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" }}{PARA 0 "" 0 "" {TEXT -1 123 "They can be rea d into a Maple session by commands similar to those that follow, where the file paths gives their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/intg.m\";\nread \"K:\\\\Ma ple/procdrs/gkdata.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 40 "load root-finding procedures including: " }{TEXT 0 6 "secant" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 285 7 "roots.m" }{TEXT -1 37 " contains the code for the procedure " } {TEXT 0 6 "secant" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its locati on. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/p rocdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 59 "Maple's numerical integration procedures available thro ugh " }{TEXT 0 9 "evalf/Int" }}{PARA 15 "" 0 "" {TEXT -1 33 "Maple wil l evaluate the integral " }{XPPEDIT 18 0 "Int(f(x),x=a..b)" "6#-%$IntG 6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 55 " using a numerical integr ation method with the command " }{TEXT 0 22 "evalf(Int(f(x),x=a..b)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(exp(-x^2),x=0..1);\nevalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"/F,; \"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G8Cou!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 35 "If a lower case \+ \"i\" is used as in " }{TEXT 0 22 "evalf(int(f(x),x=a..b)" }{TEXT -1 110 ", an attempt is made to find an analytical form for the integral, which is subsequently evaluated nunerically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(exp(-x^2 ),x=0..1);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"/F,;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$erfG6#\"\"\"F(-%%sqrtG6#%#PiGF(#F(\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+I8Cou!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 85 "In a situation where Map le cannot find an explicit analytical form for the integral, " }{TEXT 0 5 "evalf" }{TEXT -1 44 " will use a numerical integration procedure. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int(exp(-x*sin(x)),x=0..1);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#,$*&%\"xG\"\"\"-%$sinG6#F+F,! \"\"/F+;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$expG6#, $*&%\"xG\"\"\"-%$sinG6#F+F,!\"\"/F+;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kq[Ow!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 " The following information comes from Maple 7's help page for " }{TEXT 0 9 "evalf/int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 264 4 "Note" } {TEXT -1 45 ": This is changed substantially from Maple 6." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "Information on " }{TEXT 0 9 "evalf/int" } }{PARA 4 "" 0 "usage" {TEXT -1 17 "Calling Sequences" }}{PARA 0 "" 0 " " {TEXT -1 15 " evalf(Int(" }{TEXT 35 1 "f" }{TEXT -1 4 ", x=" } {TEXT 35 4 "a..b" }{TEXT -1 2 "))" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ evalf(Int(" }{TEXT 35 1 "f" }{TEXT -1 2 ", " }{TEXT 35 4 "a..b" } {TEXT -1 2 "))" }}{PARA 0 "" 0 "" {TEXT -1 15 " evalf(Int(" } {TEXT 35 1 "f" }{TEXT -1 4 ", x=" }{TEXT 35 4 "a..b" }{TEXT -1 2 ", " }{TEXT 35 4 "opts" }{TEXT -1 2 "))" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ evalf(Int(" }{TEXT 35 1 "f" }{TEXT -1 2 ", " }{TEXT 35 4 "a..b" } {TEXT -1 2 ", " }{TEXT 35 4 "opts" }{TEXT -1 2 "))" }}{PARA 0 "" 0 "" {TEXT -1 15 " evalf(int(" }{TEXT 35 1 "f" }{TEXT -1 4 ", x=" } {TEXT 35 4 "a..b" }{TEXT -1 2 "))" }}{PARA 4 "" 0 "" {TEXT -1 10 "Para meters" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 7 "f - " } {TEXT -1 44 "algebraic expression or procedure; integrand" }}{PARA 0 " " 0 "" {TEXT -1 5 " " }{TEXT 23 7 "x - " }{TEXT -1 29 "name; va riable of integration" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 7 "a, b - " }{TEXT -1 40 "endpoints of the interval of integration" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 7 "opts - " }{TEXT -1 40 "(optional) name or equation of the form " }{TEXT 35 11 "option=name" }{TEXT -1 9 "; options" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT -1 11 "Des cription" }}{PARA 15 "" 0 "" {TEXT -1 112 "In the case of a definite i ntegral, which returns unevaluated, numerical integration can be invok ed by applying " }{TEXT 35 5 "evalf" }{TEXT -1 130 " to the unevaluate d integral. To invoke numerical integration without first invoking sym bolic integration, use the inert function " }{TEXT 35 3 "Int" }{TEXT -1 8 " as in: " }{TEXT 35 23 "evalf( Int(f, x=a..b) )" }{TEXT -1 2 ". \+ " }}{PARA 15 "" 0 "" {TEXT -1 17 "If the integrand " }{TEXT 35 1 "f" } {TEXT -1 93 " is specified as a procedure or a Maple operator, then th e second argument must be the range " }{TEXT 35 4 "a..b" }{TEXT -1 79 " and not an equation. (I.e., a variable of integration must not be sp ecified.) " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Optional Parameter s" }}{PARA 15 "" 0 "" {TEXT -1 199 "Additional options can be specifie d as equations. (For backward compatibility some options are accepted \+ as values rather than equations, as specified below.) An option is one of the following forms: " }}{PARA 17 "" 0 "wmitable" {TEXT -1 123 " \+ method = or \n digits = or

\n epsilon = " }}{PARA 15 "" 0 "" {TEXT -1 18 "The specification " }{TEXT 35 15 "method = " } {TEXT -1 12 " (or simply " }{TEXT 35 6 "" }{TEXT -1 192 ") indic ates a particular numerical integration method to be applied. The list of methods which may be specified are described below. By default, a \+ hybrid symbolic-numeric strategy is applied. " }}{PARA 15 "" 0 "" {TEXT -1 18 "The specification " }{TEXT 35 17 "digits = " } {TEXT -1 12 " (or simply " }{TEXT 35 8 "" }{TEXT -1 167 ") ind icates the number of digits of precision for the computation. Some add itional guard digits are carried during the computation to attempt to \+ achieve a result with " }{TEXT 35 8 "" }{TEXT -1 76 " correct \+ digits (although a larger tolerance can be specified by using the '" } {TEXT 35 7 "epsilon" }{TEXT -1 54 "' option). By default, the Maple en vironment variable " }{TEXT 35 6 "Digits" }{TEXT -1 46 " specifies the precision for the computation. " }}{PARA 15 "" 0 "" {TEXT -1 18 "The \+ specification " }{TEXT 35 19 "epsilon = " }{TEXT -1 259 " spe cifies the relative error tolerance for the computed result. The routi nes attempt to achieve a final result with a relative error less than \+ this value. By default, the relative error tolerance which the routine s attempt to achieve for the final result is " }}{PARA 17 "" 0 "wmitab le" {TEXT -1 33 " eps = 0.5 * 10^(1-digits) " }}{PARA 14 "" 0 " " {TEXT -1 6 "where " }{TEXT 35 6 "digits" }{TEXT -1 110 " is the prec ision specified for the computation. In attempting to achieve this acc uracy, the working value of " }{TEXT 35 6 "Digits" }{TEXT -1 62 " is i ncreased as deemed necessary. It is an error to specify '" }{TEXT 35 7 "epsilon" }{TEXT -1 40 "' smaller than the default value above. " }} {PARA 14 "" 0 "" {TEXT 37 5 "Note:" }{TEXT -1 188 " For some integrand s, the numerical accuracy attained when computing values of the integr and may be insufficient to allow the value of the integral to be compu ted to the default tolerance " }{TEXT 35 3 "eps" }{TEXT -1 142 " (even though the computation is using some number of guard digits). In such cases, specifying a larger tolerance (relative to the setting of " } {TEXT 35 6 "digits" }{TEXT -1 11 ") via the '" }{TEXT 35 7 "epsilon" } {TEXT -1 28 "' parameter may be helpful. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Outline of the Numerical Integration Polyalgorithm" }} {PARA 15 "" 0 "" {TEXT -1 111 "In the default case (no specific method specified), the problem is first passed to NAG integration routines i f " }{TEXT 35 6 "Digits" }{TEXT -1 28 " is not too large (i.e., if " } {TEXT 35 24 "Digits <= evalhf(Digits)" }{TEXT -1 378 "). The NAG routi nes are in a compiled C library and hence operate at hardware floating -point speed. If the NAG routines cannot perform the integration, then some singularity-handling may be performed and control may pass back \+ to the NAG routines with a modified problem. Native Maple routines are invoked if the NAG routines cannot solve the problem (e.g., for large r values of " }{TEXT 35 6 "Digits" }{TEXT -1 57 " and for integrands i nvolving functions unknown to NAG). " }}{PARA 15 "" 0 "" {TEXT -1 461 "The native Maple hybrid symbolic-numeric solution strategy is as foll ows. The default numerical method applied is Clenshaw-Curtis quadratur e (_CCquad). If slow convergence is detected, then there must be singu larities in or near the interval of integration. Some techniques of sy mbolic analysis are used to deal with the singularities. For problems \+ with non-removable endpoint singularities, an adaptive double-exponent ial quadrature method (_Dexp) is applied. " }}{PARA 15 "" 0 "" {TEXT -1 358 "If singularities interior to the interval are suspected, then \+ an attempt is made to locate the singularities in order to break up th e interval of integration. Finally, if still unsuccessful, then the in terval is subdivided and the _Dexp method is applied, or if the method was already _Dexp or _Sinc, then an adaptive Newton-Cotes rule (_NCru le) is applied. " }}{PARA 15 "" 0 "" {TEXT -1 42 "For the limits of in tegration, the values " }{TEXT 35 8 "infinity" }{TEXT -1 8 " and/or " }{TEXT 35 9 "-infinity" }{TEXT -1 15 " are valid for " }{TEXT 35 1 "a " }{TEXT -1 8 " and/or " }{TEXT 35 1 "b" }{TEXT -1 235 ", and it attem pts to handle singularities in the integrand. For these cases, it uses techniques such as variable transformations, subtracting off the sing ularity, and integration of a truncated generalized series near the si ngularity. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "The Case of a Pro cedure (or an Operator Expression)" }}{PARA 15 "" 0 "" {TEXT -1 105 "I f the first argument is a procedure or an operator expression, then th e second argument must be a range " }{TEXT 35 4 "a..b" }{TEXT -1 26 " \+ (rather than an equation " }{TEXT 35 8 "x = a..b" }{TEXT -1 60 "). In \+ this case, no singularity handling will be attempted. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "The Method Names" }}{PARA 15 "" 0 "" {TEXT -1 23 "The optional parameter " }{TEXT 35 15 "method = " } {TEXT -1 12 " (or simply " }{TEXT 35 6 "" }{TEXT -1 38 ") accept s the following method names. " }}{PARA 17 "" 0 "wmitable" {TEXT -1 320 "method = _DEFAULT -- equivalent to not specifying the \n method; the solution strategy outlined \n \+ above will be followed. \nmethod = _ NoNAG -- indicates to avoid calling NAG routines; \n \+ otherwise follow the _DEFAULT strategy. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "NAG Methods" }}{PARA 15 "" 0 "" {TEXT -1 37 "Spec ifying a method indicates to try " }{TEXT 22 4 "only" }{TEXT -1 76 " t hat method (in particular, no singularity-handling and no Maple method s). " }}{PARA 17 "" 0 "wmitable" {TEXT -1 657 "method = _d01ajc -- for finite interval of integration; allows \n \+ for badly behaved integrands; uses adaptive \n \+ Gauss 10-point and Kronrod 21-point rules. \nmethod = _d 01akc -- for finite interval of integration, oscillating \n \+ integrands; uses adaptive Gauss 30-point and \n \+ Kronrod 61-point rules. \+ \nmethod = _d01amc -- for semi-infinite/infinite interval of integrati on. \nmethod = _d01gbc -- for multiple integrals; Monte Carlo method o ver \n hyper-rectangle (for low accuracy only). " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Maple Methods" }}{PARA 15 "" 0 "" {TEXT -1 37 "Specifying a method indicates to try \+ " }{TEXT 22 4 "only" }{TEXT -1 74 " that method (in particular, no NAG methods and no singularity-handling). " }}{PARA 17 "" 0 "wmitable" {TEXT -1 497 "method = _CCquad -- Clenshaw-Curtis quadrature method. \+ \nmethod = _Dexp -- adaptive double-exponential method . \nmethod = _Sinc -- adaptive sinc quadrature method. \nmethod = _NCrule -- adaptive Newton-Cotes method \+ \"quanc8\". Note that \n \"quanc8\" is a fixed-ord er method and hence it is \n not recommended for v ery high \n precisions (e.g., D igits > 15). " }}{PARA 15 "" 0 "" {TEXT -1 75 "Vari ous levels of user information are displayed during the computation if " }{TEXT 35 22 "infolevel[`evalf/int`]" }{TEXT -1 28 " is assigned va lues between " }{TEXT 35 1 "1" }{TEXT -1 5 " and " }{TEXT 35 1 "4" } {TEXT -1 2 ". " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "From the information in \+ the previous subsection we see that it is possible to specify the meth od to be applied by using one of the following options." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 16 "\"met hod=_CCquad\"" }}{PARA 0 "" 0 "" {TEXT -1 141 " . . . indicates to use the Clenshaw-Curtis integration method which uses Chebyshev polynomia ls to approximate the function to be integrated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 0 "" }{TEXT 23 16 "\"method= _NCrule\"" }}{PARA 0 "" 0 "" {TEXT -1 85 " . . . indicates to use the \+ 8 interval Newton-Cotes formula in an adaptive procedure." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 14 "\"m ethod=_Dexp\"" }}{PARA 0 "" 0 "" {TEXT -1 65 " . . . indicates to use \+ adaptive double exponential integration. " }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 14 "\"method=_Dexp \"" }}{PARA 0 "" 0 "" {TEXT -1 50 " . . . indicates to use adaptive si nc quadrature. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" } {TEXT 0 9 "evalf/Int" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(exp(-x*sin(x )),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#,$*&%\"xG\"\"\"-%$sinG6#F+F,!\" \"/F+;\"\"!F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := x -> exp(-x*sin(x));\np lot(f(x),x=0..2,y=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6# %\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*&9$\"\"\"-%$sinG6#F1F2!\"\" F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 278 159 159 {PLOTDATA 2 "6%-%'CUR VESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"39LLLL3VfV!#>$\"3sq@z<%>5)**!#=7$$\"3' pmm;H[D:)F/$\"3#pO=SXHQ$**F27$$\"3LLLLe0$=C\"F2$\"3KVi*)3(et%)*F27$$\" 3ILLL3RBr;F2$\"3[IVb<[#es*F27$$\"3Ymm;zjf)4#F2$\"3kZGs)oLAd*F27$$\"3=L L$e4;[\\#F2$\"3$p!om(R8ES*F27$$\"3p****\\i'y]!HF2$\"3^R>,RPb,#*F27$$\" 3,LL$ezs$HLF2$\"3q0SAS=+p*)F27$$\"3_****\\7iI_PF2$\"3,=W,1P>:()F27$$\" 3#pmmm@Xt=%F2$\"3)y!y89+YM%)F27$$\"3QLLL3y_qXF2$\"3=![dVP-M<)F27$$\"3i ******\\1!>+&F2$\"3Y&=N@0lr'yF27$$\"3()******\\Z/NaF2$\"3)**o$3Gix\\vF 27$$\"3'*******\\$fC&eF2$\"3x&>++s@wB(F27$$\"3ELL$ez6:B'F2$\"3EPA^N:@^ pF27$$\"3Smmm;=C#o'F2$\"3?%eMr]I)4mF27$$\"3-mmmm#pS1(F2$\"3H#e/C15@K'F 27$$\"3]****\\i`A3vF2$\"3woC&Q36:*fF27$$\"3slmmm(y8!zF2$\"3(f>^#=MO/dF 27$$\"3V++]i.tK$)F2$\"3wP#3kdLqR&F27$$\"39++](3zMu)F2$\"3A&QhS$*GL6&F2 7$$\"3#pmm;H_?<*F2$\"3nvhInn*y#[F27$$\"3emm;zihl&*F2$\"3)\\kNyM*>wXF27 $$\"39LLL3#G,***F2$\"3)o_\\(QJk;VF27$$\"3i:HoCF27$$ \"3'******\\)Hxe9Fcs$\"3M'ei!fDdYBF27$$\"3Ymm\"H!o-*\\\"Fcs$\"3E6*3*\\ (4@C#F27$$\"3))***\\7k.6a\"Fcs$\"33W7BC:!H9#F27$$\"3emmmT9C#e\"Fcs$\"3 [\"RyAK_`0#F27$$\"3\"****\\i!*3`i\"Fcs$\"3q\"4S?*oEt>F27$$\"3QLLL$*zym ;Fcs$\"3'y@t\"=R2.>F27$$\"3GLL$3N1#4xHLFcs$\"33hr;8y.[;F27$$\"3/++v.Uac>Fcs$\"34'4+f16?j\"F27$$\" \"#F)$\"32e*e&3e`A;F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q \"x6\"Q\"yFd[l-%%VIEWG6$;F(Fez;F(F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f := x -> exp(-x*sin(x));\n evalf(Int(f(x),x=0..1,method=_CCquad));\nevalf(Int(f(x),x=0..1,method= _NCrule));\nevalf(Int(f(x),x=0..1,method=_Dexp));\nevalf(Int(f(x),x=0. .1,method=_Sinc));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(-%$expG6#,$*&9$\"\"\"-%$sinG6#F1F2!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kq[Ow!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kq[Ow!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kq[ Ow!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iq[Ow!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1+sin(x)*ln(x),x = 0 .. Pi);" "6#- %$IntG6$,&\"\"\"F'*&-%$sinG6#%\"xGF'-%#lnG6#F,F'F'/F,;\"\"!%#PiG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "g := x -> 1+sin(x)*ln(x);\nplot(g(x),x=0..Pi,num points=200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%) operatorG%&arrowGF(,&\"\"\"F-*&-%$sinG6#9$F--%#lnGF1F-F-F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 275 182 182 {PLOTDATA 2 "6%-%'CURVESG6$7dw7 $$\"1+++I`s^;!#<$\"1*)\\uQ,FA$*!#;7$$\"1+++?SIqBF*$\"10!oAFzI6*F-7$$\" 1+++4F)))3$F*$\"1#\\2/&[0E*)F-7$$\"1+++)[;^q%F*$\"1\"o!3jCSi&)F-7$$\"1 +++:P1KjF*$\"1)***44m!QD)F-7$$\"1+++X$y7&zF*$\"1P\"Q0:')*))zF-7$$\"1** ****zv\\_%*F*$\"1#=D1kxNx(F-7$$\"1+++:Bp+6F-$\"1#QNHNjgd(F-7$$\"1+++H7 Xh7F-$\"1qYqt,J&R(F-7$$\"1+++(e%p@9F-$\"1SP.QH)fB(F-7$$\"1+++yX_'e\"F- $\"1#G\"f9QQ\"4(F-7$$\"1+++BpqJhIW'F-7$$\"1+++\"4^r:$F-$\"1z*z&H8C?kF-7$$\"1+++Yyx7 LF-$\"1e&y#>ni1kF-7$$\"1+++Q!e^Z$F-$\"1'G&y$=M/S'F-7$$\"1+++WPFCOF-$\" 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\\*F-$\"3L)*H8c\"[.^*F-7$$\"3%)*****zjP![&*F-$\"3cGXQM,&zb*F-7$$\"3s** ***4`%Hl*F-7$$\"31+ ++OsE(p*F-$\"3W)pd4[e " 0 "" {MPLTEXT 1 0 183 "evalf(Int(1 /(1-ln(x)),x=0..1,method=_CCquad));\nevalf(Int(1/(1-ln(x)),x=0..1,meth od=_Dexp));\nevalf(Int(1/(1-ln(x)),x=0..1,method=_Sinc));\nevalf(Int(1 /(1-ln(x)),x=0..1,method=_NCrule));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+BOZjf!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BOZjf!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BOZjf!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6%*&\"\"\"F',&$F'\"\"!F'*&$F'F*F'-%#lnG6#%\"xGF' !\"\"F1/F0;$F*F*F)/%'methodG%(_NCruleG" }}}{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 57 "An interface for various numerical integration routines: " }{TEXT 0 8 "q uad/Int" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "quad: usage " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " " }{TEXT -1 19 " quad( intg ) or " }{TEXT 262 0 "" }{TEXT -1 16 " quad ( intg, d )" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 11 " intg - " }{TEXT -1 46 "an integral given in the form Int(fx,x=a..b). " }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {TEXT 23 8 "d - " }{TEXT -1 84 "a positive integer giving the numb er of digits required in the value of the integral" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The pro cedure " }{TEXT 0 4 "quad" }{TEXT -1 101 " applies a numerical integra tion (quadrature) scheme to an integral given in the form Int(fx,x=a.. b)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "T his procedure is a \"catch-all\" procedure for the specific procedures considered separately in various worksheets. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "info=true/false\nThe \+ option info=true causes some information regarding the progress of the computation to be given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 72 "method=Newton_Cotes/Gauss_Legendre/Gauss_Kronrod/C lenshaw_Curtis/Romberg" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "This option allows a choice of four quadrature metho ds: " }}{PARA 15 "" 0 "" {TEXT -1 24 "Newton_Cotes . . (using " } {TEXT 0 5 "NCint" }{TEXT -1 75 ")\nAlternative names: NC, nc, NCquad, \+ NewtonCotes, newtoncotes, newton_cotes" }}{PARA 15 "" 0 "" {TEXT -1 27 "Clenshaw_Curtis . . (using " }{TEXT 0 5 "CCint" }{TEXT -1 84 ")\nA lternative names: CC, cc, CCquad, ClenshawCurtis, clenshawcurtis, clen shaw_curtis" }}{PARA 15 "" 0 "" {TEXT -1 25 "Gauss-Legendre. . (using \+ " }{TEXT 0 5 "GLint" }{TEXT -1 81 ")\nAlternative names: GL, gl, GLqua d, GaussLegendre, gausslegendre, gauss_legendre" }}{PARA 15 "" 0 "" {TEXT -1 25 "Gauss-Kronrod . . (using " }{TEXT 0 5 "GKint" }{TEXT -1 79 ")\nAlternative names: GK, gk, GKquad, Gauss_Kronrod, gausskronrod, gauss_kronrod" }}{PARA 15 "" 0 "" {TEXT -1 19 "Romberg . . (using " } {TEXT 0 5 "RBint" }{TEXT -1 44 ")\nAlternative names: RB, rb, RBquad, \+ romberg" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The Newton_Cotes, Gauss-Legendre and Gauss-Kronrod methods are all available in " }{TEXT 256 8 "adaptive" }{TEXT -1 158 " mode in that i f the desired accuracy is not achieved on a specific interval, the int erval is bisected and the scheme is applied to each of the new interva ls." }}{PARA 0 "" 0 "" {TEXT -1 80 "They can also be applied in the no n-adaptive mode via the option adaptive=false." }}{PARA 0 "" 0 "" {TEXT -1 63 "In this case there is guarantee of the accuracy of the re sults." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 5 "N Cint" }{TEXT -1 5 " and " }{TEXT 0 5 "GLint" }{TEXT -1 88 " use bisect ion to obtain error estimates, so at least one bisection is always per formed." }}{PARA 0 "" 0 "" {TEXT 0 5 "GLint" }{TEXT -1 183 " always \+ \"wastes\" function evaluations, that is, when bisection occurs, the p revious evaluations on the original interval do not contribute directl y to the the value of the integral.\n" }{TEXT 0 5 "GKint" }{TEXT -1 154 " will not waste any function evaluations if it can find the value of the integral from the first error estimate using the Kronrod point s. Otherwise, like " }{TEXT 0 5 "GLint" }{TEXT -1 78 ", as soon as any bisections are performed, it will waste function evaluations." }} {PARA 0 "" 0 "" {TEXT -1 12 "The schemes " }{TEXT 0 5 "CCint" }{TEXT -1 5 " and " }{TEXT 0 5 "RBint" }{TEXT -1 81 " are not adaptive in the same sense as the other schemes, but they are iterative." }}{PARA 0 " " 0 "" {TEXT -1 321 "They compute successive approximations for the va lue of the integral, requiring progressively more evaluations of the i ntegrand spread over the whole interval, until the the desired accurac y is achieved.\nIf the integrand is badly behaved in some sections of \+ the interval of integration, the option \"split\" can be used. \n" }} {PARA 0 "" 0 "" {TEXT -1 9 "split=n/[" }{XPPEDIT 18 0 "x[0],x[1];" "6$ &%\"xG6#\"\"!&F$6#\"\"\"" }{TEXT -1 8 ", . . . " }{XPPEDIT 18 0 "x[n]; " "6#&%\"xG6#%\"nG" }{TEXT -1 1 "]" }}{PARA 0 "" 0 "" {TEXT -1 50 "Thi s option can be used to split up the integral " }{XPPEDIT 18 0 "Int(f x,x=a..b)" "6#-%$IntG6$%#fxG/%\"xG;%\"aG%\"bG" }{TEXT -1 24 " into a \+ sum of the form" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 " Int(fx,x=a..x[1])+Int(fx,x=x[1]..x[2])" "6#,&-%$IntG6$%#fxG/%\"xG;%\"a G&F)6#\"\"\"F.-F%6$F'/F);&F)6#F.&F)6#\"\"#F." }{TEXT -1 12 " + . . . + " }{XPPEDIT 18 0 "Int(fx,x = x[n-1] .. b);" "6#-%$IntG6$%#fxG/%\"xG; &F(6#,&%\"nG\"\"\"F.!\"\"%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 158 "One can specify the actual points to be used for the sub division, or, if equally spaced points are required, the number of int ervals can be specified instead." }}{PARA 0 "" 0 "" {TEXT -1 61 "The d efault is to provide no subdivision, that is, \"split=1\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Options for speci fic" }{TEXT 265 1 " " }{TEXT -1 84 "methods are also available. For in formation on these, see the appropriate worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 17 "How to activate:\n" }{TEXT -1 133 "To make the procedure active place the cursor anywhere after the prompt [ > and press [Ent er].\nYou can then close up the subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 5 "Note:" }{TEXT -1 139 " This pro cedure only works in a particular situation if the corresponding proce dure mentioned under the method option above is also loaded." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "quad: implementation" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5183 "quad := \+ proc(ff)\n local fx,a,b,eq,la,x,mthd,Options,dv,saveDigits,rs,\n \+ aa,bb,lims,area,term,i,prntflg,width,h,\n implementedMeth ods,NCnames,GLnames,GKnames,\n CCnames,RBnames,startopts;\n\n \+ if type(ff,'function') and op(0,ff)='Int' and nops(ff)=2 \n and \+ type(op(1,ff),algebraic) and type(op(2,ff),name=realcons..realcons) th en\n fx := op(1,ff);\n eq := op(2,ff);\n x := op(1,eq); \n else\n error \"the 1st argument, %1, is invalid .. it should be a definite integral described using 'Int'\",ff;\n end if;\n\n \+ startopts := 2; \n if nargs>1 and not type(args[2],equation) then\n \+ if type(args[2],posint) then\n Digits := args[2];\n \+ startopts := 3;\n else\n error \"if the 2nd argument i s not an option, it must be a positive integer giving the number of di gits required in the value of the integral\"\n end if;\n end if ;\n if not type(x,name) then\n error \"left side of range equat ion in integral must be a variable\"\n end if;\n if not type(indet s(fx,name) minus \{x\},set(realcons)) then\n error \"integrand, % 1, must depend only on the variable %2\",fx,x;\n end if;\n\n # Get the options.\n # Set the default values to start with.\n mthd := \+ 3; # Gauss-Kronrod is the default method\n dv := 1;\n prntflg := f alse;\n Options:=[args[startopts..nargs]];\n if nargs>=startopts t hen\n if not type(Options,list(equation)) then\n error \" each optional argument must be an equation\"\n end if;\n if \+ hasoption(Options,'method','mthd','Options') then\n NCnames := \{'NC','nc','NCquad','NewtonCotes','newtoncotes',\\\n \+ 'Newton_Cotes','newton_cotes'\};\n GLnames := \{'GL','gl',' GLquad','GaussLegendre','gausslegendre',\\\n 'Gauss_ Legendre','gauss_legendre'\};\n GKnames := \{'GK','gk','GKquad ','GaussKronrod','gausskronrod',\\\n 'Gauss_Kronrod' ,'gauss_kronrod'\};\n CCnames := \{'CC','cc','CCquad','Clensha wCurtis','clenshawcurtis',\\\n 'Clenshaw_Curtis','cl enshaw_curtis'\};\n RBnames := \{'RB','rb','RBquad','Romberg', 'romberg'\};\n implementedMethods := NCnames union GLnames uni on GKnames\n union CCnames union RBnames;\n \+ if (not member(mthd,implementedMethods)) then\n error \"m ethod not implemented\"\n end if;\n if member(mthd,NCn ames) then mthd := 1\n elif member(mthd,GLnames) then mthd := \+ 2\n elif member(mthd,GKnames) then mthd := 3\n elif me mber(mthd,CCnames) then mthd := 4\n elif member(mthd,RBnames) \+ then mthd := 5 end if;\n end if;\n if hasoption(Options,'spl it','dv','Options') then\n if not type(dv,integer) and not typ e(dv,list(realcons)) then\n error \"\\\"split\\\" must be a positive integer or a list of real constants\" \n end if;\n \+ end if;\n if hasoption(Options,'info','prntflg') then\n \+ if prntflg>0 or prntflg=true then prntflg := true\n else pr ntflg := false end if;\n end if;\n end if; \n\n if dv=1 the n\n if mthd=1 then\n return NCint(fx,eq,op(Options));\n \+ elif mthd=2 then\n return GLint(fx,eq,op(Options));\n \+ elif mthd=3 then\n return GKint(fx,eq,op(Options));\n el if mthd=4 then\n return CCint(fx,eq,op(Options));\n else \+ # mthd=5\n return RBint(fx,eq,op(Options));\n end if\n \+ else\n rs := rhs(eq);\n if not type(rs,realcons..realcons) t hen\n error \"2nd argument equation right side, %1, must be a \+ range of real values\",rs;\n end if;\n # Increase precision \+ for the computation by about 25%.\n saveDigits := Digits;\n \+ Digits := Digits + max(5,trunc(Digits*0.25));\n\n aa := op(1,rs); \n bb := op(2,rs);\n a := evalf(aa);\n b := evalf(bb); \n if a=b then return 0.0\n elif a>=b then\n Digits \+ := saveDigits;\n return -quad(Int(fx,x=bb..aa),args[2..nargs]) \n end if;\n if type(dv,posint) then\n width := b-a; \n h := width/dv;\n lims := [seq(a+i*h,i=0..dv)];\n \+ else\n lims := sort([op(\{a,op(evalf(dv)),b\})]);\n \+ if op(1,lims)b then\n error \"in termediate limits must lie in %1\",aa..bb;\n end if;\n en d if;\n Digits := saveDigits;\n if prntflg then\n pr int(`Splitting interval of integration using the points`);\n p rint(evalf(op(lims)));print(``);\n end if;\n area := 0;\n \+ for i from 1 to nops(lims)-1 do\n if mthd=1 then\n \+ term := NCint(fx,x=lims[i]..lims[i+1],op(Options));\n elif \+ mthd=2 then\n term := GLint(fx,x=lims[i]..lims[i+1],op(Opti ons));\n elif mthd=3 then\n term := GKint(fx,x=lims [i]..lims[i+1],op(Options));\n elif mthd=4 then\n t erm := CCint(fx,x=lims[i]..lims[i+1],op(Options));\n else # mt hd=5\n term := RBint(fx,x=lims[i]..lims[i+1],op(Options)); \n end if;\n if prntflg then\n print(`integ ral over`,evalf(lims[i]..lims[i+1]),`-->`,term)\n end if; \n \+ area := evalf(evalf(area+term,Digits+2));\n end do; \n \+ end if;\n return area;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 ": " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 0 8 "quad/Int" }{TEXT -1 10 ": ex amples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Int(exp(-x^3),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#,$ *$%\"xG\"\"$!\"\"/F+;\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "intg := Int (exp(-x^3),x=0..1);\nevalf(intg);\nquad(intg,method=Newton_Cotes);\nqu ad(intg,method=Clenshaw_Curtis);\nquad(intg,method=Gauss_Legendre);\nq uad(intg,method=Gauss_Kronrod);\nquad(intg,method=Romberg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%intgG-%$IntG6$-%$expG6#,$*$)%\"xG\"\"$\" \"\"!\"\"/F.;\"\"!F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@=6v!)!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@=6v!)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@=6v!)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@= 6v!)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@=6v!)!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+@=6v!)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The Romberg integration is rather slow \+ when high precision is required." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "intg := Int(exp(-x^3),x=0.. 1);\nDigits := 50;\nevalf(intg);\nquad(intg,method=NC,numpoints=20);\n quad(intg,method=CC);\nquad(intg,method=GL);\nquad(intg,method=GK);\nD igits := 10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%intgG-%$IntG6$-%$ex pG6#,$*$)%\"xG\"\"$\"\"\"!\"\"/F.;\"\"!F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S \\6N&o/-bZ34Q$=L$eGXr'R@=6v!)!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"S\\6N&o/-bZ34Q$=L$eGXr'R@=6v!)!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"S\\6N&o/-bZ34Q$=L$eGXr'R@=6v!)!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S\\6N&o/-bZ34Q$=L$eGXr'R@=6v!)!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S\\6N&o/-bZ34Q$=L$eGXr'R@=6v!)!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1+sin(x)*ln(x),x = 0 .. Pi);" "6#-%$IntG6$,&\"\"\"F'*&-%$sin G6#%\"xGF'-%#lnG6#F,F'F'/F,;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "g := x -> 1+sin(x)*ln(x);\nquad(Int(g(x),x=0..Pi),20,method=NC,info=1);\nqua d(Int(g(x),x=0..Pi),20,method=GL,info=1);\nquad(Int(g(x),x=0..Pi),20,m ethod=GK,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&\"\"\"F-*&-%$sinG6#9$F--%#lnGF1F-F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~w ith~13~nodesG" }}{PARA 7 "" 1 "" {TEXT -1 39 "Warning, reached max sub division depth\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~functi on~evaluations~-->~G\"%l9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5!Qg3% e'yuFy$!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%HGauss-Legendre~quadrat ure~with~23~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.adaptive~modeG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~- ->~G\"%8I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5!Qg3%e'yuFy$!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptive~22-45~node~Gauss-Kronrod~qu adratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~eval uations~-->~G\"%lM" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5!Qg3%e'yuFy$ !#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "T he Clenshaw-Curtis method is rather slow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "quad(Int(1+sin(x)*ln( x),x=0..Pi),method=CC,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Eit erative~Clenshaw-Curtis~quadratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%ioinserting~2~new~evaluation~points~between~each~previous~pair~at~ea ch~iterationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coand~calculating~th e~Chebyshev~coefficients~using~a~fast~cosine~transformG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Fstep~1:~approx~value~of~integral~-->~G$\".PS'o :$y$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~2:~approx~value~of~i ntegral~-->~G$\".3gZxFy$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~ 3:~approx~value~of~integral~-->~G$\".+s\"[x#y$!#7" }}{PARA 7 "" 1 "" {TEXT -1 105 "Warning, integrand may have a singularity near the inter val of integration - using simple error estimate\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Fstep~4:~approx~value~of~integral~-->~G$\".gpyuFy$!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~5:~approx~value~of~integra l~-->~G$\".!f'yuFy$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~6:~ap prox~value~of~integral~-->~G$\".%e'yuFy$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"%f9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(yuFy$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(sin(3*x)^x,x = 0 .. 1);" "6#-%$IntG6$)-%$sinG6#*&\" \"$\"\"\"%\"xGF,F-/F-;\"\"!F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "h := proc(x: :realcons)\n if x=0 then 1 else sin(3*x)^x end if\nend proc;\nplot(' h(x)',x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#'%\"xG%)re alconsG6\"F*F*@%/9$\"\"!\"\"\")-%$sinG6#,$F-\"\"$F-F*F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 310 223 223 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$ \"\"\"F)7$$\"3emmm;arz@!#>$\"3&y8,Kr\"pA%*!#=7$$\"3[LL$e9ui2%F/$\"3#yv -Y**\\\"y\"*F27$$\"3nmmm\"z_\"4iF/$\"3n;A/B\"*)e+*F27$$\"3[mmmT&phN)F/ $\"37B[CgXR+*)F27$$\"3CLLe*=)H\\5F2$\"3y?irbAWU))F27$$\"3gmm\"z/3uC\"F 2$\"3&)oJYQ:E?))F27$$\"3%)***\\7LRDX\"F2$\"3Jz$HxtrC#))F27$$\"3]mm\"zR 'ok;F2$\"3wH^$Hd)\\Y))F27$$\"3w***\\i5`h(=F2$\"3%)>**owEM))))F27$$\"3W LLL3En$4#F2$\"3OLh\\%)R`Y*)F27$$\"3qmm;/RE&G#F2$\"36B/s*)y73!*F27$$\"3 \")*****\\K]4]#F2$\"3;u8iVcp'3*F27$$\"3$******\\PAvr#F2$\"3y!Q\"HT%QJ< *F27$$\"3)******\\nHi#HF2$\"3gjNdvi^h#*F27$$\"3jmm\"z*ev:JF2$\"3r>#\\& =nWW$*F27$$\"3?LLL347TLF2$\"3&[hhnoVVW*F27$$\"3,LLLLY.KNF2$\"3h6rlFNQG &*F27$$\"3w***\\7o7Tv$F2$\"3!Hc?$QiRB'*F27$$\"3'GLLLQ*o]RF2$\"3PWr>=.A 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0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "quad(Int(sin(3*x)^x,x=0..1), method=GKquad,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptiv e~10-21~node~Gauss-Kronrod~quadratureG" }}{PARA 13 "" 1 "" {GLPLOT2D 445 84 84 {PLOTDATA 2 "62-%'CURVESG6%7$7$$\"3++++++++]!#=$\"\"!F,7$F($ \"\"\"F,-%&COLORG6&%$RGBG$\"\")!\"\"F,F4-%*THICKNESSG6#\"\"#-F$6%7$7$$ \"3++++++++DF*F+7$F?F.F0F7-F$6%7$7$$\"3+++++++]7F*F+7$FFF.F0F7-F$6%7$7 $$\"3+++++++]i!#>F+7$FMF.F0F7-F$6%7$7$$\"3+++++++DJFOF+7$FUF.F0F7-F$6% 7$7$$\"3++++++]i:FOF+7$FfnF.F0F7-F$6%7$7$$\"3++++++]7y!#?F+7$F]oF.F0F7 -F$6%7$7$$\"3++++++D1RF_oF+7$FeoF.F0F7-F$6%7$7$$\"3+++++]7`>F_oF+7$F\\ pF.F0F7-F$6%7$7$$\"3-++++]il(*!#@F+7$FcpF.F0F7-F$6%7'7$F+F+7$F+F.7$F.F .7$F.F+Fjp-F16&F3$\"\"&F6F`qF`qF7-%%TEXTG6$7$$\".++++++&!#8$\"\"$F,Q1b isection~points6\"-Fcq6$7$Ffq$!\"#F,Q&~---~F\\r-%+AXESLABELSG6%Q!F\\rF fr-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$FjrFjr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C 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4" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(1/(1-ln(x)),x = 0 .. 1);" "6#-%$IntG6$*&\"\"\"F',&F 'F'-%#lnG6#%\"xG!\"\"F-/F,;\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "quad(Int(1/( 1-ln(x)),x=0..1),method=GL,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%HGauss-Legendre~quadrature~with~15~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.adaptive~modeG" }}{PARA 7 "" 1 "" {TEXT -1 39 "Warnin g, reached max subdivision depth\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %Dnumber~of~function~evaluations~-->~G\"%X7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BOZjf!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "quad(Int(1/(1-ln(x)),x=0..1) ,method=GK,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptive~10-2 1~node~Gauss-Kronrod~quadratureG" }}{PARA 7 "" 1 "" {TEXT -1 39 "Warni ng, reached max subdivision depth\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%Dnumber~of~function~evaluations~-->~G\"$h)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BOZjf!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The Newton-Cotes value is not very accurate." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "quad(Int(1/(1-ln(x)),x=0..1),method=NC,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~with~13~nodesG " }}{PARA 7 "" 1 "" {TEXT -1 39 "Warning, reached max subdivision dept h\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluation s~-->~G\"$&Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MMZjf!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Splitting the interval of integration, we get a better result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "quad(Int( 1/(1-ln(x)),x=0..1),method=NC,split=[0.001],info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SSplitting~interval~of~integration~using~the~points G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"!F$$\"\"\"!\"$$F&F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Oadaptive~Newton-Cotes~quadrature~with~13~nodesG" }}{PARA 7 "" 1 "" {TEXT -1 39 "Warning, reached max subdivision depth\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"$&Q" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%.integral~overG;$\"\"!F&$\"\"\"!\"$%$ -->G$\"+A]lM6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton- Cotes~quadrature~with~13~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dn umber~of~function~evaluations~-->~G\"$<#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%.integral~overG;$\"\"\"!\"$$F&\"\"!%$-->G$\"+o*QB'f!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BOZjf!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Clenshaw-Curtis is also n ot very accurate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "quad(Int(1/(1-ln(x)),x=0..1),method=CC,info=2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Eiterative~Clenshaw-Curtis~quad ratureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ioinserting~2~new~evaluati on~points~between~each~previous~pair~at~each~iterationG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%coand~calculating~the~Chebyshev~coefficients~us ing~a~fast~cosine~transformG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep ~1:~approx~value~of~integral~-->~G$\".`J')[O%f!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~2:~approx~value~of~integral~-->~G$\".%o/n#='f!#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~3:~approx~value~of~integral ~-->~G$\".I+?DL'f!#8" }}{PARA 7 "" 1 "" {TEXT -1 105 "Warning, integra nd may have a singularity near the interval of integration - using sim ple error estimate\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~4:~appr ox~value~of~integral~-->~G$\".i.ufM'f!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~5:~approx~value~of~integral~-->~G$\".o=HsM'f!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Fstep~6:~approx~value~of~integral~-->~G$\". aJ\\tM'f!#8" }}{PARA 7 "" 1 "" {TEXT -1 46 "Warning, reached maximum n umber of iterations\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~f unction~evaluations~-->~G\"%f9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ $\\tM'f!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Again, splitting the interval of integration gives a better res ult." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "quad(Int(1/(1-ln(x)),x=0..1),method=CC,split=[0.001], info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SSplitting~interval~of~in tegration~using~the~pointsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"!F $$\"\"\"!\"$$F&F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%Eiterative~Clenshaw-Curtis~quadratureG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%ioinserting~2~new~evaluation~points~b etween~each~previous~pair~at~each~iterationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coand~calculating~the~Chebyshev~coefficients~using~a~f ast~cosine~transformG" }}{PARA 7 "" 1 "" {TEXT -1 105 "Warning, integr and may have a singularity near the interval of integration - using si mple error estimate\n" }}{PARA 7 "" 1 "" {TEXT -1 46 "Warning, reached maximum number of iterations\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dn umber~of~function~evaluations~-->~G\"%f9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%.integral~overG;$\"\"!F&$\"\"\"!\"$%$-->G$\"+j]lM6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Eiterative~Clenshaw-Curtis~quadratureG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%ioinserting~2~new~evaluation~points~b etween~each~previous~pair~at~each~iterationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coand~calculating~the~Chebyshev~coefficients~using~a~f ast~cosine~transformG" }}{PARA 7 "" 1 "" {TEXT -1 105 "Warning, integr and may have a singularity near the interval of integration - using si mple error estimate\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~f unction~evaluations~-->~G\"$([" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%.in tegral~overG;$\"\"\"!\"$$F&\"\"!%$-->G$\"+o*QB'f!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+BOZjf!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Solving equations involving integrals" }}{PARA 0 "" 0 "" {TEXT 270 8 "Questio n" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 20 "Solve the equation \+ " }{XPPEDIT 18 0 "Int(1/(t+exp(t)),t = 0 .. x) = 1-x;" "6#/-%$IntG6$*& \"\"\"F(,&%\"tGF(-%$expG6#F*F(!\"\"/F*;\"\"!%\"xG,&F(F(F2F." }{TEXT -1 6 " for " }{TEXT 278 1 "x" }{TEXT -1 17 " between 0 and 1." }} {PARA 258 "" 0 "" {TEXT 269 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 13 "Define g by " }{XPPEDIT 18 0 "g(x) = Int(1/(t+exp(t)),t = 0 .. x) ;" "6#/-%\"gG6#%\"xG-%$IntG6$*&\"\"\"F,,&%\"tGF,-%$expG6#F.F,!\"\"/F.; \"\"!F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "We set up the function g for this integral and plot the graph of " }{XPPEDIT 18 0 " y = g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 38 " along with that of th e straight line " }{XPPEDIT 18 0 "y = 1 -x" "6#/%\"yG,&\"\"\"F&%\"xG! \"\"" }{TEXT -1 16 ". The procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 20 " is used below, but " }{TEXT 0 9 "evalf/Int" }{TEXT -1 31 " would \+ produce the same result." }}{PARA 0 "" 0 "" {TEXT -1 16 "Simply omitti ng " }{TEXT 0 4 "quad" }{TEXT -1 26 " would cause Maple to use " } {TEXT 0 9 "evalf/Int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g := x -> Int(1/(t + exp(t )),t=0..x);\nplot(['quad(g(x))',1-x],x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$*&\" \"\"F0,&%\"tGF0-%$expG6#F2F0!\"\"/F2;\"\"!9$F(F(F(" }}{PARA 13 "" 1 " " {GLPLOT2D 235 185 185 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\" +;arz@!#6$\"+!GyL8#F-7$$\"+XTFwSF-$\"+!\\8w\"RF-7$$\"+\"z_\"4iF-$\"+`x V\\eF-7$$\"+S&phN)F-$\"+#ye#>xF-7$$\"+*=)H\\5!#5$\"+W,X5&*F-7$$\"+[!3u C\"FB$\"+,KF66FB7$$\"+J$RDX\"FB$\"+=Sgr7FB7$$\"+)R'ok;FB$\"+UeyJ9FB7$$ \"+1J:w=FB$\"+\"*=2'e\"FB7$$\"+3En$4#FB$\"+7sZRG#FB7$$\"+(*ev:JFB$\"+l2]'R#FB7$$\"+347TLFB$\"+cTxGDFB7$$\"+LY. KNFB$\"+B\"Rwj#FB7$$\"+\"o7Tv$FB$\"+.:wgFFB7$$\"+$Q*o]RFB$\"+DcsmGFB7$ $\"+\"=lj;%FB$\"+6>')zHFB7$$\"+V&R$FB7$$\"+R\"3Gy%FB$\"+=qR'G$FB7$$\"+.T1&*\\FB$\"+;-b'Q$FB7$$\"+(R Qb@&FB$\"+AH%y[$FB7$$\"+=>Y2aFB$\"+(eJQd$FB7$$\"+yXu9cFB$\"+u$)\\kOFB7 $$\"+\\y))GeFB$\"+$)y%ev$FB7$$\"+i_QQgFB$\"+Xq,VQFB7$$\"+!y%3TiFB$\"+^ *f`#RFB7$$\"+O![hY'FB$\"+]dc9SFB7$$\"+#Qx$omFB$\"+KTz#4%FB7$$\"+u.I%)o FB$\"+'QuV<%FB7$$\"+(pe*zqFB$\"+n%=mC%FB7$$\"+C\\'QH(FB$\"+E7$QK%FB7$$ \"+8S8&\\(FB$\"+,o%[R%FB7$$\"+0#=bq(FB$\"+5eVnWFB7$$\"+2s?6zFB$\"+_.$o `%FB7$$\"+IXaE\")FB$\"+\"fiyg%FB7$$\"+l*RRL)FB$\"+\"\\dZn%FB7$$\"+`<.Y &)FB$\"+DFnTZFB7$$\"+8tOc()FB$\"+&f%e1[FB7$$\"+\\Qk\\*)FB$\"+*)\\*\\'[ FB7$$\"+p0;r\"*FB$\"+oP_I\\FB7$$\"+lxGp$*FB$\"+pn)y)\\FB7$$\"+!oK0e*FB $\"+NmyZ]FB7$$\"+<5s#y*FB$\"+$eJR5&FB7$$\"\"\"F)$\"+Mw+j^FB-%'COLOURG6 &%$RGBG$\"#5!\"\"F(F(-F$6$7S7$F(Fbz7$$\"3emmm;arz@!#>$\"3!RLL$e%G?y*!# =7$$\"3[LL$e9ui2%Fd[l$\"3OmmT&esBf*Fg[l7$$\"3nmmm\"z_\"4iFd[l$\"3yKL$3 s%3z$*Fg[l7$$\"3[mmmT&phN)Fd[l$\"33LL$e/$Qk\"*Fg[l7$$\"3CLLe*=)H\\5Fg[ l$\"3!pm;/\"=q]*)Fg[l7$$\"3gmm\"z/3uC\"Fg[l$\"3'GL$3_>f_()Fg[l7$$\"3%) ***\\7LRDX\"Fg[l$\"3))***\\(o1YZ&)Fg[l7$$\"3]mm\"zR'ok;Fg[l$\"30ML3-OJ N$)Fg[l7$$\"3w***\\i5`h(=Fg[l$\"3p***\\P*o%Q7)Fg[l7$$\"3WLLL3En$4#Fg[l $\"3ammm\"RFj!zFg[l7$$\"3qmm;/RE&G#Fg[l$\"3JLL$e4OZr(Fg[l7$$\"3\")**** *\\K]4]#Fg[l$\"3=+++v'\\!*\\(Fg[l7$$\"3$******\\PAvr#Fg[l$\"33+++DwZ#G (Fg[l7$$\"3)******\\nHi#HFg[l$\"3[*****\\KqP2(Fg[l7$$\"3jmm\"z*ev:JFg[ l$\"3OLL3-TC%)oFg[l7$$\"3?LLL347TLFg[l$\"3Onmm\"4z)emFg[l7$$\"3,LLLLY. KNFg[l$\"3+nmmm`'zY'Fg[l7$$\"3w***\\7o7Tv$Fg[l$\"3!3+](=t)eC'Fg[l7$$\" 3'GLLLQ*o]RFg[l$\"3qnmm;1J\\gFg[l7$$\"3A++D\"=lj;%Fg[l$\"3y***\\(=[jLe Fg[l7$$\"31++vV&R<_Fg[l7$$\"3cmm;/T1&*\\Fg[l$ \"3VLL$e*e$\\+&Fg[l7$$\"3&em;zRQb@&Fg[l$\"3;ML3-;Y%y%Fg[l7$$\"3\\***\\ (=>Y2aFg[l$\"3]++D\"3QDf%Fg[l7$$\"39mm;zXu9cFg[l$\"3'QLL3Ub_Q%Fg[l7$$ \"3l******\\y))GeFg[l$\"3O+++]@6rTFg[l7$$\"3'*)***\\i_QQgFg[l$\"3/,+]P ZhhRFg[l7$$\"3@***\\7y%3TiFg[l$\"3y++v=_\"*ePFg[l7$$\"35****\\P![hY'Fg [l$\"3*3++D'>&Q`$Fg[l7$$\"3kKLL$Qx$omFg[l$\"3Pnmm;EiJLFg[l7$$\"3!)**** *\\P+V)oFg[l$\"3?+++D'*p:JFg[l7$$\"3?mm\"zpe*zqFg[l$\"3zLL3-8/?HFg[l7$ $\"3%)*****\\#\\'QH(Fg[l$\"3<+++v]81FFg[l7$$\"3GKLe9S8&\\(Fg[l$\"3snmT &)f'[]#Fg[l7$$\"3R***\\i?=bq(Fg[l$\"3g++v$z\"[%H#Fg[l7$$\"3\"HLL$3s?6z Fg[l$\"33nmm\"z#z)3#Fg[l7$$\"3a***\\7`Wl7)Fg[l$\"3Y++voaXt=Fg[l7$$\"3# pmmm'*RRL)Fg[l$\"33LLLL+1m;Fg[l7$$\"3Qmm;a<.Y&)Fg[l$\"3iLL$eCoRX\"Fg[l 7$$\"3=LLe9tOc()Fg[l$\"3$om;aoKOC\"Fg[l7$$\"3u******\\Qk\\*)Fg[l$\"3E+ ++]hN]5Fg[l7$$\"3CLL$3dg6<*Fg[l$\"3anmm\"H%R)G)Fd[l7$$\"3ImmmmxGp$*Fg[ l$\"30PLLLB72jFd[l7$$\"3A++D\"oK0e*Fg[l$\"3i(***\\(=tY>%Fd[l7$$\"3A++v =5s#y*Fg[l$\"3#y***\\7)*ys@Fd[l7$FbzF(-Fgz6&FizF(FjzF(-%+AXESLABELSG6$ Q\"x6\"Q!6\"-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "secant('quad(g(x))'=1-x,x=0.5..0.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+dcbAh!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We can also use " }{TEXT 0 6 "fsolve " }{TEXT -1 36 " , which automatically uses Maple's " }{TEXT 0 9 "eval f/Int" }{TEXT -1 11 " procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "g := x -> Int(1/(t + exp(t)) , t=0..x);\nfsolve(g(x)=1-x,x =0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$*&\"\"\"F0,&%\"tG F0-%$expG6#F2F0!\"\"/F2;\"\"!9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+dcbAh!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Dis ection of a region into pieces with equal area" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 282 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "f(x) = cos(x^2" "6#/-%\"fG6#%\"xG-%$cosG6#*$F'\"\"#" }{TEXT -1 69 ", find the area of the region enclosed beween the graph of f and t he " }{TEXT 275 1 "x" }{TEXT -1 18 " axis between the " }{TEXT 276 1 " y" }{TEXT -1 64 " axis and the first positive intersection of the grap h with the " }{TEXT 277 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "(b) Find the value " } {XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 20 " such that the lin e " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 72 " divides the \+ region described in (a) into two pieces having equal areas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) find the va lues " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x = c" "6#/%\"xG%\"cG" }{TEXT -1 29 " such that the ver ical lines " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "x = c" "6#/%\"xG%\"cG" }{TEXT -1 82 " divide the reg ion decribed in (a) into three pieces which all have the same area." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := x -> cos(x^2);\nplot(f(x),x=0..1.5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$cosG6#*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 287 163 163 {PLOTDATA 2 "6%-%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$\"1+++DJdpK!#< $\"0kzgG%******!#:7$$\"1++v=7T9hF.$\"15)RV6I*****!#;7$$\"1++](=HPJ*F.$ \"1p]#GwB'****F77$$\"1++DJaU`7F7$\"1Xsc')ew)***F77$$\"1+]P%GZRd\"F7$\" 1?lPC;$p***F77$$\"1+](=276(=F7$\"1,939>(Q***F77$$\"1+](o**3)y@F7$\"1;1 ,@Tt))**F77$$\"1+](ofHq\\#F7$\"19Bi1xc!)**F77$$\"1+]Pf'HU\"GF7$\"1$R'H sSlo**F77$$\"1++]7*309$F7$\"1&Hc!R?S^**F77$$\"1++Dce*yU$F7$\"1ANUDE/J* *F77$$\"1++]([D9v$F7$\"1/yD\\f8,**F77$$\"1++]iNGwSF7$\"1&y(**R,Fi)*F77 $$\"1++]7XM*Q%F7$\"1?Rget(\\\")*F77$$\"1+](o%QjtYF7$\"1B;sZ=Ri(*F77$$ \"1++]i8o6]F7$\"1')*R/?Cio*F77$$\"1+++]>0)H&F7$\"1Y-4Aaj3'*F77$$\"1+]( =-p6j&F7$\"1A<8qYV,&*F77$$\"1+++vS.EfF7$\"1z5KG&y'*Q*F77$$\"1+](=xZ&\\ iF7$\"1U%4P$y#pC*F77$$\"1+]i:$4wb'F7$\"1181=Tc*3*F77$$\"1++v=#R!zoF7$ \"1!4&f$4\"4,*)F77$$\"1+]P4A@urF7$\"1PwmpyV/()F77$$\"1++Dchf#\\(F7$\"1 a/h!ff^Y)F77$$\"1+](of2L#yF7$\"1j$\\2\\oZ=)F77$$\"1+]7yG>6\")F7$\"1X8z $o\"p7zF77$$\"1++voo6A%)F7$\"1>&RJ0Y!)e(F77$$\"1*****\\xJLu)F7$\"1')p( ycsv@(F77$$\"1++v$*ydd!*F7$\"1uy$4i2$>oF77$$\"1+](=1\"F1$\"1*ReG;1iG%F77$$\"1++v Q(zS4\"F1$\"1XGsa[U^OF77$$\"1+v=-,FC6F1$\"1$4wQm?--$F77$$\"1+v$4tFe:\" F1$\"1,^4LX0FBF77$$\"1++D\"3\"o'=\"F1$\"1/!e#4+p=;F77$$\"1+voz;)*=7F1$ \"1PmE%o5yZ)F.7$$\"1+++&*44]7F1$\"1_.;gMuo!)!#=7$$\"1+]7jZ!>G\"F1$!1+! HX>/?C(F.7$$\"1+v=(4bMJ\"F1$!1[[c6hbP:F77$$\"1++]xlWU8F1$!1_a)R7yIH#F7 7$$\"1+]i&3ucP\"F1$!1z!plZN;;$F77$$\"1+++lJR09F1$!1WX%z-iS$RF77$$\"1+v =-*zqV\"F1$!1E$oTDc]u%F77$$\"1+D\"G:3uY\"F1$!1(*=$\\G`5]&F77$$\"1+++++ ++:F1$!1!RFsAO " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 36 "(a) The first intersection with the " }{TEXT 284 1 "x" }{TEXT -1 19 " axis occurs where " }{XPPEDIT 18 0 "x^2=Pi/2" "6#/*$%\" xG\"\"#*&%#PiG\"\"\"F&!\"\"" }{TEXT -1 17 ", that is, where " } {XPPEDIT 18 0 "x = sqrt(Pi/2)" "6#/%\"xG-%%sqrtG6#*&%#PiG\"\"\"\"\"#! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 58 "The area of the r egion enclosed between the graph and the " }{TEXT 272 1 "x" }{TEXT -1 11 " axis from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " \+ to " }{XPPEDIT 18 0 "x = sqrt(Pi/2)" "6#/%\"xG-%%sqrtG6#*&%#PiG\"\"\" \"\"#!\"\"" }{TEXT -1 10 " is . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "A := quad(Int(cos(x^2),x=0 ..sqrt(Pi/2)),15);\n#A := evalf(Int(cos(x^2),x=0..sqrt(Pi/2)),15);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"0I8HC9Xx*!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "(b) We set up a func tion g to give the area under the graph starting at the " }{TEXT 274 1 "y" }{TEXT -1 31 " axis, and going as far as the " }{TEXT 273 1 "x" }{TEXT -1 13 "-coordinate \"" }{TEXT 285 1 "x" }{TEXT -1 2 "\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g := x -> Int(cos(t^2),t=0..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$-%$cosG6#*$)%\"tG\" \"#\"\"\"/F4;\"\"!9$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "We need to find the value " }{XPPEDIT 18 0 " x \+ = a" "6#/%\"xG%\"aG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "g(a) = A/2 " "6#/-%\"gG6#%\"aG*&%\"AG\"\"\"\"\"#!\"\"" }{TEXT -1 6 ", and " } {TEXT 271 1 "A" }{TEXT -1 26 " is the area found in (a)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "a := evalf[15](secant('quad(g(x))'=A/2,x=0.4..0.6));\n#a := evalf[15](fsol ve('evalf(g(x))'=A/2,x=0.4..0.6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"aG$\"0F^)p!))e\"\\!#:" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We can check that the 2 integrals " }{XPPEDIT 18 0 " Int(cos(x^2),x = 0 .. a),Int(cos(x^2),x = a .. sqrt(Pi/2));" "6$-%$Int G6$-%$cosG6#*$%\"xG\"\"#/F*;\"\"!%\"aG-F$6$-F'6#*$F*F+/F*;F/-%%sqrtG6# *&%#PiG\"\"\"F+!\"\"" }{TEXT -1 12 " are equal." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "quad(Int(cos(x^2),x=0..a),15);\nquad(Int(cos(x ^2),x=a..sqrt(Pi/2)),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0lc97d s)[!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0lc97ds)[!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "(c) We need to fi nd the value " }{TEXT 266 5 "x = b" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "g(b) = A/3" "6#/-%\"gG6#%\"bG*&%\"AG\"\"\"\"\"$!\"\"" }{TEXT -1 5 " and " }{TEXT 267 5 "x = c" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "g (c) = 2/3*A" "6#/-%\"gG6#%\"cG*(\"\"#\"\"\"\"\"$!\"\"%\"AGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "b := evalf[15](secant('quad(g(x))'=A/3,x=0.2..0.4)); \nc := evalf[15](secant('quad(g(x))'=2*A/3,x=0.6..0.8));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"bG$\"0VIG/i=E$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"0bN!e@rWm!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We can check that the 3 integrals " } {XPPEDIT 18 0 "Int(cos(x^2),x=0..b), Int(cos(x^2),x=b..c),Int(cos(x^2) ,x=c..sqrt(Pi/2))" "6%-%$IntG6$-%$cosG6#*$%\"xG\"\"#/F*;\"\"!%\"bG-F$6 $-F'6#*$F*F+/F*;F/%\"cG-F$6$-F'6#*$F*F+/F*;F7-%%sqrtG6#*&%#PiG\"\"\"F+ !\"\"" }{TEXT -1 12 " are equal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "quad(Int(cos(x^2),x=0..b),1 5);\nquad(Int(cos(x^2),x=b..c),15);\nquad(Int(cos(x^2),x=c..sqrt(Pi/2) ),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0V/VTr\"eK!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0W/VTr\"eK!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0V/VTr\"eK!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 21 "An arc length example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 29 "The arc length alon g a curve " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 14 " from a point " }{XPPEDIT 18 0 "``(a,f(a))" "6#-%!G6$%\"aG-%\"f G6#F&" }{TEXT -1 25 " on the curve to a point " }{XPPEDIT 18 0 "``(b,f (b))" "6#-%!G6$%\"bG-%\"fG6#F&" }{TEXT -1 26 " is given by the integra l:" }}{PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Int(sqrt(1+( dy/dx)^2),x = a .. b);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$*&%#dyGF*%#d xG!\"\"\"\"#F*/%\"xG;%\"aG%\"bG" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 30 " provided that the derivative " }{XPPEDIT 18 0 "dy/dx;" " 6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 37 " exists throughout the interv al from " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 "For example, the length of the arc of a quarter of a circle of radius 1 is given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "f := x -> sqrt(1-x^2) :\n'f(x)'=f(x);\nInt(sqrt(1+D(f)(x)^2),x=0..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,&\"\"\"F**$)F'\"\"#F*!\"\"# F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*&%\"xG\" \"#,&F(F(*$)F*F+F(!\"\"F/F(#F(F+/F*;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now consider the function f giv en by " }{XPPEDIT 18 0 "f(x)=sin(x+x*sin(x))" "6#/-%\"fG6#%\"xG-%$sinG 6#,&F'\"\"\"*&F'F,-F)6#F'F,F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "(taken from: Maple V By Example, 2nd edition, Academic Pr ess, page 188)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "f := x -> sin(x+x*sin(x)):\n'f(x)'=f(x);\nplot (f(x),x=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$ sinG6#,&F'\"\"\"*&F'F,-F)F&F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 381 195 195 {PLOTDATA 2 "6%-%'CURVESG6$7fs7$$\"\"!F)F(7$$\"3i]cC&eb&p8!#=$\"3y L#=DIg-b\"F-7$$\"3q5)>63x`'>F-$\"3%4?&>5'=wK#F-7$$\"3YqR*pd)>hDF-$\"3' *p-7M$z^:$F-7$$\"3Y@%HT;i7B$F-$\"3^$[#R]4&)HTF-7$$\"3Zs[E^dK,RF-$\"33X lH)RD&G^F-7$$\"3+9+\"[&4$ed%F-$\"39)3!)*pA2HhF-7$$\"3ab^NehL]_F-$\"3!o R-#*3q34(F-7$$\"3Ls#3kbN;#fF-$\"3kDEDt#[A(zF-7$$\"35*QhW&\\$Hf'F-$\"3b +')RVz!*Q()F-7$$\"3]:5wGaJ:sF-$\"3LLs(RU$f8$*F-7$$\"3zS11.fpPyF-$\"38: tnWH6G(*F-7$$\"3'QF(yo`\"*f\")F-$\"3!HbTpH$er)*F-7$$\"3#e!R^M[8#[)F-$ \"3eV44t?vi**F-7$$\"3!=AxtcWKk)F-$\"3%>C!*4#ev*F-7$$\"3gu\\JE$HE,\"!#<$\"3+jw))*)z2^&*F-7$$\"3!Q Ra&4L&f/\"Fgq$\"3#)\\M'[JIaG*F-7$$\"3_y%f^`(Q76Fgq$\"3UQXy!>R:e)F-7$$ \"3YjXwg<#)y6Fgq$\"3Yix=30)[m(F-7$$\"3!or.w_drC\"Fgq$\"3OnS*otwO`'F-7$ $\"36qGW%H$\\:8Fgq$\"3+#G(Q/D^a_F-7$$\"3SH22vMov8Fgq$\"3#zd)4N.0WSF-7$ 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$\"3ojhgJa1,A!#A7$$\"3ns[!z6bes%Fgq$\"3c1(H:%[Fgq$\"3'*[)*\\Jz')HSFgdl7$$\"3;@9vt*Qh!\\Fgq$\"39zHc^R$p;*Fgd l7$$\"3Ou\"4%z!e2(\\Fgq$\"3)Gj)3b/UU;F-7$$\"3t:eW%H3%Q]Fgq$\"3!\\W'46o %Hi#F-7$$\"3*zX#[4&eg5&Fgq$\"3+))yirG>2QF-7$$\"3#Q_\")pq87<&Fgq$\"39\\ +#\\JKq4&F-7$$\"3n*e![/*ojB&Fgq$\"3+&G`[!GdhkF-7$$\"3Dtq\\/&**HI&Fgq$ \"3`G86(RPG#yF-7$$\"3#ob8X5I'p`Fgq$\"3]`%)3YY)[**)F-7$$\"3@)>%e&p#o**F-7$$\"37-jF= \"o_[&Fgq$\"3!RvV3%\\A****F-7$$\"3PA\"GX$yy,bFgq$\"3#e*fIu@>%)**F-7$$ \"3wxVy[u]ibFgq$\"3-aDy!eh!*[*F-7$$\"39L1/jqABcFgq$\"30#f(y=0ZD#)F-7$$ \"3CW>6GG-ecFgq$\"3!y`lOi)HPrF-7$$\"3WaK=$f=Gp&Fgq$\"3_*z[WA&>#z&F-7$$ \"3blXDeVhFdFgq$\"3eNvXOvd&eFgq$!3uEz9#=l&HGF-7$$\"3'oIe;6(*o)eFgq$!3sHk[7oTbXF-7 $$\"3(y8!=&Qz+#fFgq$!3k$oIjj\">biF-7$$\"3yp>qe;E`fFgq$!3=$egFgq$!3[Uyw2*f8m*F-7$$ \"3)y<21/1b.'Fgq$!3U93R&3kx()*F-7$$\"3)Rsoa(eQ^gFgq$!3Sq7m!H/p)**F-7$$ \"3e'\\**GzD$fgFgq$!2%>5.,Q)*****Fgq7$$\"31q-L5dEngFgq$!3gaN8!*Q)[)**F -7$$\"3oU5wFc?vgFgq$!3fw@$Q6>8%**F-7$$\"3G:=>Xb9$3'Fgq$!39EWyJ34p)*F-7 $$\"3Y2\\\"\\@0\\6'Fgq$!3i1I&H'ov#H*F-7$$\"3w)*zj%)[mYhFgq$!37Eldy]7m# )F-7$$\"3O\\d)e$*HP;'Fgq$!3[U29s!=\"RvF-7$$\"31*\\Lr)\\z!='Fgq$!3i'Fgq$!3IMH#))Rtuv&F-7$$\"3P***G'*3D\\@'Fgq$!3 NdCmzDLDZF-7$$\"3'*\\n(39!*>B'Fgq$!38W/l$3?rh$F-7$$\"3o*\\C@>b!\\iFgq$ !3QR9`%fX&[CF-7$$\"3Q\\APV-7miFgq$!3TvcKN:uO7F-7$$\"3)****>YH&=$G'Fgq$ !39)4:%*ezt9*!#D-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6 \"Q!Fcgm-%%VIEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The total arc length shown in t his graph can be computed by the following integral." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := x -> \+ sin(x+x*sin(x)):\nInt(sqrt(1+D(f)(x)^2),x=0..2*Pi);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*&)-%$cosG6#,&%\" xGF(*&F/F(-%$sinG6#F/F(F(\"\"#F(),(F(F(F1F(*&F/F(-F,F3F(F(F4F(F(#F(F4/ F/;\"\"!,$*&F4F(%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+9Gk07 !\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Maple 5's standard numerical procedure was unable to give a 20 digit \+ answer, but later versions eventually succeed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "f := x -> s in(x+x*sin(x)):\nInt(sqrt(1+D(f)(x)^2),x=0..2*Pi);\nevalf[20](Int(sqrt (1+D(f)(x)^2),x=0..2*Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*$,&\"\"\"F(*&)-%$cosG6#,&%\"xGF(*&F/F(-%$sinG6#F/F(F(\"\"#F(),(F(F(F 1F(*&F/F(-F,F3F(F(F4F(F(#F(F4/F/;\"\"!,$*&F4F(%#PiGF(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"5PX$4+W\"Gk07!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " Newton-Cotes fails . ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "evalf[20](Int(sqrt(1+D(f)(x)^2),x=0..2*Pi,method=_NCrule));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&$\"\"\"\"\"!F)*&)-%$cosG6 #,&%\"xGF)*&F1F)-%$sinG6#F1F)F)\"\"#F)),(F(F)F3F)*&F1F)-F.F5F)F)F6F)F) #F)F6/F1;$F*F*$\"5qZ'ezrI&=$G'!#>" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 99 "Clenshaw-Curtis now succeeds with later \+ versions of Maple, where it previously failed with Maple 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ev alf(Int(sqrt(1+D(f)(x)^2),x=0..2*Pi,20,_CCquad));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5PX$4+W\"Gk07!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 14 "Now let's try " }{TEXT 0 8 "quad/Int" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f := x -> sin(x+x*sin(x)):\nquad(Int(sqrt(1+D(f) (x)^2),x=0..2*Pi),20,method=GK,numpoints=31);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5PX$4+W\"Gk07!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 19 "Find the value of " } {XPPEDIT 18 0 "Int(``(sin(x^3)/x+2),x = 0 .. 4);" "6#-%$IntG6$-%!G6#,& *&-%$sinG6#*$%\"xG\"\"$\"\"\"F/!\"\"F1\"\"#F1/F/;\"\"!\"\"%" }{TEXT -1 190 ", to an accuracy of about 10 digits, by using (a) the Newton-C otes method, (b) the Gauss-Legendre method, (c) the Gauss-Kronrod meth od, and (d) the Clenshaw-Curtis method, via the procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 83 "Which meth od requires the least number of function evaluations in the default mo de?" }}{PARA 0 "" 0 "" {TEXT -1 38 "__________________________________ ____" }}{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 38 "______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 19 "Find the value o f " }{XPPEDIT 18 0 "Int(x^4*ln(x+sqrt(x^2+1)),x = 0 .. 2);" "6#-%$Int G6$*&%\"xG\"\"%-%#lnG6#,&F'\"\"\"-%%sqrtG6#,&*$F'\"\"#F-F-F-F-F-/F';\" \"!F3" }{TEXT -1 74 ", to an accuracy of about 50 digits, by using a s uitable numerical method." }}{PARA 0 "" 0 "" {TEXT -1 43 "The default \+ method for this integral using " }{TEXT 0 9 "evalf/Int" }{TEXT -1 34 " is essentially the same as using " }{TEXT 0 8 "quad/Int" }{TEXT -1 7 " with \"" }{TEXT 285 22 "method=Clenshaw_Curtis" }{TEXT -1 29 "\". Ca n you find a method via " }{TEXT 0 8 "quad/Int" }{TEXT -1 40 " which t akes less time than this method?" }}{PARA 0 "" 0 "" {TEXT -1 38 "_____ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "____________ __________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 24 "(a) The error function \"" }{TEXT 285 3 "erf" }{TEXT -1 17 "\" is defined by " }{XPPEDIT 18 0 "erf(x)= 2/sqrt(Pi)" "6#/-%$erf G6#%\"xG*&\"\"#\"\"\"-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-t^2),t = 0 .. x);" "6#-%$IntG6$-%$expG6#,$*$%\"tG\"\"#! \"\"/F+;\"\"!%\"xG" }{TEXT -1 21 ". Write a procedure " }{TEXT 0 5 "m yerf" }{TEXT -1 50 " using numerical integration to return its values. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "(b) P lot the graph of your version of \"" }{TEXT 285 3 "erf" }{TEXT -1 32 " \" over the interval from 0 to 2." }}{PARA 0 "" 0 "" {TEXT -1 44 "____ ________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "_________________________________ ___________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x) = Int(t^3/(t-arctan(t)),t=0..x)" "6#/-%\"fG6#%\"xG -%$IntG6$*&%\"tG\"\"$,&F,\"\"\"-%'arctanG6#F,!\"\"F3/F,;\"\"!F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Plot the graphs of \+ " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "y = exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 21 " \+ in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 43 "(b) Find the two \+ solutions of the equation " }{XPPEDIT 18 0 "f(x) = exp(x)" "6#/-%\"fG6 #%\"xG-%$expG6#F'" }{TEXT -1 22 " in the interval from " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 4" "6 #/%\"xG\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "_________ ___________________________________" }}{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 44 "_______________________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " } {XPPEDIT 18 0 "f(x) = 2*exp(-x^3)-1;" "6#/-%\"fG6#%\"xG,&*&\"\"#\"\"\" -%$expG6#,$*$F'\"\"$!\"\"F+F+F+F2" }{TEXT -1 69 ", find the area of th e region enclosed beween the graph of f and the " }{TEXT 279 1 "x" } {TEXT -1 18 " axis between the " }{TEXT 280 1 "y" }{TEXT -1 64 " axis \+ and the first positive intersection of the graph with the " }{TEXT 281 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "(b) Find the value " }{XPPEDIT 18 0 "x = \+ a" "6#/%\"xG%\"aG" }{TEXT -1 20 " such that the line " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 72 " divides the region described in (a) into two pieces having equal areas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "(c) find the values " }{XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = c " "6#/%\"xG%\"cG" }{TEXT -1 29 " such that the verical lines " } {XPPEDIT 18 0 "x = b" "6#/%\"xG%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = c" "6#/%\"xG%\"cG" }{TEXT -1 82 " divide the region decribed in (a) into three pieces which all have the same area." }}{PARA 0 "" 0 "" {TEXT -1 38 "______________________________________" }}{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 38 "______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 36 "Find the arc length along the curv e " }{XPPEDIT 18 0 "y =exp(-(x-2)^2*cos(Pi*x))" "6#/%\"yG-%$expG6#,$*& ,&%\"xG\"\"\"\"\"#!\"\"F--%$cosG6#*&%#PiGF,F+F,F,F." }{TEXT -1 7 " fr om " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x= 4" "6#/%\"xG\"\"%" }{TEXT -1 22 " correct to 15 digi ts." }}{PARA 0 "" 0 "" {TEXT 264 4 "Note" }{TEXT -1 42 ": The curve is symmetrical about the line " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "________________________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Int(ln(abs(x-7/10)),x=0..1)" "6#-%$IntG6$-%#lnG6#-%$absG6#,&%\"xG\" \"\"*&\"\"(F.\"#5!\"\"F2/F-;\"\"!F." }{TEXT -1 118 " by numerical inte gration correct to about 15 digits by splitting the interval of integr ation in a suitable way using " }{TEXT 0 8 "quad/Int" }{TEXT -1 17 " v ia the option \"" }{TEXT 285 8 "split=??" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 59 "Try both Gauss-Legendre and Gauss-Kronrod quadrat ure using " }{TEXT 0 8 "quad/Int" }{TEXT -1 18 " via the options \"" } {TEXT 285 21 "method=Gauss_Legendre" }{TEXT -1 7 "\" and \"" }{TEXT 285 20 "method=Gauss_Kronrod" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 58 "Check your values by evaluating the integral analytically ." }}{PARA 0 "" 0 "" {TEXT -1 44 "____________________________________ ________" }}{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 44 "____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }