progressively more accurate values for the integral by using the tra pezoidal rule with a progressively larger number of intervals - doubli ng the number of intervals each time." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "2^6;\ntrap(ln(2*sin(x)+c os(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[Av;a!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "2^7;\ntrap(l n(2*sin(x)+cos(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'*4/ " 0 "" {MPLTEXT 1 0 50 "2^8;\ntrap(ln(2*sin(x)+cos(x)),x=0..1,intervals=%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'=8 rT&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "2^9;\ntrap(ln(2*sin(x)+cos(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+L78 " 0 "" {MPLTEXT 1 0 51 "2^10;\ntrap(ln(2*sin(x)+cos(x)),x=0..1,in tervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Xd8 " 0 "" {MPLTEXT 1 0 51 "2^11;\ntrap(ln(2*sin(x)+cos( x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%[?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+to8 " 0 "" {MPLTEXT 1 0 51 "2^12;\ntrap( ln(2*sin(x)+cos(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%'4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+br8 " 0 "" {MPLTEXT 1 0 51 "2^13;\ntrap(ln(2*sin(x)+cos(x)),x=0..1,intervals=%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"%#>)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Es 8 " 0 "" {MPLTEXT 1 0 51 "2^14;\ntrap(ln(2*sin(x)+cos(x)),x=0..1,intervals=%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&%Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Vs8 " 0 "" {MPLTEXT 1 0 51 "2^15;\ntrap(ln(2*sin(x)+cos( x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&oF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[s8 " 0 "" {MPLTEXT 1 0 54 "2^16;\ntrap( ln(2*sin(x)+cos(x)),x=0..1,intervals=2^16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&Ob'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\s8 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "This process can be autom ated by using the procedure " }{TEXT 0 4 "trap" }{TEXT -1 18 " with th e option \"" }{TEXT 281 12 "iterate=true" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 73 "The end result is a value for the integral which \+ is correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 249 "The trapezoidal rule is used with a progressively larger number of intervals until the difference between successively \+ computed estimates is small enough to be able to conclude that the int egral has been determined to the required number of digits. " }}{PARA 0 "" 0 "" {TEXT -1 125 "Since the number of intervals is doubled with \+ each iteration, it is only necessary to calculate new values for the i ntegrand " }{XPPEDIT 18 0 "ln(2*sin(x)+cos(x))" "6#-%#lnG6#,&*&\"\"#\" \"\"-%$sinG6#%\"xGF)F)-%$cosG6#F-F)" }{TEXT -1 150 " at the mid points of the intervals of the previous step. The addition of values previou sly obtained (except at the end points) need not be repeated. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" } {TEXT -1 10 ": This is " }{TEXT 265 16 "very inefficient" }{TEXT -1 132 " because of the large number of evaluations of the integrand need ed, but the calculation is completed quickly because the procedure " } {TEXT 0 4 "trap" }{TEXT -1 40 " performs most of the calculation using " }{TEXT 265 34 "hardware floating point arithmetic" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "trap(ln(2*sin(x)+cos(x)),x=0..1,iterate=true,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Fapproximation~with~1~interval~--->~~ ~G$\"0lkAdP[*R!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~w ith~2~intervals~--->~~~G$\"00]?/Yl.&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~~~G$\"0b!eF-d>`!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~--->~ ~~G$\"0=<1&=c#R&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation ~with~16~intervals~--->~~~G$\"0b/d8\")4T&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$\"0mF*Ruf:a! #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~interval s~--->~~~G$\"0n@zC_nT&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproxi mation~with~128~intervals~--->~~~G$\"0Z+f*4/~~~G$\"0isd=8rT& !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~512~interv als~--->~~~G$\"0d?MBJrT&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Jappro ximation~with~1024~intervals~--->~~~G$\"0\"QMXd8~~~G$\"0MD L(o8~~~G$\"0v?`:PrT&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %Japproximation~with~8192~intervals~--->~~~G$\"0i>eAPrT&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~16384~intervals~--->~~~ G$\"0LWMCPrT&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~wi th~32768~intervals~--->~~~G$\"0\\]yCPrT&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~65536~intervals~--->~~~G$\"0/_*[s8 ~~~G$\"0UF#\\s8~~~G$\"0B'H\\s8 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }{TEXT 297 41 ".. using the trapezoidal rule iteratively" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We start by finding an approximation for " } {XPPEDIT 18 0 "Int(1/(x+exp(x)),x = 0 .. 1);" "6#-%$IntG6$*&\"\"\"F',& %\"xGF'-%$expG6#F)F'!\"\"/F);\"\"!F'" }{TEXT -1 46 " using the trapezo idal rule with 64 intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "drawtrap(1/(x+exp(x)),x=[0. .1,-.1..1.2],intervals=64,\n scheme=stripes,shading=[tan,yel low]);" }}{PARA 13 "" 1 "" {GLPLOT2D 596 351 351 {PLOTDATA 2 "6_o-%'CU RVESG6%7S7$$!3/+++++++5!#=$\"3;'**QR'p[U7!#<7$$!3uLLLe*pj;(!#>$\"3Q%30 -@+R;\"F-7$$!3!pm;/hV3q%F1$\"3)G_s'Q'\\C5\"F-7$$!3hLL$3P,\"G>F1$\"3g[) *HO*3*R5F-7$$\"3bsmmTS?I')!#?$\"3@DN)**oj*H)*F*7$$\"3PJLekk(3k$F1$\"3( o&pdoRU:$*F*7$$\"3vlm\"Hi/j@'F1$\"3gi`Ynzjy))F*7$$\"3O***\\iI6I)))F1$ 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$$\"+++++5!\"*Fa[l7$F``n$\"+9UT*o#Fc\\lFi_nFi]l-%+AXESLABELSG6$Q\"x6\" Q!Fj`n-%%VIEWG6$;$F`[lF`[l$\"#7F`[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36 " "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "C urve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55 " "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "C urve 62" "Curve 63" "Curve 64" "Curve 65" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "trap64 := trap(1/( x+exp(x)),x=0..1,intervals=64);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' trap64G$\"+)yfL;&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A value which is accurate to 10 digits can be obtained using Maple's numerical integration via " }{TEXT 0 9 "evalf/Int" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(1/(x+exp(x)),x=0..1);\narea := evalf(evalf(% ,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&%\"xGF' -%$expG6#F)F'!\"\"/F);\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ar eaG$\"+Mw+j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The absolute error in the approximate value for the inte gral obtained using 64 intervals is about 3.5e-5" }{XPPEDIT 18 0 "`` = 3.5*`.`*10^(-5);" "6#/%!G*(-%&FloatG6$\"#N!\"\"\"\"\"%\".GF+)\"#5,$\" \"&F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "abserr64 := abs(trap64-area);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr64G$\"'a@N!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "Assuming that the absolute error is pr oportional to the square of the width of the intervals used, we can es timate how many intervals will be needed to obtain a value which is co rrect to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 30 "If we aim for a t olerance of " }{XPPEDIT 18 0 "epsilon = 2*`.`*10^(-11);" "6#/%(epsilo nG*(\"\"#\"\"\"%\".GF')\"#5,$\"#6!\"\"F'" }{TEXT -1 95 ", this is unli kely to affect the last decimal place, since a unit in the last place \+ (1 ulp) is " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eps := 2e-11;\nabserr64/eps;\nevalf(log[4](%),6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"#!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++xg " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "log [4](1760770)" "6#-&%$logG6#\"\"%6#\"(q2w\"" }{TEXT -1 1 " " }{TEXT 300 1 "~" }{TEXT -1 28 " 10.3739 means that 1760770 " }{TEXT 299 1 "~ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "4^10.3739" "6#)\"\"%-%&FloatG6$\"'RP 5!\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The desired tolerance is reached when the absolute e rror is divided by 4 about 10 times. " }}{PARA 0 "" 0 "" {TEXT -1 70 " Thus we should double the number of intervals 10 times to reach about \+ " }{XPPEDIT 18 0 "64*`.`*2^10 = 2^16;" "6#/*(\"#k\"\"\"%\".GF&\"\"#\"# 5*$F(\"#;" }{XPPEDIT 18 0 "``=65536" "6#/%!G\"&Ob'" }{TEXT -1 12 " int ervals. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "The following calculations give progressively more accurate value s for the integral by using the trapezoidal rule with a progressively \+ larger number of intervals - doubling the number of intervals each tim e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^7;\ntrap(1/(x+exp(x)),x=0..1,intervals=%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yc4j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^8;\ntrap(1/(x+exp(x)),x=0..1,intervals=%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+X'HI;&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2^9;\ntrap(1/(x+exp(x)),x=0. .1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PJ,j^!#5" }}}{PARA 0 "" 0 "" {TEXT 298 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^10;\ntrap(1/(x+exp(x)) ,x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%C5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4!4I;&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^11;\ntrap( 1/(x+exp(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"%[?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yz+j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^12;\n trap(1/(x+exp(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%'4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?x+j^!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^13;\ntrap(1/(x+exp(x)),x=0..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%#>)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bw+j^!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^14;\ntrap(1/(x+exp(x)),x=0..1,intervals=%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"&%Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Rw+j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^15;\ntrap(1/(x+exp(x)),x=0..1,intervals=% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&oF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Nw+j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "2^16;\ntrap(1/(x+exp(x)),x=0 ..1,intervals=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&Ob'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mw+j^!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "This process can be automated by using the procedure " } {TEXT 0 4 "trap" }{TEXT -1 18 " with the option \"" }{TEXT 281 12 "ite rate=true" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 73 "The end re sult is a value for the integral which is correct to 10 digits." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" } {TEXT -1 10 ": This is " }{TEXT 265 16 "very inefficient" }{TEXT -1 132 " because of the large number of evaluations of the integrand need ed, but the calculation is completed quickly because the procedure " } {TEXT 0 4 "trap" }{TEXT -1 40 " performs most of the calculation using " }{TEXT 265 34 "hardware floating point arithmetic" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "trap(1/(x+exp(x)),x=0..1,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fapproximation~with~1~interval~--->~~~G$\"0)* \\o52ZM'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~ intervals~--->~~~G$\"0+9'H*=$*\\&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%Gapproximation~with~4~intervals~--->~~~G$\"039BH]8D&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~--->~~~G$\" 0o*p&*oU&=&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with ~16~intervals~--->~~~G$\"0*R+F\\jo^!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$\"0ZU<*eTk^!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~intervals~---> ~~~G$\"0[i$)yfL;&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximatio n~with~128~intervals~--->~~~G$\"0%4+yc4j^!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~intervals~--->~~~G$\"0m?^kHI;& !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~512~interv als~--->~~~G$\"0%)pl88I;&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Jappr oximation~with~1024~intervals~--->~~~G$\"0Y6%4!4I;&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~2048~intervals~--->~~~G$\"0c? w(z+j^!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4096 ~intervals~--->~~~G$\"0vs'>x+j^!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %Japproximation~with~8192~intervals~--->~~~G$\"0#e=bw+j^!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~16384~intervals~--->~~~ G$\"03k!Rw+j^!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~wi th~32768~intervals~--->~~~G$\"0jL]j2I;&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~65536~intervals~--->~~~G$\"0-ESj2I ;&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mw+j^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }{TEXT 289 44 ".. a t ough example for numerical integration" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "We find an approximation for " }{XPPEDIT 18 0 "Int(s qrt(1-x^2),x = 0 .. 1)" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#! \"\"/F,;\"\"!F*" }{TEXT -1 46 " using the trapezoidal rule with 32 int ervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "drawtrap(sqrt(1-x^2),x=0..1,intervals=32,\n \+ scheme=stripes,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 418 338 338 {PLOTDATA 2 "6F-%'CURVESG6%7Y7$$\"\"!F)$\"\"\"F) 7$$\"3emmm;arz@!#>$\"3e(=:\"QTi(***!#=7$$\"3[LL$e9ui2%F/$\"3!e2E0a)o\" ***F27$$\"3nmmm\"z_\"4iF/$\"3)f\\s^f/2)**F27$$\"3[mmmT&phN)F/$\"3)37fk 0E]'**F27$$\"3CLLe*=)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$$\"3WLLL 3En$4#F2$\"3_.Ayu2Py(*F27$$\"3qmm;/RE&G#F2$\"3wIBk#=x`t*F27$$\"3\")*** **\\K]4]#F2$\"31q9\"H%H@#o*F27$$\"3$******\\PAvr#F2$\"3B_^A([sOi*F27$$ \"3)******\\nHi#HF2$\"3m:#e7,zAc*F27$$\"3jmm\"z*ev:JF2$\"3)>#\\;*p8A]* F27$$\"3?LLL347TLF2$\"3pW!GHZL`U*F27$$\"3,LLLLY.KNF2$\"3#=1W(>`Yb$*F27 $$\"3w***\\7o7Tv$F2$\"3!QZNcQ$eo#*F27$$\"3'GLLLQ*o]RF2$\"3=`?'yv9l=*F2 7$$\"3A++D\"=lj;%F2$\"3]kxU3;t!4*F27$$\"31++vV&R5j')F27$$\"3&em;zRQb@&F2$\"3e0&=5t$=K&) F27$$\"3\\***\\(=>Y2aF2$\"3ld&yZ.e=T)F27$$\"39mm;zXu9cF2$\"3]/X@$oS\\F )F27$$\"3l******\\y))GeF2$\"3k!yShX>b7)F27$$\"3'*)***\\i_QQgF2$\"35nMl Um1rzF27$$\"3@***\\7y%3TiF2$\"3bX$z*QqP8yF27$$\"35****\\P![hY'F2$\"3QX hIao;GwF27$$\"3kKLL$Qx$omF2$\"3q\"p\\+YH?X(F27$$\"3!)*****\\P+V)oF2$\" 378IrHv-`sF27$$\"3?mm\"zpe*zqF2$\"3QynnKd;iqF27$$\"3%)*****\\#\\'QH(F2 $\"3G\"QTZ_=5%oF27$$\"3GKLe9S8&\\(F2$\"3a%Rp()p\"*)>mF27$$\"3R***\\i?= bq(F2$\"3=&F27$$\"3=LLe9tOc()F2$\"3PnBuRIqH[F27$$\"3u******\\Qk \\*)F2$\"3+@I:/fPhWF27$$\"3CLL$3dg6<*F2$\"3au)*4GH?')RF27$$\"3ImmmmxGp $*F2$\"3s!RL6R._\\$F27$$\"3A++D\"oK0e*F2$\"3!yTAE(=!f'GF27$$\"3C+++]oi \"o*F2$\"3?UJ1O#=K]#F27$$\"3A++v=5s#y*F2$\"37vO>U4Dt?F27$$\"3;+D1k2/P) *F2$\"3mLW(eL^zz\"F27$$\"35+]P40O\"*)*F2$\"31H.f%oH+Z\"F27$$\"3k]7.#Q? 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Fj_m-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16 " "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "C urve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "student[trap ezoid](sqrt(1-x^2),x=0..1,32);\ntrap32 := evalf(evalf(%,15));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"#kF%*&#F%\"#KF%-%$SumG6$*$, &F%F%*&\"%C5!\"\"%\"iG\"\"#F1#F%F3/F2;F%\"#JF%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trap32G$\"+dgvPy!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "The same result can be obtained with \+ the procedure " }{TEXT 0 4 "trap" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "trap(sqrt(1- x^2),x=0..1,intervals=32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+dgvP y!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "We can obtain a numerical value for this integral which is accurate t o about 10 digits by first evaluating the integral analytically." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(sqrt(1-x^2),x=0..1);\nvalue(%);\narea := evalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*$)%\"xG\"\"#F( !\"\"#F(F,/F+;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\" \"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"+M;)R&y! #5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 265 14 "absolute error" }{TEXT -1 97 " in the value for the \+ integral obtained by using the trapezoidal rule with 32 intervals is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "abserr := abs(trap32-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\")ybA;!#5" }}}{PARA 0 "" 0 "" {TEXT -1 15 " . . . and the " }{TEXT 265 14 "relative error" }{TEXT -1 10 " is . . \+ . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "relerr := abserr/abs(area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+aZ!f1#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " . . . which is about 0.2%." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The trape zoidal rule does " }{TEXT 265 29 "not give a very good estimate" } {TEXT -1 206 " for the integral in this case, even though we have used a reasonably large number of intervals. The problem occurs towards th e right end of the interval where the gradient of the curve approaches infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The error can be made small by using a very number of intervals . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "bigtrap := trap(sqrt(1-x^2),x=0..1,intervals=2^16);\n abserr := abs(bigtrap-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(big trapG$\"+f9)R&y!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"$w \"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " If " }{TEXT 0 18 "student[trapezoid]" }{TEXT -1 20 " is used instead o f " }{TEXT 0 4 "trap" }{TEXT -1 32 ", the computation takes longer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(student[trapezoid](sqrt(1-x^2),x=0..1,2^16));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e9)R&y!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "The calculation can be performed more speedily with " }{TEXT 0 18 "student[trapezoid]" }{TEXT -1 10 " by us ing " }{TEXT 0 6 "evalhf" }{TEXT -1 98 " after the symbolic expression is constructed. This enables the calculation to be performed using " }{TEXT 265 34 "hardware floating point arithmetic" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "student[trapezoid](sqrt(1-x^2),x=0..1,2^16);\nevalhf(%);\nevalf(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"'s58F%*&#F%\"&Ob'F%-%$ SumG6$*$,&F%F%*&\"+'Hn\\H%!\"\"%\"iG\"\"#F1#F%F3/F2;F%\"&Nb'F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3U)fR(e9)R&y!#=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+f9)R&y!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 4 "Note" }{TEXT -1 28 ": It is not possible \+ to use " }{TEXT 0 6 "evalhf" }{TEXT -1 15 " composed with " }{TEXT 0 18 "student[trapezoid]" }{TEXT -1 10 " directly." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalhf(stud ent[trapezoid](sqrt(1-x^2),x=0..1,2^16));" }}{PARA 8 "" 1 "" {TEXT -1 66 "Error, unable to evaluate function `student[trapezoid]` in evalhf \n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Th e procedure " }{TEXT 0 4 "trap" }{TEXT -1 18 " with the option \"" } {TEXT 281 12 "iterate=true" }{TEXT -1 148 "\" manages to obtain a valu e which is correct to 10 digits provided that the allowed maximum numb er of iterations is increased to 23 via the option \"" }{TEXT 281 16 " maxiterations=23" }{TEXT -1 6 "\" or \"" }{TEXT 281 10 "maxiter=23" } {TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "trap(sqrt(1-x^2),x=0..1,iterate=true,maxiterat ions=23,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fapproximation ~with~1~interval~--->~~~G$\"0+++++++&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~--->~~~G$\"0?A*=q7Io!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~ ~~G$\"05c-ns#*[(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation ~with~8~intervals~--->~~~G$\"0%H*3'yaCx!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~16~intervals~--->~~~G$\"0QpXfK\"3y !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~interva ls~--->~~~G$\"0%G>dgvPy!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happrox imation~with~64~intervals~--->~~~G$\"0A\\>GU#[y!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Iapproximation~with~128~intervals~--->~~~G$\"0c\"*4) >&>&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~i ntervals~--->~~~G$\"04$RdRE`y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%I approximation~with~512~intervals~--->~~~G$\"09*z\")ys`y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~1024~intervals~--->~~~G $\"0bZj\">*Q&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~w ith~2048~intervals~--->~~~G$\"0hGN\"*\\R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4096~intervals~--->~~~G$\"0Q>!>/(R &y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~8192~int ervals~--->~~~G$\"0Cu)ow(R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Ka pproximation~with~16384~intervals~--->~~~G$\"0B(4K-)R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~32768~intervals~--->~~~ G$\"0aN$Q6)R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~w ith~65536~intervals~--->~~~G$\"0eR(e9)R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~131072~intervals~--->~~~G$\"0e>?d \")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~2621 44~intervals~--->~~~G$\"07q?h\")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~524288~intervals~--->~~~G$\"08Iii\")R&y!#: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~with~1048576~inter vals~--->~~~G$\"0WO7j\")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Map proximation~with~2097152~intervals~--->~~~G$\"0[1Ij\")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~with~4194304~intervals~--->~ ~~G$\"0EKOj\")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximatio n~with~8388608~intervals~--->~~~G$\"0P`Qj\")R&y!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+M;)R&y!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 8 " }{TEXT 305 124 ".. estimating the error by \+ doubling the number of intervals used instead of calculating an accura te value by another method " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 4 "Note" }{TEXT -1 15 ": Suppose th at " }{XPPEDIT 18 0 "I[1]" "6#&%\"IG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "I[2]" "6#&%\"IG6#\"\"#" }{TEXT -1 53 " are trapezoidal \+ rule approximations for an integral " }{XPPEDIT 18 0 "I=Int(f(x),x=a.. b)" "6#/%\"IG-%$IntG6$-%\"fG6#%\"xG/F+;%\"aG%\"bG" }{TEXT -1 16 " obta ined using " }{TEXT 313 1 "n" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2*n " "6#*&\"\"#\"\"\"%\"nGF%" }{TEXT -1 29 " intervals respectively. Let \+ " }{XPPEDIT 18 0 "e[1]" "6#&%\"eG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "e[2]" "6#&%\"eG6#\"\"#" }{TEXT -1 48 " be the correspon ding absolute errors, that is, " }{XPPEDIT 18 0 "e[1]=abs(I[1]-I)" "6# /&%\"eG6#\"\"\"-%$absG6#,&&%\"IG6#F'F'F-!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "e[2]=abs(I[2]-I)" "6#/&%\"eG6#\"\"#-%$absG6#,&&%\"IG6#F '\"\"\"F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 76 "An ear lier example suggests that the error is approximately proportional to \+ " }{XPPEDIT 18 0 "h^2" "6#*$%\"hG\"\"#" }{TEXT -1 18 ". This means tha t " }{XPPEDIT 18 0 "e[1]" "6#&%\"eG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 317 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*e[2]" "6#*&\"\"%\"\"\"&%\" eG6#\"\"#F%" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "I[1]" "6#&%\"IG6#\" \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "I[2]" "6#&%\"IG6#\"\"#" } {TEXT -1 25 " are on the same side of " }{TEXT 315 1 "I" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "I[1]F-F6FA-F$6&7$F*7$F+F>F-F6FA-F$6%7$F'7$F($!\"\"F)-F.6&F0F)F)F)-%* LINESTYLEGF8-F$6%7$FH7$FI$!3++++++++]F@FVFX-F$6%7$F*7$F+FTFVFX-F$6'7$7 $$\"$v#!\"#$!\"$FU7$F(Feo7%7$$\"*+++5\"!\"*$!++++]O!#5Fgo7$Fjo$!++++]B F_p-FB6#%,PATCHNOGRIDG-F.6&F0$\")#)eqkF3$\"))eqk\"F3Fjp-FY6#\"\"$-F$6' 7$7$$\"$D$FdoFeo7$$\"$+'FdoFeo7%7$$\"++++!*eF\\pFapFeq7$FjqF]pFcpFfpF \\q-F$6'7$7$$\"#oFUFeo7$$\"#gFUFeo7%7$$\"++++'4'F\\pF]pFcr7$FhrFapFcpF fpF\\q-F$6'7$7$$\"#sFUFeo7$$\"#!)FUFeo7%7$$\"++++/zF\\pFapFas7$FfsF]pF cpFfpF\\q-F$6'7$7$$\"#QFU$F3FU7$F(F_t7%7$$\"*+++9\"F\\p$!++++]')F_pF`t 7$Fct$!++++]tF_pFcpFfpF\\q-F$6'7$7$$\"#UFUF_t7$FbsF_t7%7$$\"++++')yF\\ pFhtF`u7$FcuFetFcpFfpF\\q-%%TEXTG6%7$F($F,FUQ\"I6\"-%%FONTG6%%&TIMESG% &ROMANG\"#7-Fgu6%7$FIFjuF[vF]v-Fgu6%7$F+FjuF[vF]v-Fgu6%7$$F^qF)FeoQ$3~ eF\\v-F^v6$%*HELVETICAG\"#5-Fgu6%7$$\"\"(F)FeoQ\"eF\\vF^w-Fgu6%7$$\"\" %F)$!#wFdoFgwF^w-Fgu6%7$$\"#TFU$F\\pFUQ\"1F\\v-F^v6$F`wF,-Fgu6%7$$F,Fd o$FfwFUFexFfx-Fgu6%7$$\"$3'FdoF\\yQ\"2F\\vFfx-Fgu6%7$$\"$=$Fdo$!\"%FUF byFfx-Fgu6%7$$\"#rFUFhyFbyFfx-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F \\vFfz-F^v6#%(DEFAULTG-%%VIEWG6$;F(F+;FT$\"\"\"F)" 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" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23 " "Curve 24" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "It foll ows that " }{XPPEDIT 18 0 "abs(I[1]-I[2])" "6#-%$absG6#,&&%\"IG6#\"\" \"F*&F(6#\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 316 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3*e[2]" "6#*&\"\"$\"\"\"&%\"eG6#\"\"#F%" }{TEXT -1 21 ", which implies that " }{XPPEDIT 18 0 "e[2]" "6#&%\"eG6#\"\"#" } {TEXT -1 1 " " }{TEXT 314 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(I[ 1]-I[2])/3" "6#*&-%$absG6#,&&%\"IG6#\"\"\"F+&F)6#\"\"#!\"\"F+\"\"$F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We find approximations for " }{XPPEDIT 18 0 "Int(1/(1+ln( x+1)),x = 0 .. 1);" "6#-%$IntG6$*&\"\"\"F',&F'F'-%#lnG6#,&%\"xGF'F'F'F '!\"\"/F-;\"\"!F'" }{TEXT -1 53 " using the trapezoidal rule with 32 a nd 64 intervals." }}{PARA 0 "" 0 "" {TEXT -1 69 "The picture illustrat es using the trapezoidal rule with 32 intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "drawtrap(1/ (1+ln(x+1)),x=[0..1,-.1..1.2],intervals=32,\n scheme=stripes,co lor=magenta,shading=[plum,COLOR(RGB,.7,1,.7)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 596 351 351 {PLOTDATA 2 "6E-%'CURVESG6%7S7$$!3/+++++++5!#=$ \"3g=A/&oox6\"!#<7$$!3uLLLe*pj;(!#>$\"39<&>*H]L!3\"F-7$$!3!pm;/hV3q%F1 $\"3adD#*[[e]5F-7$$!3hLL$3P,\"G>F1$\"3%>/%f(*e&)>5F-7$$\"3bsmmTS?I')!# ?$\"3\"o=_$o.![\"**F*7$$\"3PJLekk(3k$F1$\"3O)fG@6JZl*F*7$$\"3vlm\"Hi/j @'F1$\"3!**[\"HloAJ%*F*7$$\"3O***\\iI6I)))F1$\"3ZCN]kzq:#*F*7$$\"3um;H b03!*F*7$$\"32+]7Q!***Q9F*$\"3X\\OZRb*[\"))F*7$$\"3cLL $3RuF*$\"37Or'>cu`Z)F*7$$\"3-++ ]AaB^AF*$\"31UEe$*QE7$)F*7$$\"3.++](3zF`#F*$\"3aKJ55w=e\")F*7$$\"3Z++] x&)4/GF*$\"3z&3bAM(3=!)F*7$$\"3Gm;HnE[]IF*$\"3%H:A_&oR(*yF*7$$\"3aLL$3 =dMM$F*$\"37YPV3tJhxF*7$$\"3+LLLB]k\"f$F*$\"3aK3My%p=l(F*7$$\"3I+]i&[Y .)QF*$\"3W*Ra(G4vIvF*7$$\"36LLL)>'*e8%F*$\"3%p^-gL!pGuF*7$$\"3&4+Dctui T%F*$\"3Y#oktui=K(F*7$$\"3;,](oShKo%F*$\"3#))zx91\"zCsF*7$$\"35MLe*)R$ ='\\F*$\"3kU)z@2.!GrF*7$$\"30M$e9e]w@&F*$\"3@#z80\"R*G/(F*7$$\"3UnmTNL e$\\&F*$\"3m#4mlcs[&pF*7$$\"3Qn;H<**>!y&F*$\"3Upd,ZeKnoF*7$$\"3*)**\\P %\\+(HgF*$\"3a\"H?!*zKTz'F*7$$\"3unm\"H&z;*H'F*$\"3N\\fH<>/=nF*7$$\"3A +++0UbxlF*$\"3D)yb\"=!oCk'F*7$$\"3k++DT3!*\\oF*$\"3K5lRO)R8d'F*7$$\"34 +]i:-T8rF*$\"3A#p,)>t+0lF*7$$\"3i++v[C*fS(F*$\"3#eNy)[k0MkF*7$$\"3ILLL )f!*)owF*$\"3uKCq+6fsjF*7$$\"3?,+]([!f\\zF*$\"3rD6+*3>#4jF*7$$\"3Im;H2 j%R?)F*$\"3S*3>j?+PD'F*7$$\"3M,+]-W-#[)F*$\"3M8v'F*7$$\"3uK$e*= UnV()F*$\"3+`DAhH[ThF*7$$\"3V,]7oO<7 M)*F*$\"3'oxP,vc`$fF*7$$\"3rm;/GT)4,\"F-$\"3'eMxvG=*z&F*7$$\"3AL$3 U(3D#4\"F-$\"39;KX')R%Hv&F*7$$\"3cmmm4u+=6F-$\"3d0oY\"oKFr&F*7$$\"3)** \\i&[#pa9\"F-$\"3k`%e[B(*4n&F*7$$\"3-+vVKPvr6F-$\"3$pe>A@0@j&F*7$$\"3% **************>\"F-$\"3QMLy64T\"f&F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$ \"\"!Fb[lF^[l-%*THICKNESSG6#\"\"#-%)POLYGONSG6$7&7$Fa[lFa[l7$$\"++++DJ !#6Fa[l7$F]\\l$\"+$op9q*!#57$Fa[l$\"\"\"Fb[l-F[[l6&F][l$\")1Zw\"*F`[l$ \")PJ%y'F`[lFi\\l-Fh[l6$7&F\\\\l7$$\"++++]iF_\\lFa[l7$Fa]l$\"+CkSG%*Fc \\lF`\\l-%&COLORG6&F][l$\"\"(!\"\"Ff\\lFi]l-Fh[l6$7&F`]l7$$\"++++v$*F_ \\lFa[l7$F`^l$\"+pudx\"*Fc\\lFc]lFg\\l-Fh[l6$7&F_^l7$$\"++++]7Fc\\lFa[ l7$Fi^l$\"+$)*zi%*)Fc\\lFb^lFf]l-Fh[l6$7&Fh^l7$$\"+++]i:Fc\\lFa[l7$Fb_ l$\"+sjBK()Fc\\lF[_lFg\\l-Fh[l6$7&Fa_l7$$\"++++v=Fc\\lFa[l7$F[`l$\"+5N ^L&)Fc\\lFd_lFf]l-Fh[l6$7&Fj_l7$$\"+++](=#Fc\\lFa[l7$Fd`l$\"+[(f%[$)Fc \\lF]`lFg\\l-Fh[l6$7&Fc`l7$$\"+++++DFc\\lFa[l7$F]al$\"+b_lv\")Fc\\lFf` lFf]l-Fh[l6$7&F\\al7$$\"+++]7GFc\\lFa[l7$Ffal$\"+lD(Q,)Fc\\lF_alFg\\l- Fh[l6$7&Feal7$$F^\\lFc\\lFa[l7$F_bl$\"+7\\/iyFc\\lFhalFf]l-Fh[l6$7&F^b l7$$\"+++]PMFc\\lFa[l7$Fgbl$\"+R1C>xFc\\lF`blFg\\l-Fh[l6$7&Ffbl7$$\"++ ++]PFc\\lFa[l7$F`cl$\"+YBk%e(Fc\\lFiblFf]l-Fh[l6$7&F_cl7$$\"+++]iSFc\\ lFa[l7$Ficl$\"+8)HvX(Fc\\lFbclFg\\l-Fh[l6$7&Fhcl7$$\"++++vVFc\\lFa[l7$ Fbdl$\"+'*eEPtFc\\lF[dlFf]l-Fh[l6$7&Fadl7$$\"+++](o%Fc\\lFa[l7$F[el$\" +8[GBsFc\\lFddlFg\\l-Fh[l6$7&Fjdl7$$\"+++++]Fc\\lFa[l7$Fdel$\"+iB3:rFc \\lF]elFf]l-Fh[l6$7&Fcel7$$\"+++]7`Fc\\lFa[l7$F]fl$\"+gFc\\lFj\\mFf]l-Fh[l6$7&F`]m7$$\"+ ++](o*Fc\\lFa[l7$Fi]m$\"+H9hhfFc\\lFb]mFg\\l-Fh[l6$7&Fh]m7$$\"+++++5! \"*Fa[l7$Fb^m$\"+!4hh!fFc\\lF[^mFf]l-%+AXESLABELSG6$Q\"x6\"Q!F\\_m-%%V IEWG6$;$F[^lF[^l$\"#7F[^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv e 31" "Curve 32" "Curve 33" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "trap32 := trap(1/(1+ln(x+1) ),x=0..1,intervals=32);\ntrap64 := trap(1/(1+ln(x+1)),x=0..1,intervals =64);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trap32G$\"+z(yAP(!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trap64G$\"+^]xrt!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Next we calculate th e difference " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 47 " bet ween these two trapezoidal rule estimates. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "delta := abs(trap3 2-trap64);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG$\"'GP]!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Using th e note above, this gives an approximate upper bound for the error in t he 64 interval estimate of " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$! \"\"" }{TEXT -1 18 " ( 0.0000503728 ) " }{TEXT 308 1 "~" }{TEXT -1 15 " 0.0000167909. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "abserr64_estimate := evalf[6](delta/3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%2abserr64_estimateG$\"'4z;!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "A value w hich is accurate to 10 digits can be obtained using Maple's numerical \+ integration via " }{TEXT 0 9 "evalf/Int" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 116 "This value can be used to calculate the actual er ror in the 64 interval trapezoidal rule estimate for the integral. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Int(1/(1+ln(x+1)),x=0..1);\narea := evalf[14](evalf(%));\nabserr 64 := abs(trap64-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*& \"\"\"F',&F'F'-%#lnG6#,&%\"xGF'F'F'F'!\"\"/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG$\"/oB'42;P(!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)abserr64G$\"'bz;!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The absolute error in the approximate value for the inte gral obtained using 64 intervals is about 0.0000167955. " }}{PARA 0 " " 0 "" {TEXT -1 209 "The error estimate calculated previously, without reference to the accurate value, agrees well with the actual error. N ote that both trapezoidal rule estimates are greater than the true val ue of the integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 225 "Either the error estimate, or the actual error in the \+ 64 interval trapezoidal estimate could be used to predict the approxim ate number of intervals needed to obtain a trapezoidal rule estimate w hich is correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 30 "If we \+ aim for a tolerance of " }{XPPEDIT 18 0 "epsilon = 2*`.`*10^(-11);" " 6#/%(epsilonG*(\"\"#\"\"\"%\".GF')\"#5,$\"#6!\"\"F'" }{TEXT -1 95 ", t his is unlikely to affect the last decimal place, since a unit in the \+ last place (1 ulp) is " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eps := 2e-11;\nabserr64/eps;\nevalf[6](log[4](%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"#!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++v(R)!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"'#)R)*!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "log[4]( 839775 )" "6#-&%$logG6#\"\"%6#\"'v(R)" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 27 " 9.83982 means that 839775 " } {TEXT 306 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4^9.83982" "6#)\"\"%-% &FloatG6$\"'#)R)*!\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 89 "The desired tolerance is reached when the absolute error is divided by 4 about 10 times. " }}{PARA 0 "" 0 " " {TEXT -1 70 "Thus we should double the number of intervals 10 times \+ to reach about " }{XPPEDIT 18 0 "64*`.`*2^10 = 2^16;" "6#/*(\"#k\"\"\" %\".GF&\"\"#\"#5*$F(\"#;" }{XPPEDIT 18 0 "``=65536" "6#/%!G\"&Ob'" } {TEXT -1 12 " intervals. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 53 "This process can be automated by using the procedu re " }{TEXT 0 4 "trap" }{TEXT -1 18 " with the option \"" }{TEXT 281 12 "iterate=true" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 73 "The end result is a value for the integral which is correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "trap(1/(1+ln(x+1)),x=0..1,iterate=true,info=true);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fapproximation~with~1~interval~--->~ ~~G$\"0A[da!3`z!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~ with~2~intervals~--->~~~G$\"0N![`93Mv!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~~~G$\"0ve[V8RT(!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~--->~ ~~G$\"0hDTT6BQ(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~ with~16~intervals~--->~~~G$\"0QN(o;Hut!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$\"0vy#z(yAP( !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~interva ls~--->~~~G$\"0\"**)30v~~~G$\"0q?q3\\;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~intervals~--->~~~G$\"0NlS f<;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~512~i ntervals~--->~~~G$\"0B-2s4;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%J approximation~with~1024~intervals~--->~~~G$\"0n`Bv2;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~2048~intervals~--->~~~G$\" 0-m-E2;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4 096~intervals~--->~~~G$\"04Ws82;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~8192~intervals~--->~~~G$\"0h)[1rgrt!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~16384~intervals~- -->~~~G$\"0u*z)42;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproxima tion~with~32768~intervals~--->~~~G$\"0`xo42;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~65536~intervals~--->~~~G$\"0-(R'42 ;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~131072~ intervals~--->~~~G$\"0(oF'42;P(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+'42;P(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 9 " }{TEXT 288 53 ".. an example where the integrand has negative v alues" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "We find an appro ximation for " }{XPPEDIT 18 0 "Int(sin(x),x = 0 .. 3*Pi/2);" "6#-%$Int G6$-%$sinG6#%\"xG/F);\"\"!*(\"\"$\"\"\"%#PiGF/\"\"#!\"\"" }{TEXT -1 46 " using the trapezoidal rule with 24 intervals." }}{PARA 0 "" 0 "" {TEXT -1 40 "In this example the associated graph of " }{XPPEDIT 18 0 "y=sin(x)" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 29 " is both above and b elow the " }{TEXT 287 1 "x" }{TEXT -1 60 " axis in different sections \+ of the interval of integration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "drawtrap(sin(x),x=[0..3*Pi/2 ,0..2*Pi],intervals=24);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 237 237 {PLOTDATA 2 "6'-%'CURVESG6%7en7$$\"\"!F)F(7$$\"3#*4@;)eb&p8!#=$\"3Ikwb #=y_O\"F-7$$\"3!\\-[Ce)>hDF-$\"3%[#\\ff*)GLDF-7$$\"3a=FdfdK,RF-$\"3gLj &[K5J!QF-7$$\"3!)4d`phL]_F-$\"3j9NXR4U7]F-7$$\"3e)*4]o\\$Hf'F-$\"3t_H- qceDhF-7$$\"3\"*f4v>fpPyF-$\"3!GO#*3qU&fqF-7$$\"3fX@S&ytl7*F-$\"3!\\@@ &\\!>8\"zF-7$$\"3MWp(Qh-a')F-7$$\"3Kc[Fj<#)y6FQ$\"3 aK+16bcT#*F-7$$\"3%R?WsH$\\:8FQ$\"3s&Q\"[M\"oen*F-7$$\"3/Q-+yMov8FQ$\" 3>-'H)pzD5)*F-7$$\"3:siveO(eV\"FQ$\"33)fVPO<\"4**F-7$$\"3RF?GYNj.:FQ$ \"37P(*[\"Gau(**F-7$$\"3i#y2QV$Rr:FQ$\"2%HhJ<#)******FQ7$$\"3ta>%*\\:V R;FQ$\"3c%4ddFbk(**F-7$$\"3gEh2m'puq\"FQ$\"3ap%>wGZn!**F-7$$\"3IWx7)*p .t=\\D`V'*F-7$$\"3'[V5\"o rod>FQ$\"3IzF#e`m3E*F-7$$\"3Uh\"\\.=)G*4#FQ$\"3vU'yI2&oN')F-7$$\"3z;2b 6GC>AFQ$\"3!G;)RQ+BqzF-7$$\"3#*[1Mk&y(eBFQ$\"3AfvOg?x_qF-7$$\"3'z:$)*[ 8H#[#FQ$\"3u3HvXQF-7$$\"3%pQg,@&[\")GFQ$\"3o$p9m#Q%=d#F-7$$\"3[ \"GhL)p70IFQ$\"31V?00]Ug8F-7$$\"3L===R8\\QJFQ$\"3vCA\\O%485$!#?7$$\"3) e)*4NU>qF$FQ$!3q#4*>c=8]8F-7$$\"39\\f9G&3wR$FQ$!3UNbFGIGKDF-7$$\"3sk#* zk![y_$FQ$!3q4&=j`Bsw$F-7$$\"3cq*4,D)RiOFQ$!3kENX')3zv\\F-7$$\"3Es$QlN HSz$FQ$!37W(H!pUCrgF-7$$\"3nRKN?#*Q@RFQ$!3Y?#o?\"yMJqF-7$$\"3#HId[j+G1 %FQ$!3)4j2yeGL'zF-7$$\"3%y%fJ!3l)*=%FQ$!3I_Mh6Mil')F-7$$\"3)*))e?(\\Lb K%FQ$!3#\\IN,%***4E*F-7$$\"3]!o*>Z#p%[WFQ$!3=V>(fp[Pl*F-7$$\"3\\mYOq)p c^%FQ$!3C8'y09Jr!)*F-7$$\"3[_'HN\\qGe%FQ$!3;$RoI*>C;**F-7$$\"35z3'G/,h k%FQ$!3S!eH0fP!y**F-7$$\"3t0@>#fJ$4ZFQ$!3W1O#>E`*****F-7$$\"3%)yyHNdUv ZFQ$!3w$HVeTQ,)**F-7$$\"3%>l.%y)>:%[FQ$!3lYZ3X=u;**F-7$$\"3i5!*>%)*Qh! \\FQ$!3POSx/6*G\")*F-7$$\"3IpV***3e2(\\FQ$!3U$[o%H'z!o'*F-7$$\"3XudN?& eg5&FQ$!36.2<#>x]B*F-7$$\"3],9j:*ojB&FQ$!3wHgb5wMe')F-7$$\"3)p9[f6I'p` FQ$!3/a*[=H2o\"zF-7$$\"3jTTCYyy,bFQ$!3)4d)eg?sUqF-7$$\"3@c_,vqABcFQ$!3 D1$H)zD(p_v\\F-7$$\"3C8W>Cr*o)eFQ$!3i&> 2ndo*fQF-7$$\"3'\\Qk&=ii>gFQ$!3%)f]nRQ=0EF-7$$\"3azss(*[mYhFQ$!3/up91t '4O\"F-7$$\"3?+++3`=$G'FQ$\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"#5!\"\" F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG637&F'7$$\"+4a\\j>!#5F(7$F`^l$\"+@ K!4&>Fb^lF'7&F_^l7$$\"+=3*p#RFb^lF(7$Fh^l$\"+DV$o#QFb^lFc^l7&Fg^l7$$\" +Fi[!*eFb^lF(7$F__l$\"+JBqbbFb^lFj^l7&F^_l7$$\"+O;)R&yFb^lF(7$Ff_l$\"+ 8y1rqFb^lFa_l7&Fe_l7$$\"+XqZ<)*Fb^lF(7$F]`l$\"+Chp9$)Fb^lFh_l7&F\\`l7$ $\"+Xs4y6!\"*F(7$Fd`l$\"+D`zQ#*Fb^lF_`l7&Fc`l7$$\"+'yYWP\"Ff`lF(7$F\\a l$\"+/G&y!)*Fb^lFg`l7&F[al7$$\"+Fjzq:Ff`lF(7$Fcal$\"\"\"F)F^al7&Fbal7$ $\"+oe9nhp9$)Fb^lFbbl7&Ffbl7 $$\"+\"\\%>cBFf`lF(7$F^cl$\"+1y1rqFb^lFibl7&F]cl7$$\"+KSa_DFf`lF(7$Fec l$\"+ABqbbFb^lF`cl7&Fdcl7$$\"+tN*)[FFf`lF(7$F\\dl$\"+8V$o#QFb^lFgcl7&F [dl7$$\"+9JCXHFf`lF(7$Fcdl$\"+3K!4&>Fb^lF^dl7%FbdlFedl7$$\"+aEfTJFf`lF (-%&COLORG6&Fc]l$\"#$)!\"#F_elF_el-F\\^l6,7%Fidl7$Fjdl$!+:w1-T!#>Fidl7 &Fidl7$$\"+&>UzL$Ff`lF(7$F[fl$!+EK!4&>Fb^lFeel7&Fjel7$$\"+Oy1rqFb^lF[gl7&F_gl7$$\"+f.MBTFf`lF(7$Ffgl$!+ Hhp9$)Fb^lFagl7&Fegl7$$\"++**o>VFf`lF(7$F]hl$!+I`zQ#*Fb^lFhgl7&F\\hl7$ $\"+T%Rg^%Ff`lF(7$Fdhl$!+2G&y!)*Fb^lF_hl7&Fchl7$$\"+#)*)Q7ZFf`lF(7$F[i l$Ff]lF)Ffhl-F]el6&Fc]l$\"#tFaelFail$\"#()Fael-%+AXESLABELSG6$Q\"x6\"Q !Fiil-%%VIEWG6$;F($\"+3`=$G'Ff`l%(DEFAULTG" 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 62 "student[trapezoid](sin(x),x=0..3*Pi/2,24);\ntrap24 := evalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#KF&*&%#PiGF&,&F&!\"\"*& \"\"#F&-%$SumG6$-%$sinG6#,$*(\"#;F+%\"iGF&F)F&F&/F7;F&\"#BF&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'trap24G$\"+A<&y'**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The exact value for this integral is 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(sin(x),x=0..3*Pi/2);\narea := v alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#%\"xG/F) ;\"\"!,$*(\"\"$\"\"\"\"\"#!\"\"%#PiGF0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "trap" }{TEXT -1 35 " gives this value with the option \"" }{TEXT 281 12 "iterate=true" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "trap(sin(x),x=0..3*Pi/2,iterate=true,info=true);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fapproximation~with~1~interval~--->~ ~~G$!0M#>!\\%>cB!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation ~with~2~intervals~--->~~~G$\"0AKrcQ)z[!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~~~G$\"0Rc*Hct:))! #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals ~--->~~~G$\"0iB1Ol\"4(*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happrox imation~with~16~intervals~--->~~~G$\"03eM83w#**!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$\"0yM%z;#> )**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~inte rvals~--->~~~G$\"03g`k\"[&***!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%I approximation~with~128~intervals~--->~~~G$\"0'o*y[q))***!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~intervals~--->~~~G$ \"0[eni<(****!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~wi th~512~intervals~--->~~~G$\"0k))pSH*****!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~1024~intervals~--->~~~G$\"0%fw^B)* ****!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~2048~i ntervals~--->~~~G$\"0qUze&******!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%Japproximation~with~4096~intervals~--->~~~G$\"0z&)p*))******!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~8192~intervals~-- ->~~~G$\"0bYUs*******!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproxim ation~with~16384~intervals~--->~~~G$\"0sh5$********!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~with~32768~intervals~--->~~~G$\"0 \\lF)********!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximation~wi th~65536~intervals~--->~~~G$\"0b\"p&*********!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~131072~intervals~--->~~~G$\"0+B*)* ********!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~26 2144~intervals~--->~~~G$\"0tI(**********!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~524288~intervals~--->~~~G$\"07L*** ********!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) Find approximate \+ values for " }{XPPEDIT 18 0 "Int(1/(x+exp(x)),x = 0 .. 1);" "6#-%$IntG 6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"/F);\"\"!F'" }{TEXT -1 53 " usi ng the trapezoidal rule with 32 and 64 intervals." }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Illustrate the approximation from (a) with 32 interva ls graphically. " }}{PARA 0 "" 0 "" {TEXT -1 72 "(c) Calculate the def inite integral in (a) by using the Maple procedure " }{TEXT 0 14 "eval f(Int(..))" }{TEXT -1 94 ", and use this value to calculate the absolu te error in each of the values found in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 188 "(d) Find the ratio of the absolute errors calculated in \+ (c) to verify experimentally that the trapezoidal rule has order 1, th at is, the error is approximately proportional to the square, " } {XPPEDIT 18 0 "h^2" "6#*$%\"hG\"\"#" }{TEXT -1 17 ", of the spacing " }{TEXT 268 1 "h" }{TEXT -1 13 " between the " }{TEXT 278 1 "x" }{TEXT -1 18 " values used (the " }{TEXT 265 9 "step-size" }{TEXT -1 2 ")." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" } {TEXT -1 112 ": The trapezoidal rule is designed to give exact answers for polynomials of degree 1, that is, linear functions." }}{PARA 0 " " 0 "" {TEXT -1 41 "_________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) Find approximate values for " } {XPPEDIT 18 0 "Int(x^2*sqrt(arctan(x)),x = 0 .. 1);" "6#-%$IntG6$*&%\" xG\"\"#-%%sqrtG6#-%'arctanG6#F'\"\"\"/F';\"\"!F/" }{TEXT -1 46 " using the trapezoidal rule with 32 intervals." }}{PARA 0 "" 0 "" {TEXT -1 55 "(b) Illustrate the approximation from (a) graphically. " }}{PARA 0 "" 0 "" {TEXT -1 72 "(c) Calculate the definite integral in (a) by u sing the Maple procedure " }{TEXT 0 14 "evalf(Int(..))" }{TEXT -1 86 " , and use this value to calculate the absolute error in the value foun d in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 34 "(d) Find an approxima te value for " }{XPPEDIT 18 0 "Int(x^2*sqrt(arctan(x)),x = 0 .. 1);" " 6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#-%'arctanG6#F'\"\"\"/F';\"\"!F/" } {TEXT -1 71 " which is correct to 10 digits using the trapezoidal rule in two ways. " }}{PARA 15 "" 0 "" {TEXT -1 257 "(i) Calculate a numbe r of approximations to the definite integral using the trapezoidal rul e, with a progressively larger number of intervals, until there is no \+ change in the 10 digits of the result. It is convenient to double the \+ intervals with each step. " }}{PARA 15 "" 0 "" {TEXT -1 23 "(ii) Use t he procedure " }{TEXT 0 4 "trap" }{TEXT -1 18 " with the option \"" } {TEXT 281 12 "iterate=true" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "__ _______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 32 "(a) Find approximate values for " } {XPPEDIT 18 0 "Int(ln(x^(2/3)+1),x = 0 .. 1);" "6#-%$IntG6$-%#lnG6#,&) %\"xG*&\"\"#\"\"\"\"\"$!\"\"F.F.F./F+;\"\"!F." }{TEXT -1 46 " using th e trapezoidal rule with 32 intervals." }}{PARA 0 "" 0 "" {TEXT -1 55 " (b) Illustrate the approximation from (a) graphically. " }}{PARA 0 "" 0 "" {TEXT -1 72 "(c) Calculate the definite integral in (a) by using \+ the Maple procedure " }{TEXT 0 14 "evalf(Int(..))" }{TEXT -1 86 ", and use this value to calculate the absolute error in the value found in \+ part (a). " }}{PARA 0 "" 0 "" {TEXT -1 34 "(d) Find an approximate va lue for " }{XPPEDIT 18 0 "Int(ln(x^(2/3)+1),x = 0 .. 1);" "6#-%$IntG6$ -%#lnG6#,&)%\"xG*&\"\"#\"\"\"\"\"$!\"\"F.F.F./F+;\"\"!F." }{TEXT -1 71 " which is correct to 10 digits using the trapezoidal rule in two w ays. " }}{PARA 15 "" 0 "" {TEXT -1 258 "(i) Calculate a number of appr oximations for the definite integral using the trapezoidal rule, with \+ a progressively larger number of intervals, until there is no change i n the 10 digits of the result. It is convenient to double the interval s with each step. " }}{PARA 15 "" 0 "" {TEXT -1 23 "(ii) Use the proce dure " }{TEXT 0 4 "trap" }{TEXT -1 18 " with the option \"" }{TEXT 281 12 "iterate=true" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(d) Could you have predicted roughly how many intervals would be n eeded to obtain the value for " }{XPPEDIT 18 0 "Int(ln(x^(2/3)+1),x = \+ 0 .. 1);" "6#-%$IntG6$-%#lnG6#,&)%\"xG*&\"\"#\"\"\"\"\"$!\"\"F.F.F./F+ ;\"\"!F." }{TEXT -1 27 " obtained in (d)? Explain. " }}{PARA 0 "" 0 " " {TEXT -1 41 "_________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "__ _______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 31 " Find an approximate value for " } {XPPEDIT 18 0 "Int(sin(x)/x,x = 0 .. 1);" "6#-%$IntG6$*&-%$sinG6#%\"xG \"\"\"F*!\"\"/F*;\"\"!F+" }{TEXT -1 49 " correct to 10 digits using th e trapezoidal rule." }}{PARA 0 "" 0 "" {TEXT 265 5 "Hints" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 43 "Replace the lower limit of the \+ integral by " }{XPPEDIT 18 0 "10^(-13);" "6#)\"#5,$\"#8!\"\"" }{TEXT -1 132 " to avoid division by zero. This value will not affect the 1st 10 digits of the calculated approximations to the definite integral. \+ " }}{PARA 15 "" 0 "" {TEXT -1 183 "Calculate a number of approximation s to the definite integral using the trapezoidal rule, with a progress ively increasing number of intervals, until the first 10 digits stop c hanging." }}{PARA 0 "" 0 "" {TEXT -1 41 "_____________________________ ____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 41 "_______________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) Find approx imate values for " }{XPPEDIT 18 0 "Int(x*sqrt(1-x^2),x = 0 .. 1/2);" " 6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F(/F';\"\"!*&F(F (F.F/" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int(x*sqrt(1-x^2),x = 0 .. \+ 1);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F(/F';\"\" !F(" }{TEXT -1 46 " using the trapezoidal rule with 32 intervals." }} {PARA 0 "" 0 "" {TEXT -1 46 "(b) Illustrate the approximation from (a) for " }{XPPEDIT 18 0 "Int(x*sqrt(1-x^2),x = 0 .. 1)" "6#-%$IntG6$*&% \"xG\"\"\"-%%sqrtG6#,&F(F(*$F'\"\"#!\"\"F(/F';\"\"!F(" }{TEXT -1 32 " \+ with 32 intervals graphically. " }}{PARA 0 "" 0 "" {TEXT -1 186 "(c) C alculate the exact analytical values of the definite integrals in (a), and use these values to calculate the absolute and relative errors in the approximate values obtained in (a). " }}{PARA 0 "" 0 "" {TEXT -1 193 "(d) Why is the relative error for the trapezoidal rule approximat ion for the second integral in (a) worse than the relative error for t he trapezoidal rule approximation for the first integral? " }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for trapezoid p ictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1000 "p1 := plot(-x^2/4+x/4+4,x=-1.5..2.5,color=red,thick ness=2):\np2 := plot([[[-1,0],[-1,3.5],[0,4]],[[0,0],[0,4],[1,4],[1,0] ],\n [[1,4],[2,3.5],[2,0]],[[-1.5,0],[3,0]]],color=black):\np3 := pl ots[polygonplot]([[-1,0],[-1,3.5],[0,4],[1,4],[2,3.5],[2,0]],\n \+ style=patchnogrid,color=COLOR(RGB,.8,.8,1)):\np4 := plot([[[-1,3.5 ],[0,4],[1,4],[2,3.5]]$3],style=point,\n symbol=[circle,d iamond,cross],color=black):\nt1 := plots[textplot]([[2.5,-0.1,`x`],\n \+ [-1.2,3.7,`(x ,y )`],[-0.25,4.25,`(x ,y )`],\n [1.2,4.25,`(x ,y )`],[2.2,3.8,`(x ,y )`],\n [-1,-0.13,`x`],[0,-0.13,`x`],\n [1,-0 .13,`x`],[2,-0.13,`x`]],font=[HELVETICA,10]):\nt2 := plots[textplot]([ [-1.16,3.63,`0 0`],[-0.22,4.18,`1 1`],\n [1.23,4.18,`2 2`],[ 2.23,3.73,`3 3`],\n [-.93,-0.2,`0`],[0.07,-0.2,`1`],\n [1.07,-0.2 ,`2`],[2.07,-0.2,`3`]],font=[HELVETICA,8]):\nt3 := plots[textplot]([2. 5,2.9,`y = f(x)`],color=red,font=[HELVETICA,10]):\nplots[display]([p1, p2,p3,p4,t1,t2,t3],view=[-1.5..2.5,-.2..4.6],axes=none);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1108 "p1 \+ := plot(-x^2/4+x/4+4,x=-1.5..0.5,color=COLOR(RGB,1,.5,.5),thickness=1) :\np2 := plot([[[-1,0],[-1,3.5],[0,4],[0,0]],[[-1.5,0],[.5,0]]],color= black):\np3 := plot([[-1,3.5],[-1,3.75],[0,3.75]],color=blue):\np4 := \+ plots[polygonplot]([[-1,0],[-1,3.5],[-.5,3.75],[0,3.75],[0,0]],\n \+ style=patchnogrid,color=COLOR(RGB,.9,.9,1)):\np5 := plots[polygo nplot]([[-1,3.5],[-1,3.75],[0,3.75],[0,4]],\n style=patchnog rid,color=COLOR(RGB,.7,.7,1)):\np6 := plot([[[-1,3.5],[0,4]]$3],style= point,\n symbol=[circle,diamond,cross],color=black):\np7 \+ := plottools[arrow]([-.4,2.2],[-.4,3.75],0,.06,.06,arrow):\np8 := plot tools[arrow]([-.4,1.55],[-.4,0],0,.06,.06,arrow):\nt1 := plots[textplo t]([[.5,-0.1,`x`],\n [-1.25,3.7,`(x ,y )`],[-0.05,4.25,`(x ,y )`] ,\n [-1,-0.13,`x`],[0,-0.13,`x`],[-.4,2.05,`y + y`],\n [-.38,1.98,` _____`],[-.38,1.75,`2`]],font=[HELVETICA,10]):\nt2 := plots[textplot]( [[-1.21,3.63,`0 0`],[-0.02,4.18,`1 1`],\n [-.93,-0.2,`0`],[0.07 ,-0.2,`1`],[-.35,1.98,`0 1`]],font=[HELVETICA,8]):\nplots[displa y]([p1,p2,p3,p4,p5,p6,p7,p8,t1,t2],view=[-1.5..0.5,-.2..4.5],axes=none );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for t rapezoidal error picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1119 "p1 := plot([[0,0],[8,0]],thicknes s=2,color=navy):\np2 := plot([[[0,0],[0,.5]],[[6,0],[6,.5]],[[8,0],[8, .5]]],\n style=line,color=navy,thickness=2):\np 3 := plot([[[0,0],[0,-1]],[[6,0],[6,-.5]],[[8,0],[8,-1]]],color=black, linestyle=2):\np4 := plottools[arrow]([2.75,-.3],[0,-.3],0,.13,.04,arr ow,color=brown,linestyle=3):\np5 := plottools[arrow]([3.25,-.3],[6,-.3 ],0,.13,.04,arrow,color=brown,linestyle=3):\np6 := plottools[arrow]([6 .8,-.3],[6,-.3],0,.13,.12,arrow,color=brown,linestyle=3):\np7 := plott ools[arrow]([7.2,-.3],[8,-.3],0,.13,.12,arrow,color=brown,linestyle=3) :\np8 := plottools[arrow]([3.8,-.8],[0,-.8],0,.13,.03,arrow,color=brow n,linestyle=3):\np9 := plottools[arrow]([4.2,-.8],[8,-.8],0,.13,.03,ar row,color=brown,linestyle=3):\nt1 := plots[textplot]([[0,.8,`I`],[6,.8 ,`I`],[8,.8,`I`]],font=[TIMES,ROMAN,12]):\nt2 := plots[textplot]([[3,- .3,`3 e`],[7,-.3,`e`],[4,-.76,`e`]],font=[HELVETICA,10]):\nt3 := plots [textplot]([[4.1,-.9,`1`],[.08,.7,`1`],[6.08,.7,`2`],\n \+ [3.18,-.4,`2`],[7.1,-.4,`2`]],font=[HELVETICA,8]):\nplots[display]([ p||(1..9),t||(1..3)],axes=none,view=[0..8,-1..1]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "6 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }