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"Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Newton-Cotes integration" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 49 "load numerical integration procedures including: " } {TEXT 0 5 "NCint" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 278 6 "intg.m" }{TEXT -1 37 " contains the code for the procedur e " }{TEXT 0 5 "NCint" }{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 it s location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "read \"K:\\ \\Maple/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "load interpolation and function approximation pro cedures including: " }{TEXT 0 9 "interpoly" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 278 10 "fcnapprx.m" }{TEXT -1 37 " co ntains the code for the procedure " }{TEXT 0 9 "interpoly" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can \+ be read into a Maple session by a command similar to the one that foll ows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.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 roo t-finding procedures including: " }{TEXT 0 20 "bisect,newton,secant" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 278 7 "roots.m " }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 13 "b isect,newton" }{TEXT -1 5 " and " }{TEXT 0 6 "secant" }{TEXT -1 25 " u sed in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be rea d into a Maple session by a command similar to the one that follows, w here the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/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 41 "Introductio n to Newton-Cotes integration " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "This introduction does not presuppose a k nowledge of interpolating polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 38 "We consider the problem of estimating " }{XPPEDIT 18 0 "Int(sin(x) ,x=0..Pi/2)" "6#-%$IntG6$-%$sinG6#%\"xG/F);\"\"!*&%#PiG\"\"\"\"\"#!\" \"" }{TEXT -1 24 " by a numerical method. " }}{PARA 0 "" 0 "" {TEXT -1 44 "We construct a sequence of 5 equally spaced " }{TEXT 340 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[0]=0,x[1]=Pi/8,x[2]=Pi/4,x[3] =3*Pi/8" "6&/&%\"xG6#\"\"!F'/&F%6#\"\"\"*&%#PiGF+\"\")!\"\"/&F%6#\"\"# *&F-F+\"\"%F//&F%6#\"\"$*(F9F+F-F+F.F/" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[4]=Pi/2" "6#/&%\"xG6#\"\"%*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 30 " which subdivide the interval " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7$\" \"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 33 " into 4 intervals of equal w idth " }{XPPEDIT 18 0 "h=Pi/8" "6#/%\"hG*&%#PiG\"\"\"\"\")!\"\"" } {TEXT -1 33 ". Then we obtain 5 corresponding " }{TEXT 341 1 "y" } {TEXT -1 8 " values " }{XPPEDIT 18 0 "y[0]=sin(0),y[1]=sin(Pi/8),y[2]= sin(Pi/4),y[3]=sin(3*Pi/8)" "6&/&%\"yG6#\"\"!-%$sinG6#F'/&F%6#\"\"\"-F )6#*&%#PiGF.\"\")!\"\"/&F%6#\"\"#-F)6#*&F2F.\"\"%F4/&F%6#\"\"$-F)6#*(F @F.F2F.F3F4" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[4]=sin(Pi/2)" "6#/& %\"yG6#\"\"%-%$sinG6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 47 " (evaluate d to 14 digits), such that the points" }{XPPEDIT 18 0 "``(x[0],y[0]),` `(x[1],y[1]),``(x[2],y[2]),``(x[3],y[3]),``(x[4],y[4])" "6'-%!G6$&%\"x G6#\"\"!&%\"yG6#F)-F$6$&F'6#\"\"\"&F+6#F1-F$6$&F'6#\"\"#&F+6#F8-F$6$&F '6#\"\"$&F+6#F?-F$6$&F'6#\"\"%&F+6#FF" }{TEXT -1 21 " lie along the cu rve " }{XPPEDIT 18 0 "y=sin(x)" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 373 "Digits := 14:\n[seq(Pi/8*i,i=0..4)];\nxvals := evalf (%);\nyvals := map(sin,%);\npts := zip((x,y)->[x,y],xvals,yvals):\nlin es := zip((x,y)->[[x,0],[x,y]],xvals,yvals):\nDigits := 10:\np1 := plo t(sin(x),x=0..Pi/2):\np2 := plot([pts$3],style=point,color=black,\n \+ symbol=[circle,diamond,cross]):\np3 := plot(lines,color=COLOR(RG B,0,.7,0),linestyle=2):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!,$*&\"\")!\"\"%#PiG\"\"\"F*,$*&\"\"%F(F)F*F*, $*(\"\"$F*F'F(F)F*F*,$*&\"\"#F(F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7'$\"\"!F'$\"/s)p\"3*p#R!#9$\"/X(Rj\")R&yF*$\"/i4Xs4y6!#8$ \"/\\zEjzq:F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7'$\"\"!F'$ \"/4lBV$o#Q!#9$\"/b'=\"y1rqF*$\"/I6D`zQ#*F*$\"/++++++5!#8" }}{PARA 13 "" 1 "" {GLPLOT2D 411 268 268 {PLOTDATA 2 "6--%'CURVESG6$7S7$$\"\"!F)F (7$$\"3NGK5j*))QU$!#>$\"3QsLeI+ABMF-7$$\"3DXXYUk*HS'F-$\"3]DS+L@i)R'F- 7$$\"3=N68yVJ`(*F-$\"3_b&ehJeyt*F-7$$\"3%3h%eRSe78!#=$\"3<&Q%f]#=)38F= 7$$\"3k+,hQPB[;F=$\"3V-yj65yS;F=7$$\"3'H#*ed(RUf>F=$\"3q)*)ys&)4p%>F=7 $$\"3kHX[TMk\"G#F=$\"3G3RBS#)*=E#F=7$$\"3Saq%HF=$\"3=7dd?+e/HF=7$$\"3)pq'4OKt)G$F=$\"3c0jx$Rp(HKF=7 $$\"3#f/N#RTo*e$F=$\"3ML>uye38NF=7$$\"3Ut79wN[GRF=$\"3V3zj5M@GQF=7$$\" 3/VrOc6q%F=7$$\"31\\,oR/A[_F=$\"3SMGBr+f5]F=7$$\"3uvW/ X(*RG,:eF=7$$\"3.n3p46^WlF=$\"33g0zi&Qs3'F=7$$\"3oHyF.C6no F=$\"3h+)=j,t*RjF=7$$\"3(*zx.5Ir.sF=$\"3Qz3MFxj'f'F=7$$\"3'pz!QUu\"G^( F=$\"3$[8q$y.wDoF=7$$\"3A\"3@7LGi%yF=$\"3wm47uKelqF=7$$\"3)3\"HIT&[D>) F=$\"3M(RvZssjI(F=7$$\"3gv)\\AI@S\\)F=$\"3C>#p\"po&)3vF=7$$\"3vn()=V,i >))F=$\"3%4zeG%\\()>xF=7$$\"3'G:[dg&*f:*F=$\"31()\\)y]!GHzF=7$$\"3sSt6 rL2&[*F=$\"3g&H%*=Uja7)F=7$$\"3mka(*HIZ.)*F=$\"3%3R7+w2pI)F=7$$\"3Em\" [l:+d,\"!#<$\"3Q?Zs+w\\)\\)F=7$$\"3c!3XyEmu/\"F_u$\"3Ih!)R3tfh')F=7$$ \"3sJ_*>P$Q\"3\"F_u$\"3VZPF_)*3E))F=7$$\"3.M\"G%4t676F_u$\"3kW#3Hj\"Qm *)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3Y(Q,up+vN*F=7$$\"34Q$[)>&*oU7F_u$\"3OZ!) zP7am%*F=7$$\"3I^lOFY^w7F_u$\"3-w%)pSt5q&*F=7$$\"3`!)f6EA448F_u$\"3-KP DS[]f'*F=7$$\"3=;T7EvSU8F_u$\"3B\"='y#[C.u*F=7$$\"3a_wiepWv8F_u$\"3$oY $>Q\"*z4)*F=7$$\"3U$obdw1eS\"F_u$\"3#4Wi\"*p+U')*F=7$$\"3s$)o#3`-1W\"F _u$\"3aZ!3__n`\"**F=7$$\"3-#*)4zFC( )p\"3*p#RF=$\"3/+4lBV$o#QF=7$$\"3c+X(Rj\")R&yF=$\"3;+b'=\"y1rqF=7$$\"3 %**>'4Xs4y6F_u$\"3]**H6D`zQ#*F=7$$\"37+\\zEjzq:F_uFez-%'SYMBOLG6#%'CIR CLEG-Fhz6&FjzF)F)F)-%&STYLEG6#%&POINTG-F$6&F`[l-Fd\\l6#%(DIAMONDGFg\\l Fi\\l-F$6&F`[l-Fd\\l6#%&CROSSGFg\\lFi\\l-F$6%7$F'F'-%&COLORG6&FjzF($\" \"(F][lF(-%*LINESTYLEG6#\"\"#-F$6%7$7$Fb[lF(Fa[lFj]lF_^l-F$6%7$7$Fg[lF (Ff[lFj]lF_^l-F$6%7$7$F\\\\lF(F[\\lFj]lF_^l-F$6%7$7$Fa\\lF(F`\\lFj]lF_ ^l-%+AXESLABELSG6%Q\"x6\"Q!Fg_l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"+Fj zq:!\"*F\\`l" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We can c onstruct a quartic polynomial " }{XPPEDIT 18 0 "p(x)=a*x^4+b*x^3+c*x^2 +d*x+c" "6#/-%\"pG6#%\"xG,,*&%\"aG\"\"\"*$F'\"\"%F+F+*&%\"bGF+*$F'\"\" $F+F+*&%\"cGF+*$F'\"\"#F+F+*&%\"dGF+F'F+F+F3F+" }{TEXT -1 33 " which p asses throught the points" }{XPPEDIT 18 0 "``(x[0],y[0])" "6#-%!G6$&% \"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(x[1],y[1])" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``( x[2],y[2])" "6#-%!G6$&%\"xG6#\"\"#&%\"yG6#F)" }{TEXT -1 1 "," } {XPPEDIT 18 0 "``(x[3],y[3])" "6#-%!G6$&%\"xG6#\"\"$&%\"yG6#F)" } {TEXT -1 1 "," }{XPPEDIT 18 0 "``(x[4],y[4])" "6#-%!G6$&%\"xG6#\"\"%&% \"yG6#F)" }{TEXT -1 38 " by solving the system of 5 equations " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([p(x[0])=y[ 0],``],[p(x[1])=y[1],``],[p(x[2])=y[2],``],[p(x[3])=y[3],``],[p(x[4])= y[4],``])" "6#-%*PIECEWISEG6'7$/-%\"pG6#&%\"xG6#\"\"!&%\"yG6#F.%!G7$/- F)6#&F,6#\"\"\"&F06#F9F27$/-F)6#&F,6#\"\"#&F06#FBF27$/-F)6#&F,6#\"\"$& F06#FKF27$/-F)6#&F,6#\"\"%&F06#FTF2" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 31 "for the 5 unknown coefficients " }{XPPEDIT 18 0 "a,b,c, d,e" "6'%\"aG%\"bG%\"cG%\"dG%\"eG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "unassign('p ','a','b','c','d','e');\np := x->a*x^4+b*x^3+c*x^2+d*x+e;\nDigits := 1 8:\neqns := \{seq(p(xvals[i])=yvals[i],i=1..5)\}:\nsols := solve(eqns, \{a,b,c,d,e\});\nassign(sols);\np := unapply(evalf[14](p(x)),x):\n'p(x )'=p(x);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,,*&%\"aG\"\"\")9$\"\"%F/F/*&%\"bGF/)F1 \"\"$F/F/*&%\"cGF/)F1\"\"#F/F/*&%\"dGF/F1F/F/%\"eGF/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<'/%\"eG$\"\"!F)/%\"cG$\"35d>Q]H9&*> !#>/%\"dG$\"3$4-PhypJ'**!#=/%\"bG$!3:S@'ygae.#F3/%\"aG$\"3pra,BDUrGF. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,**&$\"/b,BDUrG!#:\" \"\")F'\"\"%F-F-*&$\"/@'ygae.#!#9F-)F'\"\"$F-!\"\"*&$\"/?Q]H9&*>F,F-)F '\"\"#F-F-*&$\"/q8'ypJ'**F3F-F'F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Alternatively, the Maple procedure " } {TEXT 0 6 "interp" }{TEXT -1 31 " can be used to construct this " } {TEXT 338 24 "interpolating polynomial" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf[1 4](evalf[18](interp(xvals,yvals,x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&$\"/b,BDUrG!#:\"\"\")%\"xG\"\"%F(F(*&$\"/@'ygae.#!#9F()F*\"\"$F (!\"\"*&$\"/?Q]H9&*>F'F()F*\"\"#F(F(*&$\"/q8'ypJ'**F/F(F*F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "y=p(x)" "6#/%\"yG-%\"pG6#%\"xG" }{TEXT -1 46 " is v isually indistinguishable from the curve " }{XPPEDIT 18 0 "y=sin(x)" " 6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "p1 := plot(p(x),x=0..Pi/2,color=magenta):\np2 := plot([pts$3],sty le=point,color=black,\n symbol=[circle,diamond,cross]):\np3 := plot(lines,color=COLOR(RGB,0,.7,0),linestyle=2):\nplots[display]([p1, p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 397 259 259 {PLOTDATA 2 "6--%'C URVESG6$7S7$$\"\"!F)F(7$$\"3NGK5j*))QU$!#>$\"3-B\"*o/W!GT$F-7$$\"3DXXY Uk*HS'F-$\"3;a;oBxH#Q'F-7$$\"3=N68yVJ`(*F-$\"3&[>.\"=Iu<(*F-7$$\"3%3h% eRSe78!#=$\"3FMq**e%omI\"F=7$$\"3k+,hQPB[;F=$\"3&)4^(oNz'Q;F=7$$\"3'H# *ed(RUf>F=$\"3RPU=Y^(\\%>F=7$$\"3kHX[TMk\"G#F=$\"3]+B=@HAgAF=7$$\"3Saq%HF=$\"3kT:h^be.HF=7$$\"3)pq' 4OKt)G$F=$\"3,\"p&\\zI9HKF=7$$\"3#f/N#RTo*e$F=$\"3tp8'*[&pF^$F=7$$\"3U t79wN[GRF=$\"3kR!G(=Z@GQF=7$$\"3/VL6]F=7$$\"3uvW/7'o_F=7$$\"3]W`F)RY p*eF=$\"3Gvc()\\g&=c&F=7$$\"33'3B#f$Gd?'F=$\"3f#>Q5YYd\"eF=7$$\"3.n3p4 6^WlF=$\"3w;7WqY(y3'F=7$$\"3oHyF.C6noF=$\"3$fGJ/&>[SjF=7$$\"3(*zx.5Ir. sF=$\"3Ie[!R7()pf'F=7$$\"3'pz!QUu\"G^(F=$\"3pdH)F=$\"3pCF**y\"*=1tF=7$$\"3gv)\\A I@S\\)F=$\"3>]B!)[*>&3vF=7$$\"3vn()=V,i>))F=$\"3xaB5c7R>xF=7$$\"3'G:[d g&*f:*F=$\"3*p,Lw-u'GzF=7$$\"3sSt6rL2&[*F=$\"3Z=bjCDxC\")F=7$$\"3mka(* HIZ.)*F=$\"3-G!4*QX<1$)F=7$$\"3Em\"[l:+d,\"!#<$\"3G6@]@yw(\\)F=7$$\"3c !3XyEmu/\"F_u$\"3oHA]f(=4m)F=7$$\"3sJ_*>P$Q\"3\"F_u$\"3whZ9nt^D))F=7$$ \"3.M\"G%4t676F_u$\"3Y'3`w6]f'*)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3-#3C &=Dwd$*F=7$$\"34Q$[)>&*oU7F_u$\"3W%o9aL&3n%*F=7$$\"3I^lOFY^w7F_u$\"3Uw )=dPa4d*F=7$$\"3`!)f6EA448F_u$\"3kE@#*pRjg'*F=7$$\"3=;T7EvSU8F_u$\"3k \\DS!z6()p\"3*p#RF=$\"3/+4 lBV$o#QF=7$$\"3c+X(Rj\")R&yF=$\"3;+b'=\"y1rqF=7$$\"3%**>'4Xs4y6F_u$\"3 ]**H6D`zQ#*F=7$$\"37+\\zEjzq:F_u$\"\"\"F)-%'SYMBOLG6#%'CIRCLEG-Fhz6&Fj zF)F)F)-%&STYLEG6#%&POINTG-F$6&F`[l-Ff\\l6#%(DIAMONDGFi\\lF[]l-F$6&F`[ l-Ff\\l6#%&CROSSGFi\\lF[]l-F$6%7$F'F'-%&COLORG6&FjzF($\"\"(!\"\"F(-%*L INESTYLEG6#\"\"#-F$6%7$7$Fb[lF(Fa[lF\\^lFb^l-F$6%7$7$Fg[lF(Ff[lF\\^lFb ^l-F$6%7$7$F\\\\lF(F[\\lF\\^lFb^l-F$6%7$7$Fa\\lF(F`\\lF\\^lFb^l-%+AXES LABELSG6%Q\"x6\"Q!Fj_l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"+Fjzq:!\"*F_ `l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Graphing over a wi der interval shows that the two curves eventually diverge away from ea ch other. 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3WLbPFx'[8#FA7$$\"3%3h%eRSe78FY$!3)H!4N(f\"z\\@FA7$$\"3#pzi>DSXN\"FY$! 3!*R)*oJV#G:#FA7$$\"3I$)4Mkk\\'R\"FY$!3)zv;^\\uI:#FA7$$\"3lp\">nn_%Q9F Y$!3O2\")HcEk]@FA7$$\"3ubt4*))3/[\"FY$!3mj:wzwiX@FA7$$\"3?GP&QJ@Vc\"FY $!3S,=*Q5>0.#FA7$$\"3'H#*ed(RUf>FY$!3?78m%46Z$>FA7$$\"3kHX[TMk\"G#FY$!31#y 2;0>`n\"FA7$$\"3Saq%HFY$!34taqT '*oW**F07$$\"3)pq'4OKt)G$FY$!3-$[Xh!G9jiF07$$\"3#f/N#RTo*e$FY$!3Cyhj0y HjJF07$$\"3Ut79wN[GRFY$\"3^oM@J,438!#C7$$\"3/72&FY$\"3c+B)[Iw]'pF07$$\"31\\,oR/A[_FY$\"3\\WMy]=[UuF07$$ \"3%HJi$GP;)R&FY$\"3WPMR.kD>xF07$$\"3uvW/_B'oJNcFY$\"3*473JE\"=CzF07$$\"3c4*fwqEDs&FY$\"3h?*)el$Q+$zF07$$\"3 .Fw'HbO(4eFY$\"3)Hq%)3\"[!)**yF07$$\"3]W`F)RYp*eFY$\"35#f![\"))>W$yF07 $$\"3G:#\\(ytL^gFY$\"3DXlY9#ecj(F07$$\"33'3B#f$Gd?'FY$\"3t4DtO1@OtF07$ $\"3.n3p46^WlFY$\"3e?pc1l2hjF07$$\"3oHyF.C6noFY$\"3#G:`[7T$*3&F07$$\"3 (*zx.5Ir.sFY$\"3O4#*yRc'R\\$F07$$\"3'pz!QUu\"G^(FY$\"3%fhG#G!Q/(=F07$$ \"3A\"3@7LGi%yFY$\"3qwnQyJYnUF[t7$$\"3)3\"HIT&[D>)FY$!3)3cEn#yXN=F07$$ \"3gv)\\AI@S\\)FY$!3oM0poO?pLF07$$\"3vn()=V,i>))FY$!3[YD![1'F07$$\"3sSt6rL2&[*FY$!3#[Mrxes*3pF07$$\"3p- ka+KFW'*FY$!3!>ArLrs><(F07$$\"3mka(*HIZ.)*FY$!3BW#GO.6AL(F07$$\"3gj&Gx HP-)**FY$!3QLp+bu#RQ(F07$$\"3Em\"[l:+d,\"!#<$!3)y$34EAz(H(F07$$\"3IBm> 7KeJ5Fa\\l$!3Sl0\"*[Am*4(F07$$\"3c!3XyEmu/\"Fa\\l$!3O@jJe*)[&y'F07$$\" 3-c,#*>[Uk5Fa\\l$!3z:2\\&f(H@jF07$$\"3sJ_*>P$Q\"3\"Fa\\l$!3Zpm&)*G^[s& F07$$\"3.M\"G%4t676Fa\\l$!3+e;e^D::VF07$$\"3co*Q4iD&) o=BF07$$\"3[cr`&*GLx6Fa\\l$!3M;.qJ,2keF[t7$$\"3Hk&>q'*z.@\"Fa\\l$\"3R@ d%pA6#=EF07$$\"34Q$[)>&*oU7Fa\\l$\"3#)e4Pmh(4W&F07$$\"3I^lOFY^w7Fa\\l$ \"3u&*R+/-Nq%)F07$$\"3`!)f6EA448Fa\\l$\"3#Ri%Ro'H\"H6FA7$$\"3=;T7EvSU8 Fa\\l$\"310%ojh2tQ\"FA7$$\"3a_wiepWv8Fa\\l$\"3k*Gs'f6Y*e\"FA7$$\"3'zm \">ioi!R\"Fa\\l$\"3u.Z))3J/d;FA7$$\"3U$obdw1eS\"Fa\\l$\"3J;P/z)zZq\"FA 7$$\"3W$[Bqq0XT\"Fa\\l$\"3nDo7w:.AV\"Fa\\l$\"3L Q#)z90kJgIDO9Fa\\l$\"3tR)[DP3%GX:Fa\\l$\"3**= %)y*>=(GjF07$$\"3C/BSs7t`:Fa\\l$\"36'[`sF.OT%F07$$\"3+_h-)zjAc\"Fa\\l$ \"3b(eg(\\p#oI#F07$$\"3+++lBjzq:Fa\\l$\"3UH7`&=hm*))!#H-%'COLOURG6&%$R GBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F[gl-%%VIEWG6$;F($\"+Fjz q:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 78 "An estimate of the maximum absolute error can be obtained using the procedure " }{TEXT 0 7 "infnorm" }{TEXT -1 8 " in the " } {TEXT 0 9 "numapprox" }{TEXT -1 10 " package. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "numapprox[in fnorm](p(x)-sin(x),x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ %*>I`@!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The integral " } {XPPEDIT 18 0 "Int(sin(x),x=0..Pi/2)" "6#-%$IntG6$-%$sinG6#%\"xG/F);\" \"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 33 " can be estimated by calcula ting " }{XPPEDIT 18 0 "Int(p(x),x=0..Pi/2)" "6#-%$IntG6$-%\"pG6#%\"xG/ F);\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 28 " be the indefinite integral " }{XPPEDIT 18 0 "Int(p(x),x)" "6#-%$I ntG6$-%\"pG6#%\"xGF)" }{TEXT -1 49 " with the constant of integration \+ taken to be 0. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 " Int(p(x),x=0..Pi/2)=q(Pi/2)" "6#/-%$IntG6$-%\"pG6#%\"xG/F*;\"\"!*&%#Pi G\"\"\"\"\"#!\"\"-%\"qG6#*&F/F0F1F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "q := unapp ly(evalf[14](int(p(x),x)),x):\n'q(x)'=q(x);\nevalf[10](evalf[14](q(Pi/ 2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,**&$\"/5.Y]%Gu &!#;\"\"\")F'\"\"&F-F-*&$\"/_l>lj*3&!#:F-)F'\"\"%F-!\"\"*&$\"/LF,lZ]mF ,F-)F'\"\"$F-F-*&$\"/&oI*[e\")\\!#9F-)F'\"\"#F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bc\"*****!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "This value can be calculated using just the " }{TEXT 342 1 "y" } {TEXT -1 13 " coordinates " }{XPPEDIT 18 0 "y[0],y[1],y[2],y[3],y[4]" "6'&%\"yG6#\"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 53 " of the interpolation points, and the interval width " }{TEXT 343 1 "h" }{TEXT -1 18 ", via the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(7*y[0]+32*y[1]+12*y[2]+32*y[3]+7*y[4])" "6#-% !G6#,,*&\"\"(\"\"\"&%\"yG6#\"\"!F)F)*&\"#KF)&F+6#F)F)F)*&\"#7F)&F+6#\" \"#F)F)*&F/F)&F+6#\"\"$F)F)*&F(F)&F+6#\"\"%F)F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "2*h/45" "6#*(\"\"#\"\"\"%\"hGF%\"#X!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(7*(y[0]+ y[4])+32*(y[1]+y[3])+12*y[2])" "6#/%!G-F$6#,(*&\"\"(\"\"\",&&%\"yG6#\" \"!F*&F-6#\"\"%F*F*F**&\"#KF*,&&F-6#F*F*&F-6#\"\"$F*F*F**&\"#7F*&F-6# \"\"#F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*h/45" "6#*(\"\"#\"\"\"%\" hGF%\"#X!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "This formula (called " }{TEXT 338 11 "Bod e's Rule" }{TEXT -1 167 ") can be regarded as being similar to Simpson 's rule but, instead of using quadratic polynomials, it is based on fi tting a quartic polynomial curve through five points" }{XPPEDIT 18 0 " ``(x[0],y[0])" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 1 "," } {XPPEDIT 18 0 "``(x[1],y[1])" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" } {TEXT -1 1 "," }{XPPEDIT 18 0 "``(x[2],y[2])" "6#-%!G6$&%\"xG6#\"\"#&% \"yG6#F)" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(x[3],y[3])" "6#-%!G6$&%\" xG6#\"\"$&%\"yG6#F)" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(x[4],y[4])" "6 #-%!G6$&%\"xG6#\"\"%&%\"yG6#F)" }{TEXT -1 15 " along a curve " } {XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 21 " with equ ally spaced " }{TEXT 344 1 "x" }{TEXT -1 13 " coordinates " }{XPPEDIT 18 0 "x[0],x[1],x[2],x[3],x[4]" "6'&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"#& F$6#\"\"$&F$6#\"\"%" }{TEXT -1 35 " which subdivide the interval from \+ " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x[4]" "6#&%\"xG6#\"\"%" }{TEXT -1 33 " into 4 intervals of equal width " }{TEXT 345 1 "h" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "It is derived (using Maple) in the next section. " }} {PARA 0 "" 0 "" {TEXT -1 50 "Note that since we are using a Maple list for the " }{TEXT 347 1 "y" }{TEXT -1 67 " values the indices are all \+ increased by 1 in the following code. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "h := evalf[14](Pi/8): \nevalf(evalf[14]((7*(yvals[1]+yvals[5])+32*(yvals[2]+yvals[4])+12*yva ls[3])*2*h/45));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bc\"*****!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The pr ocedure " }{TEXT 0 5 "NCint" }{TEXT -1 50 " (see below) can be used to make this calculation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "NCint(sin(x),x=0..Pi/2,adaptive=fal se,numpoints=5,factor=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bc\"* ****!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "More generally we c an obtain integration rules based on using interpolating polynomials o f any degree. Such rules are called " }{TEXT 338 12 "Newton-Cotes" } {TEXT -1 20 " integration rules. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "Newton-Cotes rules can be used repeatedly . For example, a more accurate estimate of the integral " }{XPPEDIT 18 0 "Int(sin(x),x=0..Pi/2)" "6#-%$IntG6$-%$sinG6#%\"xG/F);\"\"!*&%#Pi G\"\"\"\"\"#!\"\"" }{TEXT -1 26 " can be obtained by using " }{TEXT 338 17 "Bode's rule twice" }{TEXT -1 23 ": once on the interval " } {XPPEDIT 18 0 "[0,Pi/4]" "6#7$\"\"!*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 44 " subdivided into 4 intervals of equal width " }{XPPEDIT 18 0 "Pi/1 6" "6#*&%#PiG\"\"\"\"#;!\"\"" }{TEXT -1 27 ", and once on the interval " }{XPPEDIT 18 0 "[Pi/4,Pi/2]" "6#7$*&%#PiG\"\"\"\"\"%!\"\"*&F%F&\"\" #F(" }{TEXT -1 43 ". The two results are then added together. " }} {PARA 0 "" 0 "" {TEXT -1 38 "The appropriate formula involving the " } {TEXT 346 1 "y" }{TEXT -1 9 " values: " }{XPPEDIT 18 0 "y[0],y[1],` . \+ . . `,y[8]" "6&&%\"yG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#\"\")" }{TEXT -1 5 " is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(7*( y[0]+y[8])+32*(y[1]+y[3]+y[5]+y[7])+12*(y[2]+y[6])+14*y[4]);" "6#-%!G6 #,**&\"\"(\"\"\",&&%\"yG6#\"\"!F)&F,6#\"\")F)F)F)*&\"#KF),*&F,6#F)F)&F ,6#\"\"$F)&F,6#\"\"&F)&F,6#F(F)F)F)*&\"#7F),&&F,6#\"\"#F)&F,6#\"\"'F)F )F)*&\"#9F)&F,6#\"\"%F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*h/45" "6# *(\"\"#\"\"\"%\"hGF%\"#X!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 374 "Digits := 1 4:\n[seq(Pi/16*i,i=0..8)];\nxvals := evalf(%);\nyvals := map(sin,%);\n pts := zip((x,y)->[x,y],xvals,yvals):\nlines := zip((x,y)->[[x,0],[x,y ]],xvals,yvals):\nDigits := 10:\np1 := plot(sin(x),x=0..Pi/2):\np2 := \+ plot([pts$3],style=point,color=black,\n symbol=[circle,diamond ,cross]):\np3 := plot(lines,color=COLOR(RGB,0,.7,0),linestyle=2):\nplo ts[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"!,$* &\"#;!\"\"%#PiG\"\"\"F*,$*&\"\")F(F)F*F*,$*(\"\"$F*F'F(F)F*F*,$*&\"\"% F(F)F*F*,$*(\"\"&F*F'F(F)F*F*,$*(F0F*F-F(F)F*F*,$*(\"\"(F*F'F(F)F*F*,$ *&\"\"#F(F)F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7+$\"\"!F' $\"/O\\3a\\j>!#9$\"/s)p\"3*p#RF*$\"/4[Di[!*eF*$\"/X(Rj\")R&yF*$\"/\"oC /xu\")*F*$\"/i4Xs4y6!#8$\"/b%fyYWP\"F5$\"/\\zEjzq:F5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7+$\"\"!F'$\"/8;?K!4&>!#9$\"/4lBV$o#QF*$\"/ h>IBqbbF*$\"/b'=\"y1rqF*$\"/b-Bhp9$)F*$\"/I6D`zQ#*F*$\"/A./G&y!)*F*$\" /++++++5!#8" }}{PARA 13 "" 1 "" {GLPLOT2D 317 218 218 {PLOTDATA 2 "61- %'CURVESG6$7S7$$\"\"!F)F(7$$\"3NGK5j*))QU$!#>$\"3QsLeI+ABMF-7$$\"3DXXY Uk*HS'F-$\"3]DS+L@i)R'F-7$$\"3=N68yVJ`(*F-$\"3_b&ehJeyt*F-7$$\"3%3h%eR Se78!#=$\"3<&Q%f]#=)38F=7$$\"3k+,hQPB[;F=$\"3V-yj65yS;F=7$$\"3'H#*ed(R Uf>F=$\"3q)*)ys&)4p%>F=7$$\"3kHX[TMk\"G#F=$\"3G3RBS#)*=E#F=7$$\"3Saq%HF=$\"3=7dd?+e/HF=7$$\"3)pq'4O Kt)G$F=$\"3c0jx$Rp(HKF=7$$\"3#f/N#RTo*e$F=$\"3ML>uye38NF=7$$\"3Ut79wN[ GRF=$\"3V3zj5M@GQF=7$$\"3/VrOc6q%F=7$$\"31\\,oR/A[_F=$ \"3SMGBr+f5]F=7$$\"3uvW/X(*RG,:eF=7$$\"3.n3p46^WlF=$\"33g0 zi&Qs3'F=7$$\"3oHyF.C6noF=$\"3h+)=j,t*RjF=7$$\"3(*zx.5Ir.sF=$\"3Qz3MFx j'f'F=7$$\"3'pz!QUu\"G^(F=$\"3$[8q$y.wDoF=7$$\"3A\"3@7LGi%yF=$\"3wm47u KelqF=7$$\"3)3\"HIT&[D>)F=$\"3M(RvZssjI(F=7$$\"3gv)\\AI@S\\)F=$\"3C>#p \"po&)3vF=7$$\"3vn()=V,i>))F=$\"3%4zeG%\\()>xF=7$$\"3'G:[dg&*f:*F=$\"3 1()\\)y]!GHzF=7$$\"3sSt6rL2&[*F=$\"3g&H%*=Uja7)F=7$$\"3mka(*HIZ.)*F=$ \"3%3R7+w2pI)F=7$$\"3Em\"[l:+d,\"!#<$\"3Q?Zs+w\\)\\)F=7$$\"3c!3XyEmu/ \"F_u$\"3Ih!)R3tfh')F=7$$\"3sJ_*>P$Q\"3\"F_u$\"3VZPF_)*3E))F=7$$\"3.M \"G%4t676F_u$\"3kW#3Hj\"Qm*)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3Y(Q,up+vN*F=7 $$\"34Q$[)>&*oU7F_u$\"3OZ!)zP7am%*F=7$$\"3I^lOFY^w7F_u$\"3-w%)pSt5q&*F =7$$\"3`!)f6EA448F_u$\"3-KPDS[]f'*F=7$$\"3=;T7EvSU8F_u$\"3B\"='y#[C.u* F=7$$\"3a_wiepWv8F_u$\"3$oY$>Q\"*z4)*F=7$$\"3U$obdw1eS\"F_u$\"3#4Wi\"* p+U')*F=7$$\"3s$)o#3`-1W\"F_u$\"3aZ!3__n`\"**F=7$$\"3-#*)4zFCF=$\"3)**Hh,A.4&>F=7$$\"3w*>()p\" 3*p#RF=$\"3/+4lBV$o#QF=7$$\"3%)**3[Di[!*eF=$\"3]*4'>IBqbbF=7$$\"3c+X(R j\")R&yF=$\"3;+b'=\"y1rqF=7$$\"3;+\"oC/xu\")*F=$\"35+b-Bhp9$)F=7$$\"3% **>'4Xs4y6F_u$\"3]**H6D`zQ#*F=7$$\"31+b%fyYWP\"F_u$\"3%**>KS!G&y!)*F=7 $$\"37+\\zEjzq:F_uFez-%'SYMBOLG6#%'CIRCLEG-Fhz6&FjzF)F)F)-%&STYLEG6#%& POINTG-F$6&F`[l-Fh]l6#%(DIAMONDGF[^lF]^l-F$6&F`[l-Fh]l6#%&CROSSGF[^lF] ^l-F$6%7$F'F'-%&COLORG6&FjzF($\"\"(F][lF(-%*LINESTYLEG6#\"\"#-F$6%7$7$ Fb[lF(Fa[lF^_lFc_l-F$6%7$7$Fg[lF(Ff[lF^_lFc_l-F$6%7$7$F\\\\lF(F[\\lF^_ lFc_l-F$6%7$7$Fa\\lF(F`\\lF^_lFc_l-F$6%7$7$Ff\\lF(Fe\\lF^_lFc_l-F$6%7$ 7$F[]lF(Fj\\lF^_lFc_l-F$6%7$7$F`]lF(F_]lF^_lFc_l-F$6%7$7$Fe]lF(Fd]lF^_ lFc_l-%+AXESLABELSG6%Q\"x6\"Q!F[bl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\" +Fjzq:!\"*F`bl" 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" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "h := evalf[14](Pi/16):\nevalf(evalf[14]((7*(yvals[1]+yvals[9])+32 *(yvals[2]+yvals[4]+yvals[6]+yvals[8])+\n 12*(yvals[3]+yvals[7])+14*yv als[5])*2*h/45));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i()******!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 5 "NCint" }{TEXT -1 50 " (see below) can be used \+ to make this calculation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "NCint(sin(x),x=0..Pi/2,adaptive=fal se,numpoints=5,factor=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i()** ****!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Constructi ng Newton-Cotes integration formulas" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 84 "Consider the proble m of computing a numerical approximation to the definite integral" }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) ;" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Subdivide the interval from " }{TEXT 311 1 "a" }{TEXT -1 4 " to " }{TEXT 312 1 "b" }{TEXT -1 6 " into " }{TEXT 313 1 "m" }{TEXT -1 30 " subintervals of equal length " }{XPPEDIT 18 0 "h = \+ (b-a)/m;" "6#/%\"hG*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"mGF*" }{TEXT -1 4 " b y " }{TEXT 319 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "a = x[0],x [1],` . . . `,x[m] = b;" "6&/%\"aG&%\"xG6#\"\"!&F&6#\"\"\"%(~.~.~.~G/& F&6#%\"mG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "The " }{XPPEDIT 18 0 "m+1" "6#,&%\"mG\"\"\"F%F%" }{TEXT -1 20 " subdividing \+ points " }{XPPEDIT 18 0 "x[0],x[1],` . . . `,x[m];" "6&&%\"xG6#\"\"!&F $6#\"\"\"%(~.~.~.~G&F$6#%\"mG" }{TEXT -1 49 " or endpoints of the subi ntervals are called the " }{TEXT 261 5 "nodes" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 8 "Now let " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6 #%\"xG" }{TEXT -1 59 " be the unique interpolation polynomial of degre e at most " }{TEXT 314 1 "m" }{TEXT -1 20 " through the points" } {XPPEDIT 18 0 "``(x[0],y[0]),``(x[1],y[1]),` . . . `,``(x[3],y[3]);" " 6&-%!G6$&%\"xG6#\"\"!&%\"yG6#F)-F$6$&F'6#\"\"\"&F+6#F1%(~.~.~.~G-F$6$& F'6#\"\"$&F+6#F9" }{TEXT -1 15 " on the graph " }{XPPEDIT 18 0 " y=f( x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 50 " corresponding to the nodes. Then the integral of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 42 " is our approximation to the integral of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b);" "6#-%$IntG6$-% \"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 " " }{TEXT 310 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(p(x),x = a .. b);" "6#-%$IntG6$-%\"pG6#% \"xG/F);%\"aG%\"bG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "For example, we can obtain Simpson's Rule as follows, using the Maple pro cedure " }{TEXT 0 6 "interp" }{TEXT -1 41 " to provide the interpolati ng polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "unassign('a','h','f'):\nxvals := [a,a+h,a+2*h]; \nyvals := map(f,xvals);\ninterp(xvals,yvals,x):\np := unapply(%,x);\n simplify(int(p(x),x=a..a+2*h));\nfor i from 0 to 2 do subs(f(a+i*h)=y[ i],%) end do:\n%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7%%\"aG, &F&\"\"\"%\"hGF(,&F&F(*&\"\"#F(F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7%-%\"fG6#%\"aG-F'6#,&F)\"\"\"%\"hGF--F'6#,&F)F-*&\"\"#F-F .F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operato rG%&arrowGF(,(*&#\"\"\"\"\"#F/*(,(-%\"fG6#,&%\"aGF/*&F0F/%\"hGF/F/F/*& F0F/-F46#,&F7F/F9F/F/!\"\"-F46#F7F/F/F9!\"#9$F0F/F/*&F.F/*(,.*(\"\"%F/ F;F/F7F/F/*(\"\"$F/F?F/F9F/F>*(F0F/F3F/F7F/F>*&F3F/F9F/F>*(FGF/F;F/F9F /F/*(F0F/F?F/F7F/F>F/F9FAFBF/F/F/*&F.F/*&,0*(F3F/F7F/F9F/F/**FIF/F?F/F 7F/F9F/F/*&F3F/)F7F0F/F/*(F0F/F;F/FTF/F>*&F?F/FTF/F/**FGF/F;F/F7F/F9F/ F>*(F0F/F?F/)F9F0F/F/F/F9FAF/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&%\"hGF&,(-%\"fG6#,&%\"aGF&*&\"\"#F&F)F&F&F&-F,6 #F/F&*&\"\"%F&-F,6#,&F/F&F)F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&*&%\"hGF&,(&%\"yG6#\"\"#F&&F,6#\"\"!F&*&\"\"%F &&F,6#F&F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Simpson's " }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/8" "6#*&\"\"$\"\"\"\"\")!\"\"" } {TEXT -1 2 " " }{TEXT 261 4 "Rule" }{TEXT -1 43 " can be obtained by \+ using 4 equally spaced " }{TEXT 315 1 "x" }{TEXT -1 8 " values " } {XPPEDIT 18 0 "x[0],x[1],x[2],x[3]" "6&&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\" \"#&F$6#\"\"$" }{TEXT -1 21 ", with corresponding " }{TEXT 316 1 "y" } {TEXT -1 8 " values " }{XPPEDIT 18 0 "y[0],y[1],y[2],y[3]" "6&&%\"yG6# \"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 58 ", and fitting a de gree 3 (cubic) interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 19 " through the 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 2 ". 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}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Bode's Rule" }{TEXT -1 43 " can be obtained by using 5 equally spaced " }{TEXT 317 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[0],x[1],x [2],x[3],x[4];" "6'&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\" \"%" }{TEXT -1 21 ", with corresponding " }{TEXT 318 1 "y" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "y[0],y[1],y[2],y[3],y[4];" "6'&%\"yG6#\" \"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 60 ", and fitt ing a degree 4 (quartic) interpolating polynomial " }{XPPEDIT 18 0 "p( x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 " through the points" }{XPPEDIT 18 0 "``(x[0],y[0]),``(x[1],y[1]),``(x[2],y[2]),``(x[3],y[3]),``(x[4],y[4 ])" "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?-F$6$&F'6#\"\"%&F+6#FF" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 506 470 470 {PLOTDATA 2 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "Making a substitution " }{XPPEDIT 18 0 "x = a + u*(b-a)" "6#/%\"xG,&%\"aG\"\"\"*&%\"uGF',&%\"bGF'F&!\"\"F'F'" } {TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "u = (x-a)/(b-a)" "6#/%\"uG*& ,&%\"xG\"\"\"%\"aG!\"\"F(,&%\"bGF(F)F*F*" }{TEXT -1 33 " in the right \+ hand integral gives" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(L[k](x),x = a .. b) = (b-a)*Int(L[k](a+u*(b-a)),u = 0 .. 1)" "6 #/-%$IntG6$-&%\"LG6#%\"kG6#%\"xG/F-;%\"aG%\"bG*&,&F1\"\"\"F0!\"\"F4-F% 6$-&F)6#F+6#,&F0F4*&%\"uGF4,&F1F4F0F5F4F4/F>;\"\"!F4F4" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L[k](a+u*(b-a)) = \+ lambda[k](u);" "6#/-&%\"LG6#%\"kG6#,&%\"aG\"\"\"*&%\"uGF,,&%\"bGF,F+! \"\"F,F,-&%'lambdaG6#F(6#F." }{TEXT -1 20 " is a polynomial in " } {TEXT 277 1 "u" }{TEXT -1 29 ", which has the value 1 when " } {XPPEDIT 18 0 "u = k/m;" "6#/%\"uG*&%\"kG\"\"\"%\"mG!\"\"" }{TEXT -1 23 " , and 0 at the points " }{XPPEDIT 18 0 "u = j/m;" "6#/%\"uG*&%\"j G\"\"\"%\"mG!\"\"" }{TEXT -1 23 " between 0 and 1 where " }{XPPEDIT 18 0 "j<>k" "6#0%\"jG%\"kG" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda[k](x) = product((x-j/m)/(k/m-j/m),j = \+ 0 .. m);" "6#/-&%'lambdaG6#%\"kG6#%\"xG-%(productG6$*&,&F*\"\"\"*&%\"j GF0%\"mG!\"\"F4F0,&*&F(F0F3F4F0*&F2F0F3F4F4F4/F2;\"\"!F3" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "j <> k;" "6#0% \"jG%\"kG" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 15 " \+ = " }{XPPEDIT 18 0 "product((m*x-j)/(k-j),j = 0 .. m);" "6#-%(produ ctG6$*&,&*&%\"mG\"\"\"%\"xGF*F*%\"jG!\"\"F*,&%\"kGF*F,F-F-/F,;\"\"!F) " }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "j <> k;" "6#0%\"jG%\"kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "Int(L[k](x),x = a .. b) = (b-a)*lambda[k](x) " "6#/-%$IntG6$-&%\"LG6#%\"kG6#%\"xG/F-;%\"aG%\"bG*&,&F1\"\"\"F0!\"\"F 4-&%'lambdaG6#F+6#F-F4" }{TEXT -1 5 " and " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b)" "6#-%$IntG6$-%\"fG6#% \"xG/F);%\"aG%\"bG" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(p(x),x = a .. b) = (b-a)*Int(lambda[k](x),x = 0 .. 1)*Sum(f(a+k*h),k = 0 .. m);" "6#/-%$IntG6$-%\"pG6#%\"xG/F*;%\"aG%\"b G*(,&F.\"\"\"F-!\"\"F1-F%6$-&%'lambdaG6#%\"kG6#F*/F*;\"\"!F1F1-%$SumG6 $-%\"fG6#,&F-F1*&F9F1%\"hGF1F1/F9;F=%\"mGF1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The integrals " }{XPPEDIT 18 0 "c[k] = Int(lamb da[k](u),u = 0 .. 1);" "6#/&%\"cG6#%\"kG-%$IntG6$-&%'lambdaG6#F'6#%\"u G/F0;\"\"!\"\"\"" }{TEXT -1 9 " of the " }{XPPEDIT 18 0 "lambda[k];" "6#&%'lambdaG6#%\"kG" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "k = 0, 1,` \+ . . .`, m" "6&/%\"kG\"\"!\"\"\"%'~.~.~.G%\"mG" }{TEXT -1 11 ", give th e " }{TEXT 261 25 "Newton-Cotes coefficients" }{TEXT -1 11 " of degree " }{TEXT 320 1 "m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Giv en coefficients " }{XPPEDIT 18 0 "c[0],c[1],` . . . `,c[m];" "6&&%\"cG 6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#%\"mG" }{TEXT -1 20 ", the correspond ing " }{TEXT 261 32 "Newton-Cotes integration formula" }{TEXT -1 3 " i s" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = a . . b);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 " " } {TEXT 274 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m*h*[c[0]*f(a)+c[1]*f( a+h)*`+ . . +`*c[k]*f(a+k*h)*`+ . . +`*c[m]*f(a+m*h)];" "6#*(%\"mG\"\" \"%\"hGF%7#,&*&&%\"cG6#\"\"!F%-%\"fG6#%\"aGF%F%*2&F+6#F%F%-F/6#,&F1F%F &F%F%%(+~.~.~+GF%&F+6#%\"kGF%-F/6#,&F1F%*&F;F%F&F%F%F%F8F%&F+6#F$F%-F/ 6#,&F1F%*&F$F%F&F%F%F%F%F%" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = m*h*Sum(c[k]*f(a+k*h),k = 0 .. m);" "6#/ %!G*(%\"mG\"\"\"%\"hGF'-%$SumG6$*&&%\"cG6#%\"kGF'-%\"fG6#,&%\"aGF'*&F0 F'F(F'F'F'/F0;\"\"!F&F'" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 275 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[k]=Int(lambda[k](x),x = 0 . . 1)" "6#/&%\"cG6#%\"kG-%$IntG6$-&%'lambdaG6#F'6#%\"xG/F0;\"\"!\"\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 " x[k]=a+k*h" "6#/&%\"xG6#%\"kG,&%\"aG\"\"\"*&F'F*%\"hGF*F*" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k = 0, 1,` . . .`, m" "6&/%\"kG\"\"!\"\"\"%' ~.~.~.G%\"mG" }{TEXT -1 24 ", and the corresponding " }{TEXT 322 1 "y " }{TEXT -1 12 " values are " }{XPPEDIT 18 0 "y[k]=f(x[k])" "6#/&%\"yG 6#%\"kG-%\"fG6#&%\"xG6#F'" }{XPPEDIT 18 0 "``=f(a+k*h) " "6#/%!G-%\"fG 6#,&%\"aG\"\"\"*&%\"kGF*%\"hGF*F*" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k = 0, 1,` . . .`, m" "6&/%\"kG\"\"!\"\"\"%'~.~.~.G%\"mG" }{TEXT -1 36 ", then the expression above becomes " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m*h*Sum(c[k]*y[k],k = 0 .. m)" "6#*(%\"mG\"\" \"%\"hGF%-%$SumG6$*&&%\"cG6#%\"kGF%&%\"yG6#F.F%/F.;\"\"!F$F%" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 323 10 "__________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " } {XPPEDIT 18 0 "m*h=b-a" "6#/*&%\"mG\"\"\"%\"hGF&,&%\"bGF&%\"aG!\"\"" } {TEXT -1 46 " is the width of the interval of integration. " }}{PARA 0 "" 0 "" {TEXT -1 104 "In each case the sum of the coefficients is 1, since the formula will be exact on the constant function " }{XPPEDIT 18 0 "f(x) = 1" "6#/-%\"fG6#%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The following proced ure constructs the polynomials " }{XPPEDIT 18 0 "lambda[k](x);" "6#-&% 'lambdaG6#%\"kG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "lambda := proc(m::nonne gint,k::nonnegint,x)\nlocal prod,j;\n if k>m then\n error \"2nd argument must no geater than 1st argument\"\n end if;\n prod := 1 ;\n for j from 0 to m do\n if j<>k then prod := prod*(m*x-j)/(k -j) end if;\n end do;\n prod;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For example, when m = 5, \+ " }{XPPEDIT 18 0 "p[3](x)" "6#-&%\"pG6#\"\"$6#%\"xG" }{TEXT -1 10 " i s . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lambda(5,5,x);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,%\"xG\"\"\",&F%\"\"&F&!\"\"F&,&F%F(\"\"#F)F&,&F%F( \"\"$F)F&,&F%F(\"\"%F)F&#F&\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,* $)%\"xG\"\"&\"\"\"#\"$D'\"#C*&#F*\"#7F(*$)F&\"\"%F(F(!\"\"*&#\"$v)F+F( )F&\"\"$F(F(*&#\"$D\"F.F(*$)F&\"\"#F(F(F2F&F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The following picture sho ws the graphs of " }{XPPEDIT 18 0 "p[0](x),p[1](x),p[2](x),p[3](x),p[ 4](x)" "6'-&%\"pG6#\"\"!6#%\"xG-&F%6#\"\"\"6#F)-&F%6#\"\"#6#F)-&F%6#\" \"$6#F)-&F%6#\"\"%6#F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p[5](x)" " 6#-&%\"pG6#\"\"&6#%\"xG" }{TEXT -1 12 " when m = 5." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot([seq( lambda(5,k,x),k=0..5)],x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 395 249 249 {PLOTDATA 2 "6*-%'CURVESG6$7Z7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a! 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I)F27$Fgy$\"3XlcZ!3)fKHxl5Ff_l7$Faz$\"3;=kO4SJe6Ff_ l7$F\\`o$\"3Uma0/s3'>\"Ff_l7$Ffz$\"3\\sQ(QiOeA\"Ff_l7$$\"3!G$3FWch)f)F 2$\"3cE:6$)*=sB\"Ff_l7$Fd`o$\"3]_>)))e\\hC\"Ff_l7$$\"3vm\"zWU$y.()F2$ \"3/j@X'HfCD\"Ff_l7$F[[l$\"3!GU3D)Q(fD\"Ff_l7$$\"3[;/^\")o_!y)F2$\"3[% GrMU-mD\"Ff_l7$$\"3\")*\\P%[ko/))F2$\"3o\\$Q=Q&ec7Ff_l7$$\"38$ek`,Y)G) )F2$\"3UR0VhZ!fD\"Ff_l7$$\"3Ym;H#e0I&))F2$\"3!RW'fJBaa7Ff_l7$$\"34Le9; ZK,*)F2$\"3!*\\c'4%zp\\7Ff_l7$F`[l$\"3*oBC\"Ff_l7$Fb[m$\"3:.jMO>0 77Ff_l7$Fe[l$\"3'GJ.7AOR;\"Ff_l7$F_\\m$\"3$*)p\\gcbP5\"Ff_l7$Fj[l$\"3% )fOTYjyD5Ff_l7$F\\]m$\"3\\zLv=\"F67$Fcr$!3j7HE*HCx7\"F67$Fhr$!3OoXRai*Q_*F/7$F]s$!3)>d[ !eY^rsF/7$Fbs$!33Ef^Ck#4-%F/7$Fgs$!3\"*GJ@[?>7#)F]w7$F\\t$\"3yf(*)4$4^\"))ot)F/7$F[u$\"3k)o*3FO\\i5F 67$F`u$\"3-LH1ekoq6F67$Feu$\"37!3G%4nef6F67$Fju$\"3g![;H!*fu.\"F67$F_v $\"3IZz`(*)oE%yF/7$Fdv$\"3E!fRyf5*[RF/7$Fiv$!3qskfo_zY(*F]w7$F_w$!3E'p [Jhl0b'F/7$Fdw$!3)*obz**H4E8F67$Fiw$!3KlS8$[H`#>F67$F^x$!3;)*=()42$=\\ #F67$Fcx$!3_R[^WGGmGF67$Fhx$!3Wq!y+7!4EIF67$F]y$!32tXR$**R(HGF67$Fby$! 3]fQ#G@N28#F67$Fgy$!3]d!4%)48K3)F/7$F\\z$\"3,5Po+_xR9F67$Faz$\"3q)QCp> yLk%F67$Ffz$\"3**yAjjt:2#*F67$F[[l$\"3#*f\"e6EYx_\"F27$Fh`p$\"3?L87v_( f'=F27$F`[l$\"3-\"*f\"yPYcC#F27$Fb[m$\"3NFEzm " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The integrals " }{XPPEDIT 18 0 "c[k] = \+ Int(p[k](u),u = 0 .. 1);" "6#/&%\"cG6#%\"kG-%$IntG6$-&%\"pG6#F'6#%\"uG /F0;\"\"!\"\"\"" }{TEXT -1 9 " of the " }{XPPEDIT 18 0 "p[k]" "6#&%\" pG6#%\"kG" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "k = 0, 1,` . . .`, m" "6 &/%\"kG\"\"!\"\"\"%'~.~.~.G%\"mG" }{TEXT -1 11 ", give the " }{TEXT 261 25 "Newton-Cotes coefficients" }{TEXT -1 11 " of degree " }{TEXT 321 1 "m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Given coeffic ients " }{XPPEDIT 18 0 "c[0],c[1];" "6$&%\"cG6#\"\"!&F$6#\"\"\"" } {TEXT -1 8 ", . . . " }{XPPEDIT 18 0 "c[m]" "6#&%\"cG6#%\"mG" }{TEXT -1 55 ", the corresponding Newton-Cotes integration formula is" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b); " "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 2 " " }{TEXT 272 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m*h*[c[0]*f(a)+c[1]*f(a+h)*` + . . +`*c[k]*f(a+k*h)*`+ . . +`*c[m]*f(a+m*h)];" "6#*(%\"mG\"\"\"%\"h GF%7#,&*&&%\"cG6#\"\"!F%-%\"fG6#%\"aGF%F%*2&F+6#F%F%-F/6#,&F1F%F&F%F%% (+~.~.~+GF%&F+6#%\"kGF%-F/6#,&F1F%*&F;F%F&F%F%F%F8F%&F+6#F$F%-F/6#,&F1 F%*&F$F%F&F%F%F%F%F%" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = m*h*Sum(c[k]*f(a+k*h),k = 0 .. m);" "6#/%!G*(% \"mG\"\"\"%\"hGF'-%$SumG6$*&&%\"cG6#%\"kGF'-%\"fG6#,&%\"aGF'*&F0F'F(F' F'F'/F0;\"\"!F&F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 120 "As mentioned above, the sum of any group of coefficients is 1, since the formula will be exact on the constant function " }{XPPEDIT 18 0 "f(x) = 1" "6#/-%\"fG6#%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "for k from 0 to 5 do\n Int('lambda'(5,k,u),u=0..1)= int(lambda(5,k,u),u=0..1);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$IntG6$-%'lambdaG6%\"\"&\"\"!%\"uG/F,;F+\"\"\"#\"#>\"$)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'lambdaG6%\"\"&\"\"\"%\"uG/F,;\" \"!F+#\"#D\"#'*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'lambda G6%\"\"&\"\"#%\"uG/F,;\"\"!\"\"\"#\"#D\"$W\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'lambdaG6%\"\"&\"\"$%\"uG/F,;\"\"!\"\"\"#\" #D\"$W\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'lambdaG6%\"\" &\"\"%%\"uG/F,;\"\"!\"\"\"#\"#D\"#'*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$IntG6$-%'lambdaG6%\"\"&F*%\"uG/F+;\"\"!\"\"\"#\"#>\"$)G" }}} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 60 "Here's \+ a procedure to compute the Newton-Cotes coefficients:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "NCC := p roc(m::posint)\n local k,j,cfs,px,c,x;\n cfs :=[];\n for k from \+ 0 to m do\n # construct poly to integrate\n px := 1;\n \+ for j from 0 to m do\n if j<>k then px := px*(m*x-j)/(k-j) end if;\n end do;\n c := int(px,x=0..1);\n cfs := [op(cfs) ,c]; # add to list of coefficients\n end do;\n cfs;\nend proc:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NCC(7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*#\"$^(\"&!G<#\"%xNF& #\"#\\\"$S'#\"%*)HF&F,F)F'F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "for i from 2 to 9 do\n NCC (i);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"\"\"\"'#\"\"# \"\"$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#\"\"\"\"\")#\"\"$F&F'F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'#\"\"(\"#!*#\"#;\"#X#\"\"#\"#:F' F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(#\"#>\"$)G#\"#D\"#'*#F(\"$W\"F *F'F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)#\"#T\"$S)#\"\"*\"#N#F(\"$! G#\"#M\"$0\"F*F'F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*#\"$^(\"&!G<# \"%xNF&#\"#\\\"$S'#\"%*)HF&F,F)F'F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7+#\"$*)*\"&]$G#\"%WH\"&vT\"#!$k%F)#\"%[_F)#!$a%\"%NGF,F*F'F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7,#\"%dG\"&+'*)#\"&Td\"F&#\"#F\"%SA#\" %47\"%+c#\"%*)G\"&+[%F/F,F)F'F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 178 "For the case of degree 8, two of the coe fficients, are negative which can lead to an increase in roundoff erro r. This should be balanced against the decrease in truncation error." }}{PARA 0 "" 0 "" {TEXT -1 68 "The same problem arises with the degree s 10, 12, 14, 16, 18 and 20." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 30 "Compound Newton-Cotes formulas" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 91 "The idea of c ompound Newton-Cotes methods is to break up the interval of integratio n into " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 20 " subintervals whe re " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 28 " is divisible by the \+ degree " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 107 " of the Newton-C otes formula that we plan to use. Then we use the Newton-Cotes method \+ on the first set of " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 148 " s ubintervals, then on the second set, and so on. These results are adde d together to provide the numerical approximation for the definite int egral." }}{PARA 0 "" 0 "" {TEXT -1 45 "The Newton-Cotes method is ther efore applied " }{XPPEDIT 18 0 "n/m" "6#*&%\"nG\"\"\"%\"mG!\"\"" } {TEXT -1 19 " times on each of " }{XPPEDIT 18 0 "n/m" "6#*&%\"nG\"\" \"%\"mG!\"\"" }{TEXT -1 70 " subintervals of a coarse subdivision of \+ the interval of integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The following procedure " }{TEXT 0 6 "NCCint" } {TEXT -1 80 " is adapted from a Maple worksheet by Brent Petersen of O regon State University." }}{PARA 0 "" 0 "" {TEXT -1 52 "http://www.pea k.org/~petersen/maple/maple_notes.html" }}{PARA 0 "" 0 "" {TEXT -1 26 "It requires the procedure " }{TEXT 0 3 "NCC" }{TEXT -1 38 " of the pr evious section to be active." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 484 "NCCint:= proc(f,r::range,m: :posint,n::posint)\n local k,j,a,b,q,coeffs,h,hh,sum;\n if not ire m(n,m)=0 then\n error \"the degree, %1, must divide the order, %2 \",m,n;\n end if;\n a := op(1,r);\n b := op(2,r);\n q := n/m; \n h := (b-a)/n; # fine subdivision\n hh := (b-a)/q; # coarse sudi vision \n coeffs := NCC(m);\n sum := 0;\n for j from 0 to q-1 \+ do\n for k from 0 to m do\n sum := sum+coeffs[k+1]*f(a+j* hh+k*h);\n end do;\n end do;\n hh*sum;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ev alf(NCCint(sin,0..Pi/2.,6,60),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"5nB+++++++5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 155 ": This procedure makes unnecessary r epetitions of function evaluations. Compound Newton-Cotes formulas can be applied more efficiently using the procedure " }{TEXT 0 5 "NCint" }{TEXT -1 27 " given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "A procedure for performing Newton-Cotes integra tion: " }{TEXT 0 5 "NCint" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "NCint: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 262 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 263 2 " " }{TEXT -1 20 " NCint( fx, rng ) " }{TEXT 264 1 "\n " }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " fx - " }{TEXT -1 55 " an expression involving a single variable, \+ say x," }}{PARA 0 "" 0 "" {TEXT -1 85 " \+ where f(x) evaluates to a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rn g - " }{TEXT 265 61 "the range x=a..b for the definite integral to \+ be aproximated." }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 " The procedure " }{TEXT 0 5 "NCint" }{TEXT -1 48 " attempts to find a n umerical approximation for " }{XPPEDIT 18 0 "Int(fx,x = a .. b);" "6#- %$IntG6$%#fxG/%\"xG;%\"aG%\"bG" }{TEXT -1 202 " by using a Newton-Co tes rule. The rule can be applied in compound form or adaptively. In t he latter case the extent of the subdivision used is assessed locally \+ by tracking the relative error locally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 112 "This option specifies whether an adaptive mechanism based on a fixed rule with \"numpoints\" nodes is to be used." }}{PARA 0 "" 0 " " {TEXT -1 7 "In the " }{TEXT 261 8 "adaptive" }{TEXT -1 7 " mode, " } {TEXT 0 5 "NCint" }{TEXT -1 182 " attempts to ensure that the answer i s accurate to the number of digits given according to the current sett ing of \"Digits\".\nAdaptive quadrature is performed if the option is \+ not set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "numpoints=n" }}{PARA 0 "" 0 "" {TEXT -1 74 "This option allows the degree of the Newton-Cotes formula to be specified." }}{PARA 0 "" 0 " " {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "numpoints=2 -- Trapezoid Rule --------- coefficients : " }{XPPEDIT 18 0 "1/2,1/2;" "6$*&\"\"\"F$\"\"#!\"\"*&F$F$F%F&" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "numpoints=3 -- Simpson' s Rule -------- coefficients: " }{XPPEDIT 18 0 "1/6,2/3,1/6" "6%*&\" \"\"F$\"\"'!\"\"*&\"\"#F$\"\"$F&*&F$F$F%F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "numpoints=4 -- Simpson's " }{XPPEDIT 18 0 "3/8 " "6#*&\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 27 " Rule ----- coefficients: \+ " }{XPPEDIT 18 0 "1/8,3/8,3/8,1/8;" "6&*&\"\"\"F$\"\")!\"\"*&\"\"$F$F %F&*&F(F$F%F&*&F$F$F%F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "numpoints=5 -- Milne Rule ----------- coefficients: " } {XPPEDIT 18 0 "7/90, 16/45, 2/15, 16/45, 7/90" "6'*&\"\"(\"\"\"\"#!*! \"\"*&\"#;F%\"#XF'*&\"\"#F%\"#:F'*&F)F%F*F'*&F$F%F&F'" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 58 "numpoints=6 ------------------------ ---- coefficients: " }{XPPEDIT 18 0 "19/288,25/96,25/144,25/144,25/9 6,19/288;" "6(*&\"#>\"\"\"\"$)G!\"\"*&\"#DF%\"#'*F'*&F)F%\"$W\"F'*&F)F %F,F'*&F)F%F*F'*&F$F%F&F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "numpoints=7 --Weddle Rule ----------- coefficients: " } {XPPEDIT 18 0 "41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840" "6)*& \"#T\"\"\"\"$S)!\"\"*&\"\"*F%\"#NF'*&F)F%\"$!GF'*&\"#MF%\"$0\"F'*&F)F% F,F'*&F)F%F*F'*&F$F%F&F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "numpoints=8 ------------------- coefficients: " }{XPPEDIT 18 0 "751/17280,3577/17280,49/640,2989/17280,2989/17280,49/640,3577/17280,7 51/17280;" "6**&\"$^(\"\"\"\"&!G= 2 " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }} {PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "NCdata" }{TEXT -1 124 " computes t he numerical values of the weights, or coefficients, of the Newton-Cot es quadrature rule with n points or nodes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the pro cedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection. \n" }{TEXT 261 4 "Note" }{TEXT -1 25 ": Both of the procedures " } {TEXT 0 5 "NCint" }{TEXT -1 5 " and " }{TEXT 0 6 "NCdata" }{TEXT -1 29 " must be active in order for " }{TEXT 0 5 "NCint" }{TEXT -1 9 " to work." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "NCint: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12233 "NCint := proc(alg_expr,eq)\nlocal Options,adapt,mthd,n,b,c,x, rs,val,ct,x0,x1,L,h,y0,y1,\n yi,s,i,xi,graph,prntflg,pts,plt,u,v, vals,area,pltcrv,\n sign,t,NCsum,NCadapt,saveDigits,eps,clr,symbl ,maxdpth,\n hasmaxdpth,fact,H,j,p,bisect,xmid,p1,p2,t1;\n\n if \+ nargs<2 then\n error \"at least 2 arguments are required; the basi c syntax is: 'NCint(f(x),x=a..b)'.\"\n end if;\n\n if not type(alg _expr,algebraic) then \n error \"the 1st argument, %1, is invalid ..it should be an algebraic expression in a single variable\",alg_exp r;\n end if; \n if not type(eq,`=`) then \n error \"the 2nd a rgument, %1, is invalid ..it should be an equation of the form 'x=a..b ' to give the required interval for the integral to be estimated\",eq; \n end if;\n x := op(1,eq);\n if not type(x,symbol) then\n \+ error \"2nd argument equation left side, %1, should be the independent variable\",x;\n end if;\n if not type(indets(alg_expr,name) minus \{x\},set(realcons)) then\n error \"the 1st argument, %1, must d epend only on the variable %2\",alg_expr,x;\n end if; \n rs := o p(2,eq);\n\n if not type(rs,realcons..realcons) then\n error \" 2nd argument equation right side, %1, should be a range of real values \",rs;\n end if; \n \n # get the requested options, but first gi ve defaults\n adapt := true;\n n := 13;\n maxdpth := max(Digits, Digits*2-4);\n fact := 1;\n clr := [gray,COLOR(RGB,0,.5,.1)];\n \+ symbl := 'circle';\n prntflg := 0;\n pltcrv := false;\n if nargs >2 then\n Options:=[args[3..nargs]];\n if not type(Options,l ist(equation)) then\n error \"each optional argument must be a n equation\"\n end if;\n if hasoption(Options,'adaptive','ad apt','Options') then\n if adapt<>true then adapt := false end \+ if; \n end if;\n if hasoption(Options,'numpoints','n','Optio ns') then\n if not type(n,posint) then\n error \"\\ \"numpoints\\\" must be a positive integer\"\n end if;\n \+ maxdpth := max(Digits,Digits*2-iquo(n,3));\n end if;\n if hasoption(Options,'maxdepth','maxdpth','Options') then\n if n ot type(maxdpth,posint) then\n error \"\\\"maxdepth\\\" mus t be a positive integer\"\n end if;\n end if;\n if h asoption(Options,'factor','fact','Options') then\n if not type (fact,posint) then\n error \"\\\"factor\\\" must be a posit ive integer\"\n end if;\n end if;\n if hasoption(Opt ions,'info','prntflg','Options') then\n if not member(prntflg, \{true,false,0,1,2,3\}) then\n error \"\\\"info\\\" must be 0 <-> false, 1, 2 or 3 <-> true\"\n end if;\n if prnt flg=false then prntflg := 0\n elif prntflg=true then prntflg : = 3 end if; \n end if;\n if hasoption(Options,'plotcurve','p ltcrv','Options') then\n if pltcrv<>true then pltcrv := false \+ end if; \n end if;\n if hasoption(Options,'colour','clr','Op tions')\n or hasoption(Options,'color','clr','Options') then\n if not type(clr,list) then clr := [clr,black] end if;\n \+ end if;\n if hasoption(Options,'symbol','symbl','Options') then\n if not member(symbl,\{'point','cross','square','diamond','cir cle',\n 'POINT','CROSS,SQUARE','DIAMOND','CIRCLE'\}) then\n symbl := 'circle';\n WARNING(\"invalid symbol f or plot\");\n end if;\n end if;\n if nops(Options)>0 then\n error \"%1 is not a valid option for %2\",op(1,Options ), procname;\n end if;\n end if;\n\n# compute weights c[i] glob al routine ncdata\n c := NCdata(n);\n if not type(c,list(numeric)) then\n error \"could not access Newton-Cotes weights via 'NCdata '\"\n end if;\n\n# NCadapt performs recursion for adaptive integrati on\n\nNCadapt := proc(alg_expr,eq,bisectionlevel,vals,area)\n local \+ x,x0,x1,L,i,term,sum,mp,m,m2,m2p,h,hh,vals1,vals2,\n xi,xstart ,yi,s1,s2,LL,mid,niseven,nm,ymid,\n area1,newarea,area2,j,last i;\n\n if not hasmaxdpth then\n if bisectionlevel>=maxdpth then \n hasmaxdpth := true;\n if prntflg>0 then\n \+ WARNING(\"reached max subdivision depth\");\n end if;\n \+ end if;\n end if;\n x := op(1,eq);\n x0 := op(1,op(2,eq));\n \+ x1 := op(2,op(2,eq));\n if x0=x1 then return 0 end if;\n \n L := evalf(x1-x0);\n m := iquo(n+1,2);\n nm := n-m;\n h := L/(n-1); \n hh := h/2;\n LL := L/2;\n mid := evalf((x0+x1)/2);\n if prn tflg>2 then\n bisect := [op(bisect),mid];\n end if;\n\n vals1 := [seq(vals[j],j=1..m)];\n xstart := x0 + hh;\n for i from 1 to \+ nm do\n xi := xstart+(i-1)*h;\n yi := traperror(evalf(eval(s ubs(x=xi,alg_expr))));\n if yi=lasterror or not type(yi,numeric) \+ then\n error \"evaluation failed at %1\",evalf(xi,saveDigits); \n end if;\n if prntflg>0 then\n ct := ct + 1;\n \+ if prntflg>1 then graph := [op(graph),[xi,yi]] end if;\n end if;\n vals1 := [op(1..2*i-1,vals1),yi,\n \+ op(2*i..nops(vals1),vals1)];\n end do;\n\n s1 := 0;\n for i from 1 to n do\n s1 := s1+c[i]*vals1[i]; # Apply the Newton-Cote s formula.\n end do; \n area1 := s1*LL;\n\n niseven := evalb(ire m(n,2)=0);\n if niseven then\n ymid := yi; # save middle value \n vals2 := [seq(vals[j],j=m+1..n)];\n xstart := x0 + LL + h ;\n lasti := nm-1;\n else\n vals2 := [seq(vals[j],j=m..n)] ;\n xstart := x0 + LL + hh;\n lasti := nm;\n end if;\n f or i from 1 to lasti do\n xi := xstart+(i-1)*h;\n yi := trap error(evalf(eval(subs(x=xi,alg_expr))));\n if yi=lasterror or not type(yi,numeric) then\n error \"evaluation failed at %1\",eva lf(xi,saveDigits);\n end if;\n if prntflg>0 then\n c t := ct + 1;\n if prntflg>1 then graph := [op(graph),[xi,yi]] \+ end if;\n end if;\n vals2:=[op(1..2*i-1,vals2),yi,\n \+ op(2*i..nops(vals2),vals2)];\n end do;\n if nis even then\n vals2 := [ymid,op(vals2)]; # insert middle value\n \+ end if;\n\n s2 := 0;\n for i from 1 to n do # Apply the Newton-Cot es formula.\n s2 := s2+c[i]*vals2[i];\n end do;\n area2 := s2 *LL;\n newarea := area1+area2;\n\n if abs((area-newarea))<=eps*abs (area)\n or bisectionlevel >= maxdpth the n\n return newarea;\n else\n return NCadapt(alg_expr,x=x0..mi d,bisectionlevel+1,vals1,area1)\n + NCadapt(alg_expr,x=mid..x1,b isectionlevel+1,vals2,area2);\n end if;\nend proc:\n\n # now do th e quadrature\n ct := 0;\n graph := [];\n bisect := [];\n\n # i ncrease precision for the computation\n saveDigits := Digits;\n Di gits := min(trunc(Digits*4/3),Digits+5);\n\n x0 := evalf(op(1,rs)); \n x1 := evalf(op(2,rs));\n if x0=x1 then return 0 end if;\n sig n := 1;\n if x10 then\n print(`adaptive Newton-Cotes quadratu re with `||n||` nodes`);\n end if;\n hasmaxdpth := false;\n \+ eps := Float(1,-saveDigits-1)*evalf(1.8^(n+2));\n L := evalf (x1-x0);\n h := L/(n-1);\n y0 := traperror(evalf(eval(subs(x =x0,alg_expr))));\n if y0=lasterror or not type(y0,numeric) then \n y0 := evalf(limit(alg_expr,x=x0,right));\n if y0=la sterror or not type(y0,numeric) then\n error \"no real valu e or right limit at %1\", evalf(x0,saveDigits);\n end if;\n \+ end if;\n if prntflg>0 then\n ct := ct + 1;\n \+ if prntflg>1 then graph := [op(graph),[x0,y0]] end if;\n end if; \n vals := [y0];\n s := c[1]*y0;\n\n for i from 2 to n- 1 do\n xi := x0+(i-1)*h;\n yi := traperror(evalf(eval( subs(x=xi,alg_expr))));\n if yi=lasterror or not type(yi,numer ic) then\n error \"evaluation failed at %1\",evalf(xi,saveD igits);\n end if;\n if prntflg>0 then\n ct \+ := ct + 1;\n if prntflg>1 then graph := [op(graph),[xi,yi]] end if;\n end if;\n vals := [op(vals),yi];\n \+ s:= s+c[i]*yi; # Apply the Newton-Cotes formula.\n end do;\n\n \+ y1 := traperror(evalf(eval(subs(x=x1,alg_expr))));\n if y1=las terror or not type(y1,numeric) then\n y1 := evalf(limit(alg_ex pr,x=x1,left));\n if y1=lasterror or not type(y1,numeric) then \n error \"no real value or left limit at %1\", evalf(x1,sa veDigits);\n end if;\n end if;\n if prntflg>0 then\n ct := ct + 1;\n if prntflg>1 then graph := [op(graph) ,[x1,y1]] end if;\n end if;\n vals := [op(vals),y1];\n \+ s := s + c[n]*y1;\n\n area := s*L; \n val := traperror(NCada pt(alg_expr,x=x0..x1,1,vals,area,area));\n if val=lasterror then \+ error \"val\" end if\n else\n L := evalf(x1-x0);\n p := fa ct*(n-1);\n if prntflg>0 then\n if fact=1 then\n \+ print(`non-adaptive Newton-Cotes quadrature with `||n||` nodes`);\n else\n print(`compound non-adaptive Newton-Cotes q uadrature with `||n||` nodes`);\n print(`repeated `||fact|| ` times over `||p||` intervals`);\n end if;\n end if;\n \+ H := L/fact; # coarse sudivision width\n h := L/p; # fine sub division width\n \n # first compute all the values needed\n \+ y0 := traperror(evalf(eval(subs(x=x0,alg_expr))));\n if y0=l asterror or not type(y0,numeric) then\n y0 := evalf(limit(alg_ expr,x=x0,right));\n if y0=lasterror or not type(y0,numeric) t hen\n error \"no real value or right limit at %1\", evalf(x 0,saveDigits);\n end if;\n end if;\n if prntflg>0 th en\n ct := ct + 1;\n if prntflg>1 then graph := [op(gr aph),[x0,y0]] end if;\n end if;\n vals := [y0];\n\n for i from 2 to p do\n xi := x0+(i-1)*h;\n yi := traperro r(evalf(eval(subs(x=xi,alg_expr))));\n if yi=lasterror or not \+ type(yi,numeric) then\n error \"evaluation failed at %1\",e valf(xi,saveDigits);\n end if;\n if prntflg>0 then\n \+ ct := ct + 1;\n if prntflg>1 then graph := [op(gr aph),[xi,yi]] end if;\n end if;\n vals := [op(vals),yi ];\n end do;\n\n y1 := traperror(evalf(eval(subs(x=x1,alg_ex pr))));\n if y1=lasterror or not type(y1,numeric) then\n \+ y1 := evalf(limit(alg_expr,x=x1,left));\n if y1=lasterror or n ot type(y1,numeric) then\n error \"no real value or left li mit at %1\", evalf(x1,saveDigits);\n end if;\n end if;\n \+ if prntflg>0 then\n ct := ct + 1;\n if prntflg>1 \+ then graph := [op(graph),[x1,y1]] end if;\n end if;\n vals : = [op(vals),y1];\n\n # now apply the compound Newton-Cotes rule\n s := 0;\n for j from 0 to fact-1 do\n for i from 1 \+ to n do\n s := s+c[i]*vals[(n-1)*j+i];\n end do;\n \+ end do;\n val := H*s;\n end if;\n if prntflg>1 then\n \+ if adapt and prntflg>2 then\n x0 := op(1,op(2,eq));\n \+ x1 := op(2,op(2,eq));\n xmid := (x0+x1)/2;\n bisect := map(u->[[u,0],[u,1]],bisect);\n p1 := plot(bisect,color=cl r[2],thickness=2);\n p2 := plot([[x0,0],[x0,1],[x1,1],[x1,0],[ x0,0]],\n color=COLOR(RGB,.5,.5,.5),thickness=2);\n \+ t1 := plots[textplot]([[xmid,3,`bisection points`],[xmid,-2,` - -- `]]);\n plt := traperror(plots[display]([p1,p2,t1],axes=non e));\n if plt = lasterror then \n WARNING(\"cannot \+ draw diagram because of the error:- %1\",lasterror)\n else\n \+ print(plt)\n end if;\n end if;\n pts := so rt(graph,(u,v) -> evalb(op(1,u)0 then\n print(`number of function evaluat ions --> `,ct);\n end if;\n\n Digits := saveDigits;\n if sign>0 \+ then\n return evalf(val);\n else\n return evalf(-val);\n \+ end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "NCdata: implementation" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4299 "NCdata : = proc(n::posint)\n local j,k,b,c;\n option remember;\n if n=1 t hen error \"argument must be greater than 1\" end if;\n if n=2 then \+ c:=[1/2,1/2]\n elif n=3 then c:=[1/6,2/3,1/6]\n elif n=4 then c:=[ 1/8,3/8,3/8,1/8]\n elif n=5 then c:=[7/90,16/45,2/15,16/45,7/90]\n \+ elif n=6 then c:=[19/288,25/96,25/144,25/144,25/96,19/288]\n elif n =7 then c:=[41/840,9/35,9/280,34/105,9/280,9/35,41/840]\n elif n=8 t hen c:=[751/17280,3577/17280,49/640,2989/17280,\n \+ 2989/17280,49/640,3577/17280,751/17280]\n elif n=9 then c:=[989/2835 0,2944/14175,-464/14175,5248/14175,\n -454/2835,52 48/14175,-464/14175,2944/14175,\n 989/28350]\n e lif n=10 then c:=[2857/89600,15741/89600,27/2240,1209/5600, \n \+ 2889/44800,2889/44800,1209/5600,27/2240,\n \+ 15741/89600,2857/89600]\n elif n=11 then c:=[16067/598 752,26575/149688,-16175/199584,\n 5675/12474,-482 5/11088,17807/24948,\n -4825/11088,5675/12474,-16 175/199584,\n 26575/149688,16067/598752]\n elif n=12 then c:=[434293/17418240, 4495513/29030400,\n \+ -3237113/87091200, 560593/1935360,\n -1599257/ 14515200, 2582261/14515200,\n 2582261/14515200, - 1599257/14515200,\n 560593/1935360, -3237113/8709 1200,\n 4495513/29030400, 434293/17418240];\n e lif n=13 then c:=[1364651/63063000, 12504/79625, -105387/875875,\n \+ 893128/1576575, -1144251/1401400,\n \+ 1215504/875875, -522602/375375,\n 1215504/87 5875, -1144251/1401400,\n 893128/1576575, -105387 /875875,\n 12504/79625, 1364651/63063000]\n eli f n=14 then c:=[8181904909/402361344000,\n 5628072966 1/402361344000, -1737125143/22353408000,\n 1114817271 1/28740096000, -6066382933/16094453760,\n 22964826443 /44706816000, -3592666051/33530112000,\n -3592666051/ 33530112000, 22964826443/44706816000,\n -6066382933/1 6094453760, 11148172711/28740096000,\n -1737125143/22 353408000, 56280729661/402361344000,\n 8181904909/402 361344000]\n elif n=15 then c:=[90241897/5003856000, 44436679/312741 000,\n -770720657/5003856000, 109420087/156370500,\n -6625093363/5003856000, 789382601/312741000,\n \+ -5600756791/1667952000, 101741867/26061750,\n \+ -5600756791/1667952000, 789382601/312741000,\n \+ -6625093363/5003856000, 109420087/156370500,\n -7 70720657/5003856000, 44436679/312741000,\n 90241897/ 5003856000]\n elif n=16 then c:=[5044289/295206912, 29505985/2296053 76,\n -25881785/229605376, 349259195/688816128,\n \+ -24806995/32800768, 273542741/229605376,\n \+ -2000332805/2066448384, 113200845/229605376,\n \+ 113200845/229605376, -2000332805/2066448384,\n \+ 273542741/229605376, -24806995/32800768,\n 349259 195/688816128, -25881785/229605376,\n 29505985/2296 05376, 5044289/295206912];\n elif n=17 then c:=[15043611773/97692469 8750,\n 63813303296/488462349375,-1997012608/10854718875 ,\n 83221185536/97692469875,-17540896432/8881133625,\n \+ 232088271872/54273594375,-3403267203968/488462349375,\n \+ 936887501824/97692469875,-37904588786/3618239625,\n \+ 936887501824/97692469875,-3403267203968/488462349375,\n \+ 232088271872/54273594375,-17540896432/8881133625,\n \+ 83221185536/97692469875,-1997012608/10854718875,\n \+ 63813303296/488462349375,15043611773/976924698750]\n else\n for j from 1 to n do\n c[j] := (n-1)^(j-1)/j;\n end do;\n \+ for k from 1 to n-1 do\n for j from n to k+1 by -1 do\n \+ c[j] := c[j]-(k-1)*c[j-1]\n end do;\n end do;\n \+ for k from n-1 to 1 by -1 do\n for j from k+1 to n do\n \+ c[j] := c[j]/k;\n end do;\n for j from k to n -1 do c[j] := c[j]-c[j+1] end do;\n end do;\n end if;\n [seq( c[j],j=1..n)];\nend proc: # of ncdata" }}}{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 0 5 "NCint " }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 4 "Note" }{TEXT -1 74 ": Questions 9 to 13 are general problems involving numerical integrat ion. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }{TEXT 287 89 ".. comparison of New ton-Cotes integration with integration of an interpolating polynomial " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f(x)=1/(x+exp(x))" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F )-%$expG6#F'F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "Fi rst we find an interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 42 " of degree 10 whose values match those of \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " at the 11 poi nts " }{XPPEDIT 18 0 "0,1/10,1/5,3/10,` . . . `,9/10,1" "6)\"\"!*&\" \"\"F%\"#5!\"\"*&F%F%\"\"&F'*&\"\"$F%F&F'%(~.~.~.~G*&\"\"*F%F&F'F%" } {TEXT -1 28 " between 0 and 1 inclusive. " }}{PARA 0 "" 0 "" {TEXT -1 91 "The coefficients in the interpolating polynomials are calculated u sing 15 digit arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "f := x -> 1/(x+exp(x));\nxvals := \+ [seq(0.1*j,j=0..10)];\nyvals := evalf(map(f,xvals),15);\np := unapply( evalf(interp(xvals,yvals,x),15),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&9$F--%$expG6#F/F -!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7-$\"\"!F'$\" \"\"!\"\"$\"\"#F*$\"\"$F*$\"\"%F*$\"\"&F*$\"\"'F*$\"\"(F*$\"\")F*$\"\" *F*$\"#5F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7-$\"\"\"\"\"!$ \"0,$>_#yvH)!#:$\"0[K6k.`.(F+$\"0$e6=Z7hgF+$\"0e!*[--fG&F+$\"0(zP_2$Rl %F+$\"0_uok;'GTF+$\"0,n-9M\\o$F+$\"0,`84%>0LF+$\"0')Q.2Ul(HF+$\"0&**p8 UT*o#F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,8*$)9$\"#5\"\"\"$\"0yFSo$)QG&!#:*&$\"0aDs03e?$!#9F1)F/ \"\"*F1!\"\"*&$\"0Eno\">m**>F8F1F/F1F;*&$\"0RQO\")zGz)F8F1)F/\"\")F1F1 *&$\"0;pk`,&*[$F8F1)F/\"\"#F1F1*&$\"0s4972xX\"!#8F1)F/\"\"(F1F;*&$\"0E fKuM$HgF8F1)F/\"\"$F1F;*&$\"0EYTH#Qg;FLF1)F/\"\"'F1F1*&$\"0Fv!*Go$\\)* F8F1)F/\"\"%F1F1*&$\"0!HifVJ=9FLF1)F/\"\"&F1F;$F1\"\"!F1F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Now we fi nd " }{XPPEDIT 18 0 "Int(p(x),x=0..1)" "6#-%$IntG6$-%\"pG6#%\"xG/F);\" \"!\"\"\"" }{TEXT -1 25 " as an approximation for " }{XPPEDIT 18 0 "In t(f(x),x=0..1)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Int('p(x)',x=0..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"pG6#%\"xG/F);\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q(3I;&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "NCint" } {TEXT -1 19 " with the options \"" }{TEXT 278 14 "adaptive=false" } {TEXT -1 7 "\" and \"" }{TEXT 278 12 "numpoints=11" }{TEXT -1 304 "\" \+ will apply a Newton-Cotes rule with appropriate coefficients just once , that is, no bisections are performed. The value obtained is the same as that just calculated because the 11 point Newton-Cotes formula is \+ based on exactly the same idea of integrating the interpolating polyno mial just constructed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "NCint(1/(x+exp(x)),x=0..1,adaptive= false,numpoints=11,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Sno n-adaptive~Newton-Cotes~quadrature~with~11~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 366 173 173 {PLOTDATA 2 "6(-%'CURVESG6%7-7$$\"\"!F)$\"\"\"F) 7$$\"3/+++++++5!#=$\"3?+5>_#yvH)F/7$$\"35+++++++?F/$\"3W**H8TOINqF/7$$ \"3))**************HF/$\"3'***f6=Z7hgF/7$$\"3A+++++++SF/$\"3Y+5*[--fG& F/7$$\"3++++++++]F/$\"3#***zP_2$Rl%F/7$$\"3w**************fF/$\"3))**R (ok;'GTF/7$$\"3a**************pF/$\"3E+!o-9M\\o$F/7$$\"3U+++++++!)F/$ \"36+SN\"4%>0LF/7$$\"3A+++++++!*F/$\"3)****Q.2Ul(HF/7$F*$\"3))****p8UT *o#F/-%'COLOURG6&%$RGBG$\")=THv!\")F[oF[o-%&STYLEG6#%%LINEG-F$6%F&-%&C OLORG6&FjnF)$\"\"&!\"\"$F+Fio-F_o6#%&POINTG-%&TITLEG6#%2evaluation~poi ntsG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q!6\"Fip-%%VIEWG6$%(DEFAULTGF ^q" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~eval uations~-->~G\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q(3I;&!#5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" } {TEXT -1 113 ": Essentially the same result can be obtained by making \+ direct use of the appropriate Newton-Cotes coefficients. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "C := \+ NCdata(11);\nf := x -> 1/(x+exp(x));\nh := 0.1;\nSum('f(h*(i-1))*C[i]' ,i=1..11);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG7-#\"&n g\"\"'_()f#\"&vl#\"')o\\\"#!&vh\"\"'%e*>#\"%vc\"&uC\"#!%D[\"&)36#\"&2y \"\"&[\\#F2F/F,F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&9$F--%$expG6#F/F-!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SumG6$*&-%\"fG6#*&%\"hG\"\"\",&%\"iGF,F,!\"\"F,F,& %\"CG6#F.F,/F.;F,\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+o(3I;&!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Alth ough this value is reasonably accurate, it is not accurate to 10 digit s." }}{PARA 0 "" 0 "" {TEXT -1 76 "We can obtain a value which is accu rate to 10 digits by using the procedure " }{TEXT 0 5 "NCint" }{TEXT -1 163 " in the (default) adaptive mode, so that the 11 point rule is \+ applied repeatedly on bisected subintervals until the integral is calc ulated to the desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "NCint(1/(x+exp(x)),x=0..1,nu mpoints=11,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~N ewton-Cotes~quadrature~with~11~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 415 88 88 {PLOTDATA 2 "6+-%'CURVESG6%7$7$$\"3++++++++]!#=$\"\"!F,7$F($ \"\"\"F,-%&COLORG6&%$RGBGF,$\"\"&!\"\"$F/F6-%*THICKNESSG6#\"\"#-F$6%7$ 7$$\"3++++++++DF*F+7$F@F.F0F8-F$6%7$7$$\"3++++++++vF*F+7$FGF.F0F8-F$6% 7'7$F+F+7$F+F.7$F.F.7$F.F+FM-F16&F3F4F4F4F8-%%TEXTG6$7$$\".++++++&!#8$ \"\"$F,Q1bisection~points6\"-FT6$7$FW$!\"#F,Q&~---~Fgn-%*AXESSTYLEG6#% %NONEG-%+AXESLABELSG6%Q!FgnFeo-%%FONTG6#%(DEFAULTG-%%VIEWG6$FioFio" 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" }}}{PARA 13 "" 1 "" {GLPLOT2D 403 253 253 {PLOTDATA 2 "6(-%'CURVESG6%7K7$$\"\"!F)$\"\"\"F) 7$$\"39+++++++D!#>$\"3w**fIX@&4_*!#=7$$\"3G+++++++]F/$\"3%****eqG;/3*F 27$$\"3s*************\\(F/$\"3u***erS)*Qn)F27$$\"3/+++++++5F2$\"3?+5>_ #yvH)F27$$\"3+++++++]7F2$\"3%***\\v_w=[zF27$$\"3%**************\\\"F2$ \"3)****3QH9Hi(F27$$\"3))************\\tF27$$\"35++++ +++?F2$\"3W**H8TOINqF27$$\"31++++++]AF2$\"3s***H`r,!pnF27$$\"3++++++++ DF2$\"3c+qr9oz=lF27$$\"3A++++++]FF2$\"3s**H(RNfKG'F27$$\"3))********** ****HF2$\"3'***f6=Z7hgF27$$\"37++++++]KF2$\"3'***\\]Z,F^eF27$$\"3w**** *********\\$F2$\"3g**pE.ap_cF27$$\"3+++++++]PF2$\"3E+5'3^1XY&F27$$\"3A +++++++SF2$\"3Y+5*[--fG&F27$$\"3))************\\UF2$\"3,+IQq8;;^F27$$ \"35+++++++XF2$\"31+!3L!\\ja\\F27$$\"3y************\\ZF2$\"3!***\\k,at +[F27$$\"3++++++++]F2$\"3#***zP_2$Rl%F27$$\"3A++++++]_F2$\"3.+!*zUwt8X F27$$\"3U+++++++bF2$\"3?+g!))4;(zVF27$$\"3c************\\dF2$\"3#)**HS SZY^UF27$$\"3w**************fF2$\"3))**R(ok;'GTF27$$\"3+++++++]iF2$\"3 =+S8Ud$3,%F27$$\"3A+++++++lF2$\"3!)**zXDO\"y*QF27$$\"3W++++++]nF2$\"3; +I'[ym#*y$F27$$\"3a**************pF2$\"3E+!o-9M\\o$F27$$\"3y********** **\\sF2$\"39+Szr[d%e$F27$$\"3++++++++vF2$\"3!)***4`\\mz[$F27$$\"3A++++ ++]xF2$\"33+5*H:.\\R$F27$$\"3U+++++++!)F2$\"36+SN\"4%>0LF27$$\"3c***** *******\\#)F2$\"3)***>B.Bm=KF27$$\"3w*************\\)F2$\"3x**RU8L9NJF 27$$\"3+++++++]()F2$\"3y**4f!4%[aIF27$$\"3A+++++++!*F2$\"3)****Q.2Ul(H F27$$\"3W++++++]#*F2$\"33+gYwU=,HF27$$\"3a*************\\*F2$\"3\"**** G!HlGGGF27$$\"3y************\\(*F2$\"3/+!*RHFtdFF27$F*$\"3))****p8UT*o #F2-%'COLOURG6&%$RGBG$\")=THv!\")FbxFbx-%&STYLEG6#%%LINEG-F$6%F&-%&COL ORG6&FaxF)$\"\"&!\"\"$F+F`y-Ffx6#%&POINTG-%&TITLEG6#%2evaluation~point sG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q!6\"F`z-%%VIEWG6$%(DEFAULTGFez " 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evalua tions~-->~G\"#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mw+j^!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "This valu e agrees with the value given by " }{TEXT 0 5 "evalf" }{TEXT -1 5 " an d " }{TEXT 0 3 "Int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(Int(1/(x+exp(x)),x=0. .1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+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 2 " }{TEXT 288 74 ".. the degree of polynomials for which Newton-Cotes integration is \"exact\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "In t(1-x^9,x = 0 .. 1);" "6#-%$IntG6$,&\"\"\"F'*$%\"xG\"\"*!\"\"/F);\"\"! F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(1-x^9,x = 0 .. 1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&\"\"\"F'*$)%\"xG\"\"*F'!\"\"/F*; \"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The Newton-Cotes 9-p oint formula is exact for polynomials of degree 9 or less." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "When the number o f points is " }{TEXT 261 3 "odd" }{TEXT -1 36 ", we get an extra degre e \"for free\"." }}{PARA 0 "" 0 "" {TEXT -1 140 "For example, Simpson' s rule is the 3-point Newton-Cotes rule. It is designed to be exact on quadratics, but turns out to be exact on cubics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "NCint(1-x^9, x=0..1,info=1,numpoints=9,adaptive=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Rnon-adaptive~Newton-Cotes~quadrature~with~9~nodesG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~ G\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++!*!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "In adaptive mode \+ one bisection is peformed in order to check the error, so (2 times \"n umpoints\" -1) function evaluations are required." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "NCint(1-x^9, x=0..1,numpoints=9,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nad aptive~Newton-Cotes~quadrature~with~9~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 325 108 108 {PLOTDATA 2 "6)-%'CURVESG6%7$7$$\"3++++++++]!#=$ \"\"!F,7$F($\"\"\"F,-%&COLORG6&%$RGBGF,$\"\"&!\"\"$F/F6-%*THICKNESSG6# \"\"#-F$6%7'7$F+F+7$F+F.7$F.F.7$F.F+F?-F16&F3F4F4F4F8-%%TEXTG6$7$$\".+ +++++&!#8$\"\"$F,Q1bisection~points6\"-FF6$7$FI$!\"#F,Q&~---~FO-%*AXES STYLEG6#%%NONEG-%+AXESLABELSG6%Q!FOFgn-%%FONTG6#%(DEFAULTG-%%VIEWG6$F[ oF[o" 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" }}}{PARA 13 "" 1 "" {GLPLOT2D 325 196 196 {PLOTDATA 2 "6(-%'CURVESG6%737$$\"\"!F)$\"\"\"F)7$$\"3+++++++]i!#> $\"2-+a)**********!#<7$$\"3+++++++]7!#=$\"2)**R\\D********F27$$\"3++++ +++v=F6$\"2'**puNr******F27$$\"3++++++++DF6$\"3e**p-`='*****F67$$\"3++ +++++DJF6$\"3;+g!Hy:(****F67$$\"3+++++++]PF6$\"3y**4A-N`)***F67$$\"3++ +++++vVF6$\"3Q+!=txFT***F67$$\"3++++++++]F6$\"3++++](o/)**F67$$\"3++++ +++DcF6$\"3W+I'))*GiV**F67$$F.F6$\"3a**frZ3[a)*F67$$\"3+++++++voF6$\"3 9+g(3Xtol*F67$$\"3++++++++vF6$\"3;+q?PJ:\\#*F67$$\"3+++++++D\")F6$\"3S +![3CUoX)F67$$\"3+++++++]()F6$\"3%****)p')>U$*pF67$$\"3+++++++v$*F6$\" 3****R\"G$\\v0WF67$F*F(-%'COLOURG6&%$RGBG$\")=THv!\")FhpFhp-%&STYLEG6# %%LINEG-F$6%F&-%&COLORG6&FgpF)$\"\"&!\"\"$F+Ffq-F\\q6#%&POINTG-%&TITLE G6#%2evaluation~pointsG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q!6\"Ffr-% %VIEWG6$%(DEFAULTGF[s" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Dnumber~of~function~evaluations~-->~G\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++!*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 114 "The Newton-Cotes 9-point rule is not exa ct on polynomials of degree 10, so more function evaluations are requi red." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "NCint(1-x^10,x=0..1,numpoints=9,info=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive~Newton-Cotes~quadrature~with~9~nodes G" }}{PARA 13 "" 1 "" {GLPLOT2D 300 65 65 {PLOTDATA 2 "6,-%'CURVESG6%7 $7$$\"1+++++++]!#;\"\"!7$F($\"\"\"F+-%&COLORG6&%$RGBGF+$\"\"&!\"\"$F.F 5-%*THICKNESSG6#\"\"#-F$6%7$7$$\"1+++++++DF*F+7$F?F-F/F7-F$6%7$7$$\"1+ ++++++vF*F+7$FFF-F/F7-F$6%7'7$F+F+7$F+F-7$F-F-7$F-F+FL-F06&F2F3F3F3F7- %%TEXTG6$7$$\".++++++&!#8$F:F+%1bisection~pointsG-FS6$7$FV$!\"#F+%!G-F S6$7$FV$\"\"$F+Fjn-%+AXESLABELSG6$FjnFjn-%*AXESSTYLEG6#%%NONEG-%%VIEWG 6$%(DEFAULTGFjo" 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" }}}{PARA 13 "" 1 "" {GLPLOT2D 312 165 165 {PLOTDATA 2 "6(-%'CURVES G6%7C7$\"\"!$\"\"\"F(7$$\"1++++++DJ!#Xb%* *F=7$$F1F=$\"1+I#)H004**F=7$$\"1+++++]ilF=$\"1++s'*G&=&)*F=7$$\"1+++++ +voF=$\"1+!yu\\+Tw*F=7$$\"1+++++](=(F=$\"1+gOn'e?j*F=7$$\"1+++++++vF=$ \"1+]!H&['oV*F=7$$\"1+++++]7yF=$\"1,gu_q'H:*F=7$$\"1++++++D\")F=$\"1+! *o?V=Y()F=7$$\"1+++++]P%)F=$\"1+!fmR>8<)F=7$$\"1++++++]()F=$\"1+?OQUCp tF=7$$\"1+++++]i!*F=$\"1++rV\"pLE'F=7$$F7F=$\"1+I^\\_RbZF=7$$\"1+++++] (o*F=$\"1+!zKVQ-s#F=7$F)F(-%'COLOURG6&%$RGBG$\")=THv!\")FcuFcu-%&STYLE G6#%%LINEG-F$6%F&-%&COLORG6&FbuF($\"\"&!\"\"$F*Fav-Fgu6#%&POINTG-%+AXE SLABELSG6$%!GFiv-%'SYMBOLG6#%'CIRCLEG-%&TITLEG6#%2evaluation~pointsG-% %VIEWG6$%(DEFAULTGFew" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Dnumber~of~function~evaluations~-->~G\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"4444*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "The Newton-Cotes 10-point rule is exact \+ for polynomials of degree 9, but not for polynomials of degree 10." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "NCint(1-x^10,x=0..1,adaptive=false,numpoints=10,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Snon-adaptive~Newton-Cotes~quadrature~with~ 10~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~eval uations~-->~G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!R&*34*!#5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }{TEXT 289 59 ".. using adaptive and non-adaptive Newton-Cotes integration" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(s in(x),x = 0 .. Pi/2);" "6#-%$IntG6$-%$sinG6#%\"xG/F);\"\"!*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "NCint(sin(x),x=0..Pi/2,numpo ints=10,info=3,color=[coral,navy]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Oadaptive~Newton-Cotes~quadrature~with~10~nodesG" }}{PARA 13 "" 1 " " {GLPLOT2D 306 86 86 {PLOTDATA 2 "6)-%'CURVESG6%7$7$$\"3;+](Rj\")R&y! #=$\"\"!F,7$F($\"\"\"F,-%'COLOURG6&%$RGBG$\")!\\DP\"!\")F4$\")viobF6-% *THICKNESSG6#\"\"#-F$6%7'7$F+F+7$F+F.7$$\"3c'*[zEjzq:!#$\"3****RwuUd:()F-7$$\"3-+S*>DH`u \"!#=$\"3%****omx\"[O)e#F37$$\"3/+! ))R]e1\\$F3$\"3!***fDL9??MF37$$\"3/+])*HJKjVF3$\"3@+gSQ_'3hF3$\"3u***4NOkdt&F37 $$\"33+g(z+<8)pF3$\"3I+S'o4wyU'F37$$\"3a**H(Rj\")R&yF3$\"3C+S'=\"y1rqF 37$$F,F3$\"3k***)=JWWgwF37$$\"3;+!pf)3J*f*F3$\"39++*GW?:>)F37$$\"37+g> ^v>Z5!#<$\"3y**4%y.a-m)F37$$\"3/+gz8SYM6F_o$\"3Y**\\Oqy2j!*F37$$\"3'** *fRw/t@7F_o$\"3!****ey?EpR*F37$$\"37+g**Qp**38F_o$\"3a**4*GEe#f'*F37$$ \"3%***\\f,ME'R\"F_o$\"3'***47Iv2[)*F37$$\"3))**\\>k)HN[\"F_o$\"3E+q\" 4)p%>'**F37$$\"3/+]zEjzq:F_o$\"\"\"F)-%'COLOURG6&%$RGBG$\"*++++\"!\")$ \")AR!)\\FfqF(-%&STYLEG6#%%LINEG-F$6%F&-Faq6&Fcq$\")!\\DP\"FfqFar$\")v iobFfq-Fjq6#%&POINTG-%+AXESLABELSG6$Q!6\"F[s-%&TITLEG6#%2evaluation~po intsG-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$%(DEFAULTGFhs" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "The result is accurate t o 10 digits with a single application of the 10 point rule. A single b isection has been performed, and " }{XPPEDIT 18 0 "2*`.`*10-1 = 19;" " 6#/,&*(\"\"#\"\"\"%\".GF'\"#5F'F'F'!\"\"\"#>" }{TEXT -1 32 " function \+ evaluations were used." }}{PARA 0 "" 0 "" {TEXT -1 104 "The same accur acy can be achieved in the non-adaptive mode by a single application o f the 10-point rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "NCint(sin(x),x=0..Pi/2,adaptive=false,num points=10,info=true,\n color=[coral,navy]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Snon-adaptive~Newton-Cotes~quadrature~with~10~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 268 162 162 {PLOTDATA 2 "6(-%'CURVESG6%7, 7$\"\"!F(7$$\"1+S*>DH`u\"!#;$\"1+!pmx\"[O^v>Z5!#:$\"1+5%y.a-m )F,7$$\"1+gRw/t@7FF$\"1+!fy?EpR*F,7$$\"1+]f,ME'R\"FF$\"1+57Iv2[)*F,7$$ \"1+]zEjzq:FF$\"\"\"F(-%'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\FhnF(- %&STYLEG6#%%LINEG-F$6%F&-FY6&Fen$\")!\\DP\"FhnFco$\")viobFhn-F\\o6#%&P OINTG-%+AXESLABELSG6$%!GF]p-%'SYMBOLG6#%'CIRCLEG-%&TITLEG6#%2evaluatio n~pointsG-%%VIEWG6$%(DEFAULTGFip" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 4 " }{TEXT 290 88 ".. attempting to minim ise the number of function evaluations required by the procedure " } {TEXT 260 5 "NCint" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "Int(arctan(x)^2,x = 0 .. 4);" "6#-%$IntG6$*$-%'arcta nG6#%\"xG\"\"#/F*;\"\"!\"\"%" }{TEXT -1 5 " and " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "NCint(arctan (x)^2,x=0..4,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive ~Newton-Cotes~quadrature~with~13~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 306 86 86 {PLOTDATA 2 "6--%'CURVESG6%7$7$$\"\"#\"\"!$F*F*7$F($\"\"\"F* -%&COLORG6&%$RGBGF*$\"\"&!\"\"$F.F5-%*THICKNESSG6#F)-F$6%7$7$F-F+7$F-F -F/F7-F$6%7$7$$\"3++++++++]!#=F+7$FCF-F/F7-F$6%7$7$$\"3++++++++:!#$ \"3%***4)RR/Tt\"!#?7$$\"3g**\\LLLLL$)F-$\"3t**4$*RWY7pF07$$\"3+++++++] 7!#=$\"35+I(oY;ka\"F-7$$\"3;+gmmmmm;F9$\"30+!z^c3us#F-7$$\"35+SLLLL$3# F9$\"3I+I/!fE(=UF-7$$\"3++++++++DF9$\"3#***RuQXX,gF-7$$\"3#***pmmmm;HF 9$\"3o***Q@k4R0)F-7$$\"33+ILLLLLLF9$\"3'***\\a#>M_.\"F97$$\"3+++++++]P F9$\"3!***RYQR;(G\"F97$$\"3C+!ommmm;%F9$\"3#****Q>G+'e:F97$$\"33+ILLLL $e%F9$\"3/+SMlh&p%=F97$$\"3o***)**********\\F9$\"33+5K`5p\\@F97$$\"3Z+ qmmmm;aF9$\"3/+5)4]bVY#F97$$\"3%)**RLLLLLeF9$\"33+!33BE')y#F97$$\"3+++ ++++]iF9$\"3()**H-^>L?JF97$$\"3;+gmmmmmmF9$\"3%)**fy<1ZdMF97$$\"3#)**R LLLL$3(F9$\"3G+!Q/:>#)z$F97$$\"3]**>+++++vF9$\"3A+I=qn$49%F97$$\"3Z+qm mmm;zF9$\"35++nvJ;%[%F97$$\"3L+?LLLLL$)F9$\"3%)**f7CFhE[F97$$\"3++++++ +]()F9$\"3%****fNolr;&F97$$\"3o**zmmmmm\"*F9$\"3O+g,&[d[]&F97$$\"3_**H LLLL$e*F9$\"3,+!fRvm)QeF97$$\"2'***)************!#<$\"3a++o]F]ohF97$$ \"3.+qmmmmT5F]s$\"3Y**4y5F>$\\'F97$$\"3'***HLLLL$3\"F]s$\"3h+Iv`%pC\"o F97$$\"3+++++++D6F]s$\"3r**zqA&ff7(F97$$\"3%***fmmmmm6F]s$\"3l**HkJ?PL uF97$$\"31+SLLLL37F]s$\"3z**pvF$)[MxF97$$F8F]s$\"3:+?>AD:H!)F97$$\"3-+ qmmmm\"H\"F]s$\"3.++!H!HE<$)F97$$\"3(***HLLLLL8F]s$\"3I+SG8Uw)f)F97$$ \"3+++++++v8F]s$\"33++vg2kt))F97$$\"3%***fmmmm;9F]s$\"3\")**4X0.\">9*F 97$$\"31+SLLLLe9F]s$\"3,+5Bs'=OS*F97$$\"3++++++++:F]s$\"3#***HaC]$)e'* F97$$\"3.+qmmmmT:F]s$\"3^******Qxk2**F97$$\"3'***HLLLL$e\"F]s$\"33+I)> 3;],\"F]s7$$\"3+++++++D;F]s$\"3.+?fz!\\'Q5F]s7$$\"3%***fmmmmm;F]s$\"31 +!QYSw;1\"F]s7$$\"31+SLLLL3F]s$\"3!***z&)HF]s$\"34+I-:`327F]s7$$\"\"#F) $\"3)***HJ$GydA\"F]s7$$\"3'***HLLLL$3#F]s$\"3#****[@w\\2#>1!p9F]s7$$\"3+++++++]FF]s$\"3-+!p0*eM$\\\"F]s7$ $\"31+SLLLLLGF]s$\"3,++^$R,m^\"F]s7$$\"3%***fmmmm;HF]s$\"31+!\\%*pP)Q: F]s7$$\"\"$F)$\"3)***\\aT`6g:F]s7$$\"3'***HLLLL$3$F]s$\"3')**4vw1\\!e \"F]s7$$\"3-+qmmmmmJF]s$\"3()***\\*oh,+;F]s7$$\"3+++++++]KF]s$\"33+g&= mS(=;F]s7$$\"3(***HLLLLLLF]s$\"37+!\\*Q'4nj\"F]s7$$\"3%***fmmmm;MF]s$ \"3/+q(*oa'Rl\"F]s7$$\"3++++++++NF]s$\"31+![ijZ0n\"F]s7$$\"3'***HLLLL$ e$F]s$\"3-+!4c&H\\'o\"F]s7$$\"3%***fmmmmmOF]s$\"3&)**RLsd$=q\"F]s7$$FW F]s$\"3&)**pVZ\"3mr\"F]s7$$\"3(***HLLLLLQF]s$\"3!***\\0H+%3t\"F]s7$$\" 3%***fmmmm;RF]s$\"3*)**Rs5%fXu\"F]s7$$\"\"%F)$\"37+SHxCzd~G\"#t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Using 9 n odes is more efficient in terms of the number of function evaluations \+ required. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "NCint(arctan(x)^2,x=0..4,numpoints=9,info=1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive~Newton-Cotes~quadrature~wit h~9~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~eva luations~-->~G\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The fol lowing loop gives an indication of what happens when we vary the numbe r of nodes used." }}{PARA 0 "" 0 "" {TEXT -1 39 "Notice that the value s obtained become " }{TEXT 278 10 "unreliable" }{TEXT -1 53 " when the number of nodes is greater than about 17. " }}{PARA 0 "" 0 "" {TEXT -1 20 "Using the procedure " }{TEXT 0 5 "NCint" }{TEXT -1 96 " with 9 \+ nodes is probably the best option, even though we can do slightly bett er with 16 nodes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 79 "for i from 6 to 25 do \n NCint(arctan(x)^2 ,x=0..4,numpoints=i,info=1);\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive~Newton-Cotes~quadrature~with~6~nodesG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"$@#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive~Newton-Cotes~quadrature~with~7~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"$@\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive~Newton-Cotes~quadrature~with~8~nodesG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G \"$8\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%Nadaptive~Newton-Cotes~quadrature~with~9~nodesG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~- ->~G\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~with~10~node sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations ~-->~G\"#t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~with~11~n odesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluati ons~-->~G\"#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~wit h~12~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~ev aluations~-->~G\"#n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~w ith~13~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~ evaluations~-->~G\"#t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature ~with~14~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~functio n~evaluations~-->~G\"#z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrat ure~with~15~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~func tion~evaluations~-->~G\"#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4Enr U!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadr ature~with~16~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~fu nction~evaluations~-->~G\"#h" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4E nrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~qua drature~with~17~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~ function~evaluations~-->~G\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ 4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~q uadrature~with~18~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~o f~function~evaluations~-->~G\"#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+KEnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cote s~quadrature~with~19~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumbe r~of~function~evaluations~-->~G\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+2EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Co tes~quadrature~with~20~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnum ber~of~function~evaluations~-->~G\"#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton- Cotes~quadrature~with~21~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dn umber~of~function~evaluations~-->~G\"#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newt on-Cotes~quadrature~with~22~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %Dnumber~of~function~evaluations~-->~G\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadap tive~Newton-Cotes~quadrature~with~23~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%Oadaptive~Newton-Cotes~quadrature~with~24~nodesG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton-Cotes~quadrature~with~25~nodesG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#\\" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8EnrU!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Knowing the value of the \+ integral correct to 10 digits we can obtain this value with the proced ure " }{TEXT 0 5 "NCint" }{TEXT -1 8 " in the " }{TEXT 278 12 "non-ada ptive" }{TEXT -1 44 " mode with even fewer function evaluations. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "NCint(arctan(x)^2,x=0..4,numpoints=7,adaptive=false,factor=9,info= 1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%encompound~non-adaptive~Newt on-Cotes~quadrature~with~7~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% Crepeated~9~times~over~54~intervalsG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%Dnumber~of~function~evaluations~-->~G\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4EnrU!\"*" }}}{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 293 74 ".. an example in which a lar ge number of function evaluations are required" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(1+sin(x^2)*exp(-x /15),x = 0 .. 10);" "6#-%$IntG6$,&\"\"\"F'*&-%$sinG6#*$%\"xG\"\"#F'-%$ expG6#,$*&F-F'\"#:!\"\"F5F'F'/F-;\"\"!\"#5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> 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}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Using 11 nodes \+ is more efficient. 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KFfoFiw7$$\"3)****piGs')H$FfoFdw7$$\"3;++h-@@xLFfo$\"3L+!*zw1Z/#*F47$$ \"3!)***[*=>vbMFfoFjv7$$\"3)****)GNVFfo$\"33++QI))eB)*F47$$\"3!***f-:(H#)R%FfoF^s7$ $\"35+gOJ&pnZ%Ffo$\"3k**>@jxMO%*F47$$\"3E+gqZ$4`b%Ffo$\"3/+Sk]Ut`\"*F4 7$$\"3c**f/k\"\\Qj%Ffo$\"3<+5G#)[I7))F47$$\"3l**\\Q!)*)Q7ZFfo$\"3&***p 2[)4ZT)F47$$\"3#)**\\s'zG4z%Ffo$\"31+q,g=MlzF47$$\"3#***R18'o%p[Ffo$\" 35+gMOC=quF47$$\"35+SSH%3![\\Ffo$\"3f**>/$[wl$pF47$$\"3G+SuX#[l-&Ffo$ \"3K+qFP,&HP'F47$$\"3d**R3i!)30^Ffo$\"3g**p0M3\\)y&F47$$\"3m**HUyyi$=& Ffo$\"3')**HBu2\"G>&F47$$\"3#***Rw%pn@E&Ffo$\"3;+gEA_h&f%F47$$\"3,+I56 vqS`Ffo$\"3x**4`*R&R1SF47$$\"3>+IWFtC>aFfo$\"3s**>MNC;MMF47$$FMFfo$\"3 )***>MOVB()GF47$$\"3e**>7gpKwbFfoFX7$$\"3w**>Ywn'[l&Ffo$\"3)***fg#>T#) *=F47$$\"3%***>!Gf1Mt&Ffo$\"3%)**>Dl4Fo9F47$$\"3.+594k%>\"eFfo$\"31+qV X!yx3\"F47$$\"3?+5[Di[!*eFfoFD7$$\"3Q+5#=/E!pfFfo$\"35++zkXR#*[F-7$$\" 3e***f\"eecZgFfo$\"3)***z8~G\"#\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g0pXS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 48 ": Sim pson's rule works well with this integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "simp(sin(1-cos(x)) ,x=0..2*Pi,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximat ion~with~2~intervals~--->~~~G$\"0!\\sah&)3Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~~~G$\"0]cuXfpZ%!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~ --->~~~G$\"0SDZ.ip.%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroxima tion~with~16~intervals~--->~~~G$\"0I:\"G-pXS!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~intervals~--->~~~G$\"0$*G-c!pXS !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~64~interva ls~--->~~~G$\"0$*G-c!pXS!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g0p XS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "In fact we can obtain a value of the integral which is correct to 10 digits using Simpson's rule with 22 intervals. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "evalf(eval f(student[simpson](sin(1-cos(x)),x=0..2*Pi,22),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g0pXS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "NCint(sin(1-cos(x)),x=0..2*P i,numpoints=3,adaptive=false,factor=11,info=2,\n color=[coral,navy]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%encompound~non-adaptive~Newton-Co tes~quadrature~with~3~nodesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Drepe ated~11~times~over~22~intervalsG" }}{PARA 13 "" 1 "" {GLPLOT2D 529 273 273 {PLOTDATA 2 "6(-%'CURVESG6%797$$\"\"!F)F(7$$\"31+]W@L*f&G!#=$ \"35+]K%)\\f\\S!#>7$$\"38++*Gk')>r&F-$\"3-+!\\&*f03e\"F-7$$\"3k**\\Lk* zzc)F-$\"3G+?LQtF$Q$F-7$$\"3.+!y&GtRU6!#<$\"3k**HIML`=bF-7$$\"3/+?sgm* zU\"F>$\"31+q\"eY-Lc(F-7$$\"3$***p'G*ff8$\"3k**\\[Xs(f4*F-7$$\"3/+? ,D`>**>F>$\"3%****eo\"f_z)*F-7$$\"31+g:dYz%G#F>$\"3)****)o!o'ok**F-7$$ \"3%)****H*)RRqDF>$\"3X+?:W%)[O'*F-7$$\"3=+]W@L*f&GF>$\"3c**>7Rz.a#*F- 7$$\"31++f`EfTJF>$\"3G+qDoU(H4*F-7$$\"3')**Rt&)>>FMF>$\"3a+!=\"Rz.a#*F -7$$\"35+!yyJ\"z7PF>$\"3_***[TW)[O'*F-7$$\"3)***H-]1R)*RF>$\"3o**zo!o' ok**F-7$$\"3K+!o@)**)RG%F>FN7$$\"35+?J9$*epXF>$\"3y***)[Xs(f4*F-7$$\"3 !***fXY')=b[F>$\"3%)**p#eY-Lc(F-7$$\"3z**4gyzyS^F>$\"33+!3VLL&=bF-7$$ \"3n**fu5tQEaF>F97$$\"3N++*Gk')>r&F>$\"3%)**>b*f03e\"F-7$$\"39+S.vfe(* fF>$\"3s**\\N%)\\f\\SF07$$\"37++=2`=$G'F>F(-%'COLOURG6&%$RGBG$\"*++++ \"!\")$\")AR!)\\FerF(-%&STYLEG6#%%LINEG-F$6%F&-F`r6&Fbr$\")!\\DP\"FerF `s$\")viobFer-Fir6#%&POINTG-%&TITLEG6#%2evaluation~pointsG-%+AXESLABEL SG6$Q!6\"F^t-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$%(DEFAULTGFgt" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G \"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g0pXS!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "SPint(s in(1-cos(x)),x=0..2*Pi,adaptive=false,factor=11,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Gusing~Simpson's~rule~with~22~intervalsG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~ G\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g0pXS!\"*" }}}{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 291 32 ".. New ton-Cotes integration via " }{TEXT 260 9 "evalf/Int" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "One can \+ instruct Maple's numerical integration procedure " }{TEXT 0 15 "evalf( Int(...))" }{TEXT -1 54 " to perform Newton-Cotes integration with the option \"" }{TEXT 278 14 "method=_NCrule" }{TEXT -1 55 "\". However t his is a fixed order method using 9 nodes. " }}{PARA 0 "" 0 "" {TEXT -1 49 "We illustrate this in calculating the integral: " }{XPPEDIT 18 0 "Int(sin(x^2),x = 0 .. Pi/2);" "6#-%$IntG6$-%$sinG6#*$%\"xG\"\"#/ F*;\"\"!*&%#PiG\"\"\"F+!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf(Int(si n(x^2),x=0..Pi/2,method=_NCrule));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+*Gj6G)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "It is possible to determine the number of function evalua tions used." }}{PARA 0 "" 0 "" {TEXT -1 38 "The following function, or procedure, " }{TEXT 0 7 "f_count" }{TEXT -1 19 " returns the value " }{TEXT 278 8 "sin(x^2)" }{TEXT -1 14 " for an input " }{TEXT 279 1 "x " }{TEXT -1 32 ", but also increments a counter " }{TEXT 278 8 "num_ev al" }{TEXT -1 40 " by 1 whenever the procedure is called. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "f_co unt := proc(x)\n global num_eval;\n num_eval := num_eval+1;\n si n(x^2);\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(f_countGf*6#% \"xG6\"F(F(C$>%)num_evalG,&F+\"\"\"F-F--%$sinG6#*$)9$\"\"#F-F(6#F+F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "num_eval := 0;\nevalf(Int('f_count(x)',x=0..Pi/2,method=_NCru le));\nprint(`The number of function evaluations used is `||num_eval|| `.`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)num_evalG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*Gj6G)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OThe~number~of~function~evaluations~used~is~43.G" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Some of these 43 e valuations may not be related to the evaluation of the integral." }} {PARA 0 "" 0 "" {TEXT -1 75 "Another way to obtain the number of funct ion evaluations is to use Maple's " }{TEXT 278 9 "infolevel" }{TEXT -1 11 " mechanism." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 104 "infolevel[`evalf/int`] := 1:\nevalf(Int(sin (x^2),x=0..Pi/2,method=_NCrule));\ninfolevel[`evalf/int`] := 0:" }} {PARA 6 "" 1 "" {TEXT -1 36 "evalf/int/control: Trying easyproc" }} {PARA 6 "" 1 "" {TEXT -1 52 "evalf/int/control: \"applying Newton-Co tes routine\"" }}{PARA 6 "" 1 "" {TEXT -1 88 "From quanc8, result = .8 281163288549 integrand evals = 33 error = .2445130364066e-11" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*Gj6G)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 3 "qua" }{TEXT 256 2 "nc" } {TEXT 260 1 "8" }{TEXT -1 11 " refers to " }{TEXT 257 3 "qua" }{TEXT -1 49 "drature (that is, numerical integration) using a " }{TEXT 256 1 "N" }{TEXT -1 6 "ewton-" }{TEXT 256 1 "C" }{TEXT -1 17 "otes method \+ with " }{TEXT 260 1 "8" }{TEXT -1 12 " intervals. " }}{PARA 0 "" 0 "" {TEXT -1 116 "The number of function or integrand evaluations needed i s 33, which is the same as the number used by the procedure " }{TEXT 0 5 "NCint" }{TEXT -1 15 " with 9 nodes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "NCint(sin(x^2),x=0.. Pi/2,numpoints=9,info=true,\n color=[grey,blue]);\nevalf(evalf(Int(s in(x^2),x=0..Pi/2),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive ~Newton-Cotes~quadrature~with~9~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 480 86 86 {PLOTDATA 2 "6+-%'CURVESG6%7$7$$\"3;+](Rj\")R&y!#=$\"\"!F,7$ F($\"\"\"F,-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%*THICKNESSG6#\"\"#-F$ 6%7$7$$\"3B+!))p\"3*p#RF*F+7$F?F.F0F7-F$6%7$7$$\"35+g4Xs4y6!#$\"3$**** 4`0p&4C!#?7$$\"3e++ZUqZ<)*F-$\"3I**HR@j8Q'*F07$$\"39++Pc:is9!#=$\"3))* *f7tUWo@F-7$$\"37+S\\3a\\j>F9$\"3Y+]zN#fV&QF-7$$\"31+qhg#pVX#F9$\"3?+! )=&)eG?gF-7$$\"3/+5u7JCXHF9$\"3%)***z#>Cej')F-7$$\"3)***R'['p6OMF9$\"3 ++!>zp[z<\"F97$$\"3B+!))p\"3*p#RF9$\"30+?s.1-O:F97$$\"3<+56pY'yT%F9$\" 3%***z@B\\QR>F97$$\"3()**\\B@&Q(3\\F9$\"3))***)oDAK'Q#F97$$\"3O+!eLP7' *R&F9$\"3#)**\\7M&\\W(GF97$$\"31+?[Di[!*eF9$\"3D+SSZsd+MF97$$\"3e+]gx+ O\"Q'F9$\"3A+IX'ef0'RF97$$\"3E+!H(HRBsoF9$\"3/+qxFJ9\\XF97$$\"3m**>&=y 2JO(F9$\"3S++to#>)f^F97$$\"3O**f(Rj\")R&yF9$\"3I+qa$*yo%y&F97$$\"3a**z 4'[b[M)F9$\"3#***fsM5M9kF97$$\"3N+?AQ$Hd$))F9$\"3W+5q')oxPqF97$$\"3u** \\M!>.mK*F9$\"3Y***ps.cBk(F97$$\"3v***pC/xu\")*F9$\"3w**R]=J\"Q@)F97$$ \"3$****e%*3N3.\"!#<$\"3!***z(H4Qjt)F97$$\"3)***>nuC#*z5F^r$\"3)****4N !>u#>*F97$$\"3%***R))f)4!H6F^r$\"3g***\\h,@Zc*F97$$\"35+g4Xs4y6F^r$\"3 &)***ftCMK$)*F97$$\"32+!3.j%=F7F^r$\"3w**>,VH+z**F97$$\"37+5_:?Fw7F^r$ \"3q**RcJ=9$)**F97$$\"32+It+%f`K\"F^r$\"3K**\\Q!\\Az#)*F97$$\"37+g%fyY WP\"F^r$\"3-+]g\"=kw\\*F97$$\"3)***p:rT`B9F^r$\"3K+SNbxtz*)F97$$\"3/++ Pc:is9F^r$\"39+5NKull#)F97$$\"3****>eT*3<_\"F^r$\"3G+qd\"yAAN(F97$$\"3 /+]zEjzq:F^r$\"3R+?RE&fEC'F9-%'COLOURG6&%$RGBG$\")=THv!\")F\\vF\\v-%&S TYLEG6#%%LINEG-F$6%F&-Fiu6&F[vF(F($\"*++++\"F^v-F`v6#%&POINTG-%'SYMBOL G6#%'CIRCLEG-%&TITLEG6#%2evaluation~pointsG-%+AXESLABELSG6$Q!6\"Fgw-%% VIEWG6$%(DEFAULTGF\\x" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Dnumber~of~function~evaluations~-->~G\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*Gj6G)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)G j6G)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Using 13 nodes is more efficient." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "NCint(sin(x^2),x=0..Pi/ 2,numpoints=13,info=true,\n color=[grey,blue]);\nevalf(evalf(Int(sin (x^2),x=0..Pi/2),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~N ewton-Cotes~quadrature~with~13~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 322 56 56 {PLOTDATA 2 "6)-%'CURVESG6%7$7$$\"3;+](Rj\")R&y!#=$\"\"!F,7$ F($\"\"\"F,-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%*THICKNESSG6#\"\"#-F$ 6%7'7$F+F+7$F+F.7$$\"3c'*[zEjzq:!#-%&COLORG6&F3$\"\"&!\"\"F HFHF7-%%TEXTG6$7$$\".vRj\")R&y!#8$\"\"$F,Q1bisection~points6\"-FL6$7$F O$!\"#F,Q&~---~FU-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!FUF]o-%%FONTG 6#%(DEFAULTG-%%VIEWG6$FaoFao" 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" }}} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7;7$ $\"\"!F)F(7$$\"3Z***z\\p%)\\a'!#>$\"3#***4)[OpOG%!#?7$$\"3*)**f**Qp**3 8!#=$\"38+?)=9*Q8F4$\"3Y+]zN#fV&QF-7$$\"3y**>*z(Q* zh#F4$\"3!)**z&Q3F&[oF-7$$\"3G++\\ZB\\sKF4$\"3-++,xZ()o5F47$$\"3B+!))p \"3*p#RF4$\"30+?s.1-O:F47$$\"3=+g['G*[\"e%F4$\"3/+5fo_i$3#F47$$\"3d**R )fv()fB&F4$\"3()**45LCM2FF47$$\"31+?[Di[!*eF4$\"3D+SSZsd+MF47$$\"3e++) \\p%)\\a'F4$\"3)***RG,/(Q:%F47$$\"3'***zZkJ[*>(F4$\"3!)***pvYjU&\\F47$ $\"3O**f(Rj\")R&yF4$\"3I+qa$*yo%y&F47$$\"3a**HZ.,[3&)F4$\"3$)**\\U:TVB mF47$$\"3O+?(HdyH;*F4$\"3&)***)GFDxVuF47$$\"3b+!pC/xu\")*F4$\"3C+?]=J \"Q@)F47$$\"3)***p>^v>Z5!#<$\"3u***=Ib.n*))F47$$\"36+g9)RZE6\"Fep$\"37 +S@YOF^%*F47$$\"35+g4Xs4y6Fep$\"3&)***ftCMK$)*F47$$\"37+g/#4ZNC\"Fep$ \"3++]4Hl-(***F47$$\"37+g**Qp**38Fep$\"3m**>_+%*Q)*)*F47$$\"37+g%fyYWP \"Fep$\"3-+]g\"=kw\\*F47$$\"37+g*Gj'*)R9Fep$\"3'***zjNPyj()F47$$\"3.+] %)zkM0:Fep$\"3M+!oE#ozywF47$$\"3/+]zEjzq:Fep$\"3R+?RE&fEC'F4-%'COLOURG 6&%$RGBG$\")=THv!\")FdsFds-%&STYLEG6#%%LINEG-F$6%F&-Fas6&FcsF(F($\"*++ ++\"Ffs-Fhs6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-%&TITLEG6#%2evaluation~poin tsG-%+AXESLABELSG6$Q!6\"F_u-%%VIEWG6$%(DEFAULTGFdu" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)Gj6G)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)Gj6G)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "A single bisection is performed so that a total of " }{XPPEDIT 18 0 "2*`.`*13-1=25" "6#/,&*(\"\"#\"\"\"%\".GF' \"#8F'F'F'!\"\"\"#D" }{TEXT -1 31 " function evaluations are used." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 " }{TEXT 294 53 ".. a problematic example for Newton-Cotes integration" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(x),x = 0 .. 2)" "6#-%$IntG6$-%%sqrtG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "NCint(sqrt(x),x=0..2,numpoints=9,info=true);\nevalf(evalf(int(sq rt(x),x=0..2),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Nadaptive~Newt on-Cotes~quadrature~with~9~nodesG" }}{PARA 7 "" 1 "" {TEXT -1 39 "Warn ing, reached max subdivision depth\n" }}{PARA 13 "" 1 "" {GLPLOT2D 460 97 97 {PLOTDATA 2 "6I-%'CURVESG6%7$7$$\"\"\"\"\"!$F*F*7$F(F(-%&COL ORG6&%$RGBGF*$\"\"&!\"\"$F)F3-%*THICKNESSG6#\"\"#-F$6%7$7$$\"3++++++++ ]!#=F+7$F=F(F-F5-F$6%7$7$$\"3++++++++DF?F+7$FEF(F-F5-F$6%7$7$$\"3+++++ ++]7F?F+7$FLF(F-F5-F$6%7$7$$\"3+++++++]i!#>F+7$FSF(F-F5-F$6%7$7$$\"3++ +++++DJFUF+7$FenF(F-F5-F$6%7$7$$\"3++++++]i:FUF+7$F\\oF(F-F5-F$6%7$7$$ \"3++++++]7y!#?F+7$FcoF(F-F5-F$6%7$7$$\"3++++++D1RFeoF+7$F[pF(F-F5-F$6 %7$7$$\"3+++++]7`>FeoF+7$FbpF(F-F5-F$6%7$7$$\"3-++++]il(*!#@F+7$FipF(F -F5-F$6%7$7$$\"3+++++D\"G)[F[qF+7$FaqF(F-F5-F$6%7$7$$\"3++++]iSTCF[qF+ 7$FhqF(F-F5-F$6%7$7$$\"3++++DJq?7F[qF+7$F_rF(F-F5-F$6%7$7$$\"3++++Dc^. 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Ln1dF[]m7$F\\dm$\"+*yg9s#F[]mFgcm7&F[dm7$$\"+DS;!G#F[]m$\"+fy,>cF[]m7$ Fedm$\"+4J$**p#F[]mF`dm7&Fddm7$$\"+&=3DQ#F[]m$\"+iTu[bF[]m7$F^em$\"+U \\2#p#F[]mFidm7&F]em7$$\"+!H0U]#F[]m$\"+mV\"H[&F[]m7$Fgem$\"+#y4(*p#F[ ]mFbem7&Ffem7$$\"+-()H2EF[]m$\"+_uhXaF[]m7$F`fm$\"+cHS@FF[]mF[fm7&F_fm 7$$\"+[3AFFF[]m$\"+8'=^U&F[]m7$Fifm$\"+O&HNw#F[]mFdfm7&Fhfm7$$\"+nAPLG F[]m$\"+#>*\\EaF[]m7$Fbgm$\"+Re,9GF[]mF]gm7&Fagm7$$\"+)>P)\\HF[]m$\"+S Y[XaF[]m7$F[hm$\"+Z!)4!)GF[]mFfgm7&Fjgm7$$\"+a$R21$F[]m$\"+e#=^Z&F[]m7 $Fdhm$\"+`I**[HF[]mF_hm7&Fchm7$$\"+>TXwJF[]m$\"+*f\"y5bF[]m7$F]im$\"+K *3;-$F[]mFhhm7&F\\im7$$\"+&R;FG$F[]m$\"+i-*3a&F[]m7$Ffim$\"+9a*Q3$F[]m Faim7&Feim7$$\"+++++MF[]m$\"+twhjbF[]m7$F_jm$\"+'yn@9$F[]mFjim-F_jl6&F a_l$\"#&)!\"#FhjmFhjm-%&STYLEG6#%,PATCHNOGRIDG-%%TEXTG6%7$$\"#CFcjl$\" #hFcjlQ)y~=~f(x)6\"F^_l-F`[n6%7$Fc[n$\"$X#FjjmQ)y~=~g(x)Fh[nFail-F`[n6 %7$$\"\"(Fcjl$\"\")FcjlQ&x~=~aFh[n-F__l6&Fa_lFf_lFf_lFf_l-F`[n6%7$$\"# MFcjlFd\\nQ&x~=~bFh[nFg\\n-%+AXESLABELSG6%Q\"xFh[nQ\"yFh[n-%%FONTG6#%( DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;Fe_l$\"$.%Fjjm;Fd\\nFe[n" 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" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f (x)>=g(x)" "6#1-%\"gG6#%\"xG-%\"fG6#F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" } {TEXT -1 57 ". Then the area of the region enclosed between to graphs \+ " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 24 " and the vertical lines " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 13 " is given by \+ " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(``(f(x)-g(x) ),x = a .. b);" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\" /F-;%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 296 9 "Question:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "Calculate (correct to 10 digits) the area of the reg ion enclosed between the curves " }{XPPEDIT 18 0 "y=1/(1+5*ln(x^2+1)) " "6#/%\"yG*&\"\"\"F&,&F&F&*&\"\"&F&-%#lnG6#,&*$%\"xG\"\"#F&F&F&F&F&! \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2*x-x^2;" "6#/%\"yG,&*& \"\"#\"\"\"%\"xGF(F(*$F)F'!\"\"" }{TEXT -1 43 " between their two poin ts of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 77 "Use Newton-Cotes \+ integration to perform any necessary numerical integration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 55 "The following picture i llustrates the region involved. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 490 "f := x -> 2*x-x^2: g := x - > 1/(1+5*ln(x^2+1)):\nxL := fsolve(f(x)=g(x),x=0..1):\nxR := fsolve(f( x)=g(x),x=1..2): m := 20:\np1 := plot(f(x),x=xL..xR,numpoints=m,adapti ve=false):\np2 := plot(g(x),x=xL..xR,numpoints=m,adaptive=false):\npts 1 := op(1,op(1,p1)): pts2 := op(1,op(1,p2)):\np1 := plots[polygonplot] ([seq([pts1[i],pts1[i+1],pts2[i+1],pts2[i]],i=1..m-1)],\n color=grey, style=patchnogrid):\np2 := plot([f(x),g(x)],x=-.3..2.3,y=-.3..1.1,colo r=[red,blue],thickness=2):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 326 292 292 {PLOTDATA 2 "6'-%)POLYGONSG677&7$$\"+j\"GZs$! #5$\"+Pj4igF*7$$\"+m>w)e%F*$\"+%H]=2(F*7$F.$\"+tN#R6&F*7$F($\"+Qj4igF* 7&F-7$$\"+eJbS`F*$\"+8b&*GyF*7$F:$\"+RnJNWF*F27&F97$$\"+\")4-'='F*$\"+ /kNX&)F*7$FC$\"+d>o;QF*F>7&FB7$$\"+9J4PqF*$\"+y#=@7*F*7$FL$\"+h**z?LF* FG7&FK7$$\"+\\-7%)yF*$\"+)G0Bb*F*7$FU$\"+QQ]EHF*FP7&FT7$$\"+IGUp')F*$ \"+Sk&H#)*F*7$Fhn$\"+0\"o(HEF*FY7&Fgn7$$\"+n\"eD[*F*$\"+&RDK(**F*7$Fao $\"+\\W\"pP#F*F\\o7&F`o7$$\"+*Q]B.\"!\"*$\"+BX`*)**F*7$Fjo$\"+\"f75;#F *Feo7&Fio7$$\"+q_<;6F\\p$\"+nI.l)*F*7$Fdp$\"+Py*>)>F*F_p7&Fcp7$$\"+j%* R-7F\\p$\"+uXM!f*F*7$F]q$\"+^fbF=F*Fhp7&F\\q7$$\"+zdMy7F\\p$\"+tiBD#*F *7$Ffq$\"+2u\\6p\\9F\\p$\"+P8xxzF*7$Fhr$\"+[8f,:F*Fcr7&Fgr7$$\"+E IUK:F\\p$\"+8sDlrF*7$Fas$\"+6=**>9F*F\\s7&F`s7$$\"+_2b2;F\\p$\"+L3#)3j F*7$Fjs$\"+Qt^a8F*Fes7&Fis7$$\"+&)[)op\"F\\p$\"+lX^V^F*7$Fct$\"+?rg&G \"F*F^t7&Fbt7$$\"+9EcsF\\p$\"+EW!\\8 \"F*7$F^v$\"+JW!\\8\"F*Fiu-%'COLOURG6&%$RGBG$\")=THv!\")FivFiv-%&STYLE G6#%,PATCHNOGRIDG-%'CURVESG6%7S7$$!3))**************H!#=$!3Y********** ****oFgw7$$!3!omm;*RFLCFgw$!33xa=:-jeaFgw7$$!3ILL3A(o,%>Fgw$!3-9<=6Hwc UFgw7$$!3ymm;u-i&Q\"Fgw$!3E/:v-\\BjHFgw7$$!31mmm\">fRF)!#>$!3i)pQ!R-DB 3YFgw$\"35!*\\13i%G4(Fgw7$$\"3oJLeM`'45&Fgw$\"3YzXqMf%**f (Fgw7$$\"3?mmmhV\"po&Fgw$\"3Ht-kF#H(R\")Fgw7$$\"39lmmY+H$='Fgw$\"3EDE@ 8DFV&)Fgw7$$\"3o***\\7(HpgnFgw$\"3S%3\\t**)o]*)Fgw7$$\"3Klmm'R#zrsFgw$ \"3m<9JF$)ob#*Fgw7$$\"3y)**\\7Z\\D$yFgw$\"3!R8U0#e@I&*Fgw7$$\"3=***\\P \"G_m$)Fgw$\"3Iy+?G_\"Q\"Fc_l$\"3]#> BT6xoa)Fgw7$$\"3Nmmm>\"yPV\"Fc_l$\"3'3^%)*GaO=\")Fgw7$$\"3%*****\\(4=* *[\"Fc_l$\"3Y2)>ud-)*f(Fgw7$$\"3&HLe9E*yS:Fc_l$\"3s4N\"pupa2(Fgw7$$\"3 %*****\\!)[S'f\"Fc_l$\"3$f!y:&=7IW'Fgw7$$\"3Cm;zV[t[;Fc_l$\"35Cg]C5V\" z&Fgw7$$\"3v**\\iLZV.R$Fgw7$$\"3MLLL\"RCo '=Fc_l$\"3)4'f*euah[#Fgw7$$\"35LL3c#o>#>Fc_l$\"3?xluZ`u*\\\"Fgw7$$\"3s m;z,blw>Fc_l$\"3Szn+#3.Wh%F\\y7$$\"3*)*****4S2p-#Fc_l$!3W([dG-\")QX&F \\y7$$\"3!om;%[<]%3#Fc_l$!3#)p!)=B/Wh[,O@Fc_l$!3%)f%\\u p'H0HFgw7$$\"3))**\\7(\\Q4>#Fc_l$!3kNNMR]M$=%Fgw7$$\"3(***\\([Y2NC#Fc_ l$!3K!\\)*>:3JY&Fgw7$$\"3#)*************H#Fc_l$!3-&*************oFgw-F fv6&Fhv$\"*++++\"F[w$\"\"!FaglF`gl-%*THICKNESSG6#\"\"#-Faw6%7fn7$Few$ \"3e())Gq#[m))pFgw7$F[x$\"3Ulf/?nLmxFgw7$F`x$\"3Dh7#\\!4bS%)Fgw7$Fex$ \"3r`A)\\R#pJ\"*Fgw7$Fjx$\"3)y/Ld7F,n*Fgw7$$!3a++DJJ5'\\&F\\y$\"3soLfI CV^)*Fgw7$F`y$\"3e9l?#>0K'**Fgw7$$!3;-+DJ+Ru?F\\y$\"3=-38I``y**Fgw7$$! 3Dpmm\"*H`I9F\\y$\"3AnLT!Qz(*)**Fgw7$$!3]jL$3_fn'y!#?$\"3Qncedn!p***Fg w7$$!3[M++D\"*=G9F]jl$\"3_%GN\\,)*)****Fgw7$$\"3a%HL3F\"Q5]F]jl$\"33K* es(\\u)***Fgw7$$\"3Oimmm^*[9\"F\\y$\"31ac!oz]M***Fgw7$$\"3E&**\\i?_()y \"F\\y$\"3I%R'pJ*HS)**Fgw7$Fey$\"3SDb4Z!30(**Fgw7$$\"3yhm;HfJ*4&F\\y$ \"3km:XG!==()*Fgw7$Fjy$\"38-AX'QD\"3(*Fgw7$F_z$\"3C)[!3dR&f>*Fgw7$Fdz$ \"3bfgjn/$H_)Fgw7$Fiz$\"359k%eH*4_xFgw7$F^[l$\"31Cz&QH!HnqFgw7$Fc[l$\" 306dZ@\"y[L'Fgw7$Fh[l$\"3%\\Be'4oBmcFgw7$F]\\l$\"3_]Q/#)Hq%4&Fgw7$Fb\\ l$\"3Pfc&))Ruuj%Fgw7$Fg\\l$\"3cC0k'zH[;%Fgw7$F\\]l$\"3Z13=LCY=QFgw7$Fa ]l$\"3k4$QU$zkpMFgw7$Ff]l$\"335@')yLo-KFgw7$F[^l$\"3-eq3v(>\"[HFgw7$F` ^l$\"3#Q3%[g()RPFFgw7$Fe^l$\"3Qjk0INPXDFgw7$Fj^l$\"3!)4[()>%=.R#Fgw7$F __l$\"3Z2p``Z\\UAFgw7$Fe_l$\"3&zSK'Q%Gr5#Fgw7$Fj_l$\"34([Vp,%>-?Fgw7$F _`l$\"3FA&G.Gg/!>Fgw7$Fd`l$\"3d&[+D#))H1=Fgw7$Fi`l$\"39%4971#[B " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The required area \+ is " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = a .. b);" "6#-%$IntG6$-%!G6 #,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;%\"aG%\"bG" }{TEXT -1 9 ", wh ere " }{XPPEDIT 18 0 "f(x) = 2*x-x^2;" "6#/-%\"fG6#%\"xG,&*&\"\"#\"\" \"F'F+F+*$F'F*!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g(x) = 1/(1+ 5*ln(x^2+1));" "6#/-%\"gG6#%\"xG*&\"\"\"F),&F)F)*&\"\"&F)-%#lnG6#,&*$F '\"\"#F)F)F)F)F)!\"\"" }{TEXT -1 6 ", and " }{TEXT 299 1 "a" }{TEXT -1 5 " and " }{TEXT 327 1 "b" }{TEXT -1 9 " are the " }{TEXT 298 1 "x " }{TEXT -1 81 " coordinates of the left and right hand points of inte rsection of the two curves " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG 6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = g(x)" "6#/%\"yG-%\"gG 6#%\"xG" }{TEXT -1 15 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "A numerical root-finding method is n eeded to calculate the " }{TEXT 300 1 "x" }{TEXT -1 62 " coordinates o f the points of intersection of the two curves. " }}{PARA 0 "" 0 "" {TEXT -1 49 "We could use the secant method via the procedure " } {TEXT 0 6 "secant" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f := x -> 2*x-x^2;\ng := x - > 1/(1+5*ln(x^2+1));\na := secant(f(x)=g(x),x=0.3..0.4,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&\"\"#\"\"\"9$F/F/*$)F0F.F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F- F-*&\"\"&F--%#lnG6#,&*$)9$\"\"#F-F-F-F-F-F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".!p2')4TP!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".gvE;Vs$!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".Z\"e(GZs$ !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".$H j\"GZs$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~ G$\".rK;GZs$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+j\"GZs$!# 5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b := secant(f(x)=g(x),x=1.8..2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".hkPVm$>!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".[6n*QT>!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".?\"* zY:%>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$ \".!okjaT>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~-> ~~~G$\".=ZOY:%>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 6~~->~~~G$\".=ZOY:%>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+l jaT>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We can check that the two functions " }{XPPEDIT 18 0 "f(x) = 2*x-x ^2;" "6#/-%\"fG6#%\"xG,&*&\"\"#\"\"\"F'F+F+*$F'F*!\"\"" }{TEXT -1 6 " \+ and " }{XPPEDIT 18 0 "g(x) = 1/(1+5*ln(x^2+1));" "6#/-%\"gG6#%\"xG*& \"\"\"F),&F)F)*&\"\"&F)-%#lnG6#,&*$F'\"\"#F)F)F)F)F)!\"\"" }{TEXT -1 73 " have essentially the same floating point value for the numerical \+ values " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 18 " calculated above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "'f(a)'=f(a);\n'g(a)'=g(a);\n'f(b)'=f(b);\n'g(b)'=g(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"aG$\"+Pj4ig!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"aG$\"+Pj4ig!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"bG$\"*V/\\8\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"bG$\"+JW!\\8\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Maple cannot find an anal ytical expression for the indefinite integral " }{XPPEDIT 18 0 "Int(` `(f(x)-g(x)),x);" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\" \"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Int(f(x)-g(x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*&\"\"#\"\"\"%\"xGF)F)*$)F*F(F)!\"\" *&F)F),&F)F)*&\"\"&F)-%#lnG6#,&F)F)F+F)F)F)F-F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"$!\"\"%\"xGF%F&*$)F'\"\"#\"\"\"F+-%$intG6$,$*&F +F+,&F+F+*&\"\"&F+-%#lnG6#,&F+F+F(F+F+F+F&F&F'F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "The required area can be calculated by numerical integration. We can employ Newton-Cotes integ ration via the procedure " }{TEXT 0 5 "NCint" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "NCint(f(x)-g(x),x=a..b,info=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Oadaptive~Newton-Cotes~quadrature~with~13~nodesG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p![ZL)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "In order to check that this last \+ value is correct to 10 digits we could repeat the calculation using a \+ higher precision." }}{PARA 0 "" 0 "" {TEXT -1 85 "Maple's procedures f or numerical root-finding and numerical integration can be used. " }} {PARA 0 "" 0 "" {TEXT -1 93 "We specify that the numerical integration is performed by Maple's 9 point Newton-Cotes rule. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "f := x -> 2*x-x^2;\ng := x -> 1/(1+5*ln(x^2+1));\nDigits := 15:\na := fsolve(f( x)=g(x),x=0..1);\nb := fsolve(f(x)=g(x),x=1..2);\nInt(f(x)-g(x),x=a..b );\nans := evalf(Int(f(x)-g(x),x=a..b,method=_NCrule));\nDigits := 10: \nevalf(ans);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&*&\"\"#\"\"\"9$F/F/*$)F0F.F/!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&\"\"\"F-,&F-F-*&\"\"&F--%#lnG6#,&*$)9$\"\"#F-F-F-F-F-F-!\"\"F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"02sK;GZs$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"0)zrkjaT>!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*&\"\"#\"\"\"%\"xGF)F)*$)F*F(F)!\"\"*&F)F),& F)F)*&\"\"&F)-%#lnG6#,&F)F)F+F)F)F)F-F-/F*;$\"02sK;GZs$!#:$\"0)zrkjaT> !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG$\"0U'>p![ZL)!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p![ZL)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 10" }{TEXT 339 41 " .. calculat ing arc length along a curve " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 10 "arc length" }{TEXT -1 15 " along a curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#% \"xG" }{TEXT -1 13 " from a point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%! G6$%\"aG-%\"fG6#F&" }{TEXT -1 24 " on the curve to a point" }{XPPEDIT 18 0 "``(b,f(b));" "6#-%!G6$%\"bG-%\"fG6#F&" }{TEXT -1 26 " is given b y the integral " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t(sqrt(1+(dy/dx)^2),x = a .. b) = Int(sqrt(1+`f '`(x)^2),x = a .. b); " "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*$*&%#dyGF+%#dxG!\"\"\"\"#F+/%\"xG; %\"aG%\"bG-F%6$-F(6#,&F+F+*$-%$f~'G6#F3F1F+/F3;F5F6" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 30 " provided that the derivative " } {XPPEDIT 18 0 "dy/dx = `f '`(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6# %\"xG" }{TEXT -1 37 " exists throughout the interval from " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = b " "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 338 4 "Note" }{TEXT -1 135 ": The integral invol ved here often cannot be found analytically, and in such situations it is appropriate to use numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 101 "Even if Maple can find an analytical expression for the \+ integral it may be horrendously complicated. " }}{PARA 0 "" 0 "" {TEXT -1 67 "Consider the problem of calculating the arc length along \+ the curve " }{XPPEDIT 18 0 "y = x-sin(x);" "6#/%\"yG,&%\"xG\"\"\"-%$si nG6#F&!\"\"" }{TEXT -1 36 " from the origin to the point where " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f := x -> x-sin(x):\n'f(x)'=f(x);\np1 := plot(f(x),x=-.4..2.3,color=red):\np2 : = plot(f(x),x=0..2,color=blue,thickness=3):\nplots[display]([p1,p2]); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"-%$sinGF &!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 400 191 191 {PLOTDATA 2 "6&-%'CUR VESG6$7S7$$!3A+++++++S!#=$!3)*\\\\8pd;e5!#>7$$!3U++]PoZ6MF*$!3a$QaxEF) yl!#?7$$!3a+]i!)fS**GF*$!3Ga!Q\\+&HXSF37$$!3R++DY(GNK#F*$!3#pn0KHp]3#F 37$$!38++vBU$Qu\"F*$!3'))4Mt)=yC))!#@7$$!3w+]7))[*o;\"F*$!3KY#[i2cjk#F C7$$!3p/+D1F)*>jF-$!3U7yfX9R1U!#A7$$!3?c+]i0Q9yF3$!3-<)o04-I&z!#D7$$\" 3'G**\\PuKl%\\F-$\"3hE'f#HY'p,#FN7$$\"3]**\\(oQ8c1\"F*$\"3x22X^_e:?FC7 $$\"3-++]Ug\"Hl\"F*$\"3?&>fLikj^(FC7$$\"3_***\\7a7-<#F*$\"3/V%HRTX&*p \"F37$$\"3[****\\xec_FF*$\"3w^!\\&>KriMF37$$\"3g)***\\7/JPLF*$\"3[w$[< [`0;'F37$$\"3V****\\A,#3!RF*$\"3e3ZXKXu<)*F37$$\"37**\\PC4a7WF*$\"3,o, zR?.=9F-7$$\"3C****\\_k-@]F*$\"39M\\!)*H#H$3#F-7$$\"3,******4N\\ObF*$ \"3c1NiE[W&y#F-7$$\"3r)*\\PRU5OhF*$\"3-t-z9FuyPF-7$$\"3A(****\\Lhom'F* $\"3Vc26\"=.,$[F-7$$\"3!))*\\P*)f=\\sF*$\"3KaB$>r3W='F-7$$\"3C+]7onp.y F*$\"3O/&zAu\\Fo(F-7$$\"3s***\\PfqAQ)F*$\"3kfD(o5[oZ*F-7$$\"34**\\(o(> e8*)F*$\"3jh\"p>,BV8\"F*7$$\"3u(**\\73tm[*F*$\"3S*yOxHx-O\"F*7$$\"3r* \\PuO&>35!#<$\"3!yCFT9ZKi\"F*7$$\"3#**\\i!=Z,g5Fgs$\"3=S9i0/_w=F*7$$\" 3,+]PO5)f6\"Fgs$\"33Vr?1&)Qw@F*7$$\"3%)****\\>(*zt6Fgs$\"3:;C)>0Ld^#F* 7$$\"3u**\\(3-k.B\"Fgs$\"3%pi7)=h`xGF*7$$\"3!**\\P4*G4&G\"Fgs$\"3=bWF> DHcKF*7$$\"3!)**\\7q*feM\"Fgs$\"3`;S)zl;0r$F*7$$\"3s****\\$*=Y+9Fgs$\" 3qY\\;>wL\\TF*7$$\"3#****\\75h(e9Fgs$\"3Mck%H&[I]YF*7$$\"3_*\\P%[))e6: Fgs$\"3dm]9#*4TL^F*7$$\"3!)***\\(HNMp:Fgs$\"3m4Wf]eW$p&F*7$$\"3k*\\PR= 'oB;Fgs$\"3O(*e;k_%3D'F*7$$\"3s*\\(o:**[!o\"Fgs$\"33.w+rA+loF*7$$\"3%) ***\\i%f-O)F*7$ $\"3%******4zj,&=Fgs$\"3sp\"47SP$*)))F*7$$\"3&)**\\it&Gu!>Fgs$\"3Z$f@D ahbj*F*7$$\"3)**\\P\\<>U'>Fgs$\"3fO0\"y&ohS5Fgs7$$\"38++]RQS;?Fgs$\"3< #4$G\\[096Fgs7$$\"33+]7aL@w?Fgs$\"3\\`'>nrS7?\"Fgs7$$\"3t*****pp2(H@Fg s$\"3Ocj-B[(=G\"Fgs7$$\"3))*\\PR#Qu'=#Fgs$\"3wUc85u^q8Fgs7$$\"3%)*\\i] nM8C#Fgs$\"3]w*y%Hu%yX\"Fgs7$$\"3#)*************H#Fgs$\"3czK#yy%Ha:Fgs -%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fe[lFd[l-F$6%7S7$Fd[lFd[l7$$\"39 LLLL3VfVF-$\"3)zLa9:#p!Q\"FN7$$\"3'pmm;H[D:)F-$\"3]!zBSway-*FN7$$\"3LL LLe0$=C\"F*$\"3aK5%=eS$*=$FC7$$\"3ILLL3RBr;F*$\"3O=\"=k7+)oxFC7$$\"3Ym m;zjf)4#F*$\"3_*Gf.e=q`\"F37$$\"3=LL$e4;[\\#F*$\"3Gy0ei%e*zDF37$$\"3p* ***\\i'y]!HF*$\"3WqJ+?$=!pSF37$$\"3,LL$ezs$HLF*$\"3g1(>7=io6'F37$$\"3_ ****\\7iI_PF*$\"32_&p_22Nu)F37$$\"3#pmmm@Xt=%F*$\"3$>3.q!*))H@\"F-7$$ \"3QLLL3y_qXF*$\"3amRtAhuu:F-7$$\"3i******\\1!>+&F*$\"3qUGnKPxf?F-7$$ \"3()******\\Z/NaF*$\"3]%Q'e+GeOEF-7$$\"3'*******\\$fC&eF*$\"3I,<'*QI: %G$F-7$$\"3ELL$ez6:B'F*$\"3;kSA$4Ba&RF-7$$\"3Smmm;=C#o'F*$\"3W')3=$z4J '[F-7$$\"3-mmmm#pS1(F*$\"3^-VbD2AIdF-7$$\"3]****\\i`A3vF*$\"3%f8\"HEm@ eoF-7$$\"3slmmm(y8!zF*$\"3mB%Hbrc(ozF-7$$\"3V++]i.tK$)F*$\"3F&))f'*)Gn 8$*F-7$$\"39++](3zMu)F*$\"3)f/a6eDA2\"F*7$$\"3#pmm;H_?<*F*$\"3Ls][D%)* HB\"F*7$$\"3emm;zihl&*F*$\"3?8#4CtoMR\"F*7$$\"39LLL3#G,***F*$\"3G&=^8@ c2e\"F*7$$\"3*[W8z\"F*7$$\"3!****\\PQ#\\\"3\"Fgs$ \"39Mm'[c@$))>F*7$$\"3BLL$e\"*[H7\"Fgs$\"3YuwN$)fn:AF*7$$\"3#*******pv xl6Fgs$\"3Gl(\\_jDoY#F*7$$\"3z****\\_qn27Fgs$\"36SEv+i$)GFF*7$$\"3%)** *\\i&p@[7Fgs$\"3o]\"4pygz*HF*7$$\"3#)****\\2'HKH\"Fgs$\"31>&*=Tc/:LF*7 $$\"3_mmmwanL8Fgs$\"3o]*fAZrlh$F*7$$\"3'******\\2goP\"Fgs$\"3YxtU<%og& RF*7$$\"3CLLeR<*fT\"Fgs$\"3!*eyF)f+&zUF*7$$\"3'******\\)Hxe9Fgs$\"31vr *\\S5/l%F*7$$\"3Ymm\"H!o-*\\\"Fgs$\"3')325*H6g,&F*7$$\"3))***\\7k.6a\" Fgs$\"3Eg>]mVW:aF*7$$\"3emmmT9C#e\"Fgs$\"3KS%*yi$pI#eF*7$$\"3\"****\\i !*3`i\"Fgs$\"3wA)p+\\VzE'F*7$$\"3QLLL$*zym;Fgs$\"3:U#osc;Rr'F*7$$\"3GL L$3N1#4I%>*F*7$$\"30++DOl5;>Fgs$\"3%=s&G2/O^( *F*7$$\"3/++v.Uac>Fgs$\"3A%piA!p-I5Fgs7$$\"\"#Fe[l$\"3G=V " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = x-si n(x);" "6#/-%\"fG6#%\"xG,&F'\"\"\"-%$sinG6#F'!\"\"" }{TEXT -1 7 ", the n " }{XPPEDIT 18 0 "`f '`(x) = 1-cos(x);" "6#/-%$f~'G6#%\"xG,&\"\"\"F) -%$cosG6#F'!\"\"" }{TEXT -1 41 ", so the required arc length is given \+ by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(1+(1- cos(x))^2),x = 0 .. 2);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**$,&F*F*-%$co sG6#%\"xG!\"\"\"\"#F*/F0;\"\"!F2" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(sqrt(2-2*cos(x)+cos(x)^2),x = \+ 0 .. 2);" "6#/%!G-%$IntG6$-%%sqrtG6#,(\"\"#\"\"\"*&F,F--%$cosG6#%\"xGF -!\"\"*$-F06#F2F,F-/F2;\"\"!F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 261 "The analytical expression which Maple gives for this int egral is so complicated that, when it evaluated numerical with 10 digi t precision, there is a significant error in the numerical value obtai ned. Numerical integration provides a better option in this case." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(sqrt(1+D(f)(x)^2),x=0..2);\nsimplify(%);\nvalue(%);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*$),&F(F(-%$c osG6#%\"xG!\"\"\"\"#F(F(#F(F1/F/;\"\"!F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,(\"\"#\"\"\"*&F(F)-%$cosG6#%\"xGF)!\"\"*$)F+F(F)F)#F )F(/F.;\"\"!F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,V*&#\"\")\"\"&\"\" \"*0F'#\"\"$\"\"%,&\"#:!\"\"*&\"\"(F(F'#F(\"\"#F(F(,&*$F'F2F(F+F/!\"#, &*&F,F(F'F2F(\"\"*F/F/,&F1F(*&F+F(F'F2F(F/,&*&F+F(F'F2F(F'F(F/-%*Ellip ticFG6$,$*(F3F(F'#F(F,,&F5F(F(F(F/F(*&F3F2,&F'F(F5F(#F/F3F(F(F/*&#F&F' F(*0F'F*F-F(F4F6F7F/F:F/FF(F(F(*&#FinF'F(*0F'F*FaoF(F4F6FjnF/F:F/FF(F^rF(F`rF(F'F2F(F/F(*&,2**\"$]#F(-F^q6 #*0F3F2F'FCFOF2FdqFGFcqFGFUF/,.*&F\\qF(,.F\\qF(*&F,F(F'F2F(*&F\\qF(FRF (F/*(F3F(F'F2FRF(F/*&F'F(FQF(F(FdtF/F/F(*(F3F(FewF/F'F2F(*(F\\qF(FewF/ FRF(F/*(FewF/F'F2FRF3F(**F3F(FewF/F'F2FRF(F/*(F'F(FewF/FRF3F(FGF(F3F2F RF(F(*0\"&SI#F(FcwF2F^rF(F`rF(FfvF(FOF2FRF(F(*(FcvF(F`wF(F3F2F/*.\"%OR F(FcwF2FcsF(F^rF(F`rF(FOF2F(*,\"$5\"F(F`wF(F3F2F'F2FRF(F/*0\"&/.\"F(Fc wF2FcsF(F^rF(F`rF(FOF2FRF(F/*.\"%+))F(FcwF2F^rF(F`rF(FfvF(FOF2F/**F]vF (F`wF(F3F2F'F2F(F(,***\"&![GF(F^rF(F`rF(FRF(F/*,\"&OF\"F(F^rF(F`rF(F'F 2FRF(F(*(FivF(F^rF(F`rF(F(**F[wF(F^rF(F`rF(F'F2F/F/F/F(F(F(*&#F,F'F(*& FcsF(FfoF(F(F(*&#F&F'F(*(F'F*F4F/FfoF(F(F(*&#F&F'F(*(F'F*F4F/-F_p6$,$* (F\\qF/F'F2F4F(F(FEF(F(F/*&#F&F'F(*0F'F*F-F(F4F6F7F/F:F/F " 0 "" {MPLTEXT 1 0 43 "NCint(sqrt(1+D(f)(x)^2),x=0. .2,info=true);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~Newton- Cotes~quadrature~with~13~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 521 70 70 {PLOTDATA 2 "6+-%'CURVESG6%7$7$$\"\"\"\"\"!$F*F*7$F(F(-%&COLORG6&%$ RGBGF+$\"\"&!\"\"$F)F3-%*THICKNESSG6#\"\"#-F$6%7$7$$\"3++++++++]!#=F+7 $F=F(F-F5-F$6%7$7$$\"3++++++++:!#$\"3%***4lw.++5!#< 7$$\"3g**\\LLLLL$)F/$\"35+?<@g++5F27$$\"3+++++++]7!#=$\"3++]tP/.+5F27$ $\"3#***pmmmmm;F;$\"3&***zS+g4+5F27$$\"35+SLLLL$3#F;$\"3.+](*\\PB+5F27 $$\"3++++++++DF;$\"31+!o?5$[+5F27$$\"3#***pmmmm;HF;$\"30++\")f9*3+\"F2 7$$\"3%)**RLLLLLLF;$\"32+5?BP^,5F27$$\"3+++++++]PF;$\"3++!)=(o6C+\"F27 $$\"3C+!ommmm;%F;$\"30++0@Ll.5F27$$\"33+ILLLL$e%F;$\"3++SUB>J05F27$$\" 3I+5+++++]F;$\"3-+?M]^Y25F27$$\"3Z+qmmmm;aF;$\"3.+Sy55F27$$\"3%)**R LLLLLeF;$\"3.+!4e=\"e85F27$$\"3+++++++]iF;$\"3&****[xj5x,\"F27$$\"3o** zmmmmmmF;$\"3(****)>v_mA5F27$$\"3#)**RLLLL$3(F;$\"3****>%o&f_G5F27$$\" 3]**>+++++vF;$\"3-+?1;*p`.\"F27$$\"3Z+qmmmm;zF;$\"30+I)yHpK/\"F27$$\"3 :+]LLLLL$)F;$\"3)***>Wf(*G_5F27$$\"3+++++++]()F;$\"3++!o\"z\"*[i5F27$$ \"3o**zmmmmm\"*F;$\"3++S#y^;R2\"F27$$\"3_**HLLLL$e*F;$\"3)***RNp3h'3\" F27$F*$\"31+STJ2g+6F27$$\"3.+qmmmmT5F2$\"3/+q::N!f6\"F27$$\"3'***HLLLL $3\"F2$\"3#***\\<-`_K6F27$$\"3+++++++D6F2$\"33+gjW3Y]6F27$$\"3-+qmmmmm 6F2$\"3#****3Nz$pp6F27$$\"31+SLLLL37F2$\"3)****Q/8(>!>\"F27$$F:F2$\"32 +])=xL>@\"F27$$\"3-+qmmmm\"H\"F2$\"3$***>')4t&[B\"F27$$\"31+SLLLLL8F2$ \"35+?9_G\"*e7F27$$\"3+++++++v8F2$\"3/+qsDz.%G\"F27$$\"3%***fmmmm;9F2$ \"37+qH5M;58F27$$\"31+SLLLLe9F2$\"3-+5msW@P8F27$$\"3++++++++:F2$\"3-+] Oy96l8F27$$\"3.+qmmmmT:F2$\"37+S)>'3x$R\"F27$$\"3'***HLLLL$e\"F2$\"34+ q$G%f5B9F27$$\"3+++++++D;F2$\"35+!\\wnFIX\"F27$$\"3-+qmmmmm;F2$\"3++!* pX`W$[\"F27$$\"31+SLLLL3F2$\"3-+5Fhf@ r;F27$$\"31+SLLLLe>F2$\"33+!zSYPDq\"F27$$\"\"#F)$\"3%)***R'y'HOt\"F2-% 'COLOURG6&%$RGBG$\")=THv!\")FjzFjz-%&STYLEG6#%%LINEG-F$6%F&-%&COLORG6& FizF($\"\"&!\"\"$F+Fh[l-F^[l6#%&POINTG-%&TITLEG6#%2evaluation~pointsG- %'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q!6\"Fh\\l-%%VIEWG6$%(DEFAULTGF]]l " 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evalua tions~-->~G\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/+U/C!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Maple's N ewton-Cotes integration gives the same result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(sqrt(1+D (f)(x)^2),x=0..2,method=_NCrule);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6%*$,&\"\"\"F(*$),&F(F(-%$cosG6#%\"xG!\"\"\"\"#F (F(#F(F1/F/;\"\"!F1/%'methodG%(_NCruleG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/+U/C!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 338 4 "Note" }{TEXT -1 258 ": Increasing the precision for the numerical evaluation of the analytical expression to 15 digits, and t hen rounding the result to 10 digits, gives a numerical value for the \+ integral which agrees with the value obtained by numerical integration to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Int(sqrt(1+D(f)(x)^2),x=0..2);\nsimplify(%):\nva lue(%):\nevalf[15](%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*$,&\"\"\"F(*$),&F(F(-%$cosG6#%\"xG!\"\"\"\"#F(F(#F(F1/F/;\"\" !F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"01BN+?WS#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/+U/C!\"*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 11" }{TEXT 324 41 " .. calculating arc length \+ along a curve " }}{PARA 0 "" 0 "" {TEXT 325 8 "Question" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 62 "Calculate correct to 10 digits the \+ arc length along the curve " }{XPPEDIT 18 0 "y=ln(x^3+1)" "6#/%\"yG-%# lnG6#,&*$%\"xG\"\"$\"\"\"F,F," }{TEXT -1 22 " from the point where " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 20 " to the point where \+ " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "Use Newton-Cotes integration to perform any necessar y numerical integration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 326 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "f := x -> l n(x^3+1):\n'f(x)'=f(x);\np1 := plot(f(x),x=-.4..2.3,color=red):\np2 := plot(f(x),x=0..2,color=blue,thickness=3):\nplots[display]([p1,p2]);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%#lnG6#,&*$)F'\"\"$ \"\"\"F/F/F/" }}{PARA 13 "" 1 "" {GLPLOT2D 338 265 265 {PLOTDATA 2 "6& -%'CURVESG6$7S7$$!3A+++++++S!#=$!3+2XX]-)Rh'!#>7$$!3U++]PoZ6MF*$!3-d]8 3WI^SF-7$$!3a+]i!)fS**GF*$!3%RCc)3zfnCF-7$$!3R++DY(GNK#F*$!3!)p#)o@xNi 7F-7$$!38++vBU$Qu\"F*$!3]BU_/n.<`!#?7$$!3w+]7))[*o;\"F*$!3w+o*>oe,f\"F B7$$!3p/+D1F)*>jF-$!3IS%GaidY_#!#@7$$!3?c+]i0Q9yFB$!3q]:5ip\"=x%!#C7$$ \"3'G**\\PuKl%\\F-$\"3Cu%*z-OD57FM7$$\"3]**\\(oQ8c1\"F*$\"3lGLfxkI47FB 7$$\"3-++]Ug\"Hl\"F*$\"3KnfAMu\"e]%FB7$$\"3_***\\7a7-<#F*$\"3zW1RSJ%p, \"F-7$$\"3[****\\xec_FF*$\"3YXR4B^1k?F-7$$\"3g)***\\7/JPLF*$\"3=0k>DBc \\OF-7$$\"3V****\\A,#3!RF*$\"3uo9eG#ehw&F-7$$\"37**\\PC4a7WF*$\"3ss!3( eZCU#)F-7$$\"3C****\\_k-@]F*$\"37rW3Cr*=>\"F*7$$\"3,******4N\\ObF*$\"3 O#)e(eO[vc\"F*7$$\"3r)*\\PRU5OhF*$\"3E&Q$odXby?F*7$$\"3A(****\\Lhom'F* $\"3a*oy*)=7`f#F*7$$\"3!))*\\P*)f=\\sF*$\"32nG$*)3:xA$F*7$$\"3C+]7onp. yF*$\"3M#z*HG\">\"))QF*7$$\"3s***\\PfqAQ)F*$\"3OE!4Vc!zIYF*7$$\"34**\\ (o(>e8*)F*$\"3#Qh)G<,Ta`F*7$$\"3u(**\\73tm[*F*$\"3=H9$GsBA<'F*7$$\"3r* \\PuO&>35!#<$\"3C]\\x_2laqF*7$$\"3#**\\i!=Z,g5Ffs$\"3/Lngk+)Q%yF*7$$\" 3,+]PO5)f6\"Ffs$\"3:$eydnRBr)F*7$$\"3%)****\\>(*zt6Ffs$\"3&\\wk'3/J@'* F*7$$\"3u**\\(3-k.B\"Ffs$\"3%=2o1C-<0\"Ffs7$$\"3!**\\P4*G4&G\"Ffs$\"3J ?-]%)[cQ6Ffs7$$\"3!)**\\7q*feM\"Ffs$\"3)GqG>FM[B\"Ffs7$$\"3s****\\$*=Y +9Ffs$\"3A\"=zW%*z3K\"Ffs7$$\"3#****\\75h(e9Ffs$\"3>F^5kd,79Ffs7$$\"3_ *\\P%[))e6:Ffs$\"3'pP%zVXw$\\\"Ffs7$$\"3!)***\\(HNMp:Ffs$\"3%pGxo1v?e \"Ffs7$$\"3k*\\PR='oB;Ffs$\"3E\\\"ehyUSm\"Ffs7$$\"3s*\\(o:**[!o\"Ffs$ \"3C$*)Q,#fY[Ffs7$$\"3%******4zj,&=Ffs$\"37))\\XclU#*>Ffs7$$\"3 &)**\\it&Gu!>Ffs$\"3cQxNbQ)=2#Ffs7$$\"3)**\\P\\<>U'>Ffs$\"3SG(=HyJ#\\@ Ffs7$$\"38++]RQS;?Ffs$\"3Gr`rAm.>AFfs7$$\"33+]7aL@w?Ffs$\"3K%o%3cyb(H# Ffs7$$\"3t*****pp2(H@Ffs$\"3gqv:'oikO#Ffs7$$\"3))*\\PR#Qu'=#Ffs$\"3$)3 9%Gxs&QCFfs7$$\"3%)*\\i]nM8C#Ffs$\"3Z@*GRJ.j]#Ffs7$$\"3#)************* H#Ffs$\"3C1UP*p8xd#Ffs-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fd[lFc[l-F $6%7S7$Fc[lFc[l7$$\"39LLLL3VfVF-$\"3)H>2B&pf%G)!#A7$$\"3'pmm;H[D:)F-$ \"3dr,yqd/FB7$$\"3ILLL3RBr;F*$ \"3I'4H8+Mpl%FB7$$\"3Ymm;zjf)4#F*$\"3mT>b%\\#***>*FB7$$\"3=LL$e4;[\\#F *$\"3,v_vVx'3a\"F-7$$\"3p****\\i'y]!HF*$\"3tL%G2Ej@U#F-7$$\"3,LL$ezs$H LF*$\"38,/>H%[Si$F-7$$\"3_****\\7iI_PF*$\"3!)f2,4=M[^F-7$$\"3#pmmm@Xt= %F*$\"3o_TFT:,&3(F-7$$\"3QLLL3y_qXF*$\"3$G$H1bX**=\"*F-7$$\"3i******\\ 1!>+&F*$\"3c1i&z&y4z6F*7$$\"3()******\\Z/NaF*$\"3%))Rz:KP*)[\"F*7$$\"3 '*******\\$fC&eF*$\"3DAgRs++F=F*7$$\"3ELL$ez6:B'F*$\"3!fw1hNsq;#F*7$$ \"3Smmm;=C#o'F*$\"3Lg2<(oc6h#F*7$$\"3-mmmm#pS1(F*$\"32G<^z;e>IF*7$$\"3 ]****\\i`A3vF*$\"3)z%36Q@`HNF*7$$\"3slmmm(y8!zF*$\"3bXmr`b')4SF*7$$\"3 V++]i.tK$)F*$\"3qc7g7]ClXF*7$$\"39++](3zMu)F*$\"3+.:7))>!)=^F*7$$\"3#p mm;H_?<*F*$\"3[%oXOh/*=dF*7$$\"3emm;zihl&*F*$\"3Yx$eYM#\\(G'F*7$$\"39L LL3#G,***F*$\"3l0Xdyxm;pF*7$$\"3?Lu9:\"Ffs7$$\"3_mmmwanL8Ffs$\"38.\\\\YSc:7Ffs7$$\"3' ******\\2goP\"Ffs$\"3E:I!f!fv$G\"Ffs7$$\"3CLLeR<*fT\"Ffs$\"3kZa#pYS_M \"Ffs7$$\"3'******\\)Hxe9Ffs$\"3=Opy[U.79Ffs7$$\"3Ymm\"H!o-*\\\"Ffs$\" 31Cwxi[Su9Ffs7$$\"3))***\\7k.6a\"Ffs$\"3&[_N%)eT!R:Ffs7$$\"3emmmT9C#e \"Ffs$\"3/J[b6?j,;Ffs7$$\"3\"****\\i!*3`i\"Ffs$\"3m7:;-FZm;Ffs7$$\"3QL LL$*zym;Ffs$\"3ydJ=ZKAGFfs7$$\"3lmm;9@BM=Ffs$\"3o5>zV'f+(>Ffs7$$\"3ELLL`v&Q(=Ffs$\"3w\"*y mt3[D?Ffs7$$\"30++DOl5;>Ffs$\"31*opti'z$3#Ffs7$$\"3/++v.Uac>Ffs$\"31b# y\"[H')Q@Ffs7$$\"\"#Fd[l$\"3c>iLxXA(>#Ffs-F][l6&F_[lFc[lFc[lF`[l-%*THI CKNESSG6#\"\"$-%+AXESLABELSG6%Q\"x6\"Q!Fd[m-%%FONTG6#%(DEFAULTG-%%VIEW G6$;$!\"%!\"\"$\"#BF`\\mFi[m" 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 10 "Note that " }{XPPEDIT 18 0 "`f \+ '`(x) = 3*x^2/(x^3+1);" "6#/-%$f~'G6#%\"xG*(\"\"$\"\"\"*$F'\"\"#F*,&*$ F'F)F*F*F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "'D(f)'(x)=D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"DG6#%\"fG6#%\"xG,$*(\"\"$\"\"\"F*\"\"# ,&*$)F*F-F.F.F.F.!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The required arc length is given by the definite integral " }}{PARA 257 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(sqrt(1+`f '`(x)^2),x = 0 .. 2) = Int(sqrt(1+9*x^4/((x^3+1)^2)),x = 0 .. 2);" "6#/-%$IntG6$-%%sqrt G6#,&\"\"\"F+*$-%$f~'G6#%\"xG\"\"#F+/F0;\"\"!F1-F%6$-F(6#,&F+F+*(\"\"* F+*$F0\"\"%F+*$,&*$F0\"\"$F+F+F+F1!\"\"F+/F0;F4F1" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 50 "Maple cannot evaluate this integral analy tically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(sqrt(1+D(f)(x)^2),x=0..2);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*(\"\"*F(%\"xG\"\"%,&*$ )F+\"\"$F(F(F(F(!\"#F(#F(\"\"#/F+;\"\"!F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*$,&\"\"\"F(*(\"\"*F(%\"xG\"\"%,&*$)F+\"\"$F(F (F(F(!\"#F(#F(\"\"#/F+;\"\"!F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 101 "However, this integral can be evaluated \+ numerically. Using Newton-Cotes integration via the procdure " }{TEXT 0 5 "NCint" }{TEXT -1 14 " gives . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "NCint(sqrt(1+D(f)(x)^ 2),x=0..2,info=true);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Oadaptive~ Newton-Cotes~quadrature~with~13~nodesG" }}{PARA 13 "" 1 "" {GLPLOT2D 521 70 70 {PLOTDATA 2 "6--%'CURVESG6%7$7$$\"\"\"\"\"!$F*F*7$F(F(-%&COL ORG6&%$RGBGF+$\"\"&!\"\"$F)F3-%*THICKNESSG6#\"\"#-F$6%7$7$$\"3++++++++ ]!#=F+7$F=F(F-F5-F$6%7$7$$\"3++++++++DF?F+7$FEF(F-F5-F$6%7$7$$\"3+++++ +++vF?F+7$FLF(F-F5-F$6%7$7$$\"3++++++++:!#$\"3(***\\pZ3++5!#<7$$\"3I+gmmmmmTF/$\"3****RJhN,+ 5F27$$\"3g+5++++]iF/$\"35+5oG'o++\"F27$$\"3g**\\LLLLL$)F/$\"3)***fWRn@ +5F27$$\"30+qmmmmT5!#=$\"3,+gv$[G0+\"F27$$\"3+++++++]7FE$\"3'***zmbP4, 5F27$$\"33+ILLLLe9FE$\"3\"***RXW2--5F27$$\"3#***pmmmmm;FE$\"30+?P'RMM+ \"F27$$\"3+++++++v=FE$\"3-+IxQUZ05F27$$\"35+SLLLL$3#FE$\"36++6\"[\"H35 F27$$\"3#***pmmmm\"H#FE$\"35+!))eMX?,\"F27$$\"3++++++++DFE$\"33+I8Q')* o,\"F27$$\"33+ILLLL3FFE$\"3%***psKE,B5F27$$\"3#***pmmmm;HFE$\"3/+S()R9 aI5F27$$\"3+++++++DJFE$\"32+qn!4E'R5F27$$\"3%)**RLLLLLLFE$\"3!*****)\\ l)Q]5F27$$\"3;+gmmmmTNFE$\"33+!o([n#H1\"F27$$\"3+++++++]PFE$\"35++Q'z3 t2\"F27$$\"33+ILLLLeRFE$\"31+SMy.d$4\"F27$$\"3C+!ommmm;%FE$\"3))**>A=> r66F27$$\"3+++++++vVFE$\"3,+!yN&ypJ6F27$$\"33+ILLLL$e%FE$\"31+5T6tX`6F 27$$\"3#***pmmmm\"z%FE$\"3)***>8(=')o<\"F27$$\"3I+5+++++]FE$\"3'***\\: D/&=?\"F27$$\"3`**HLLLL3_FE$\"3$***4G7F27$$\"3Z+qmmmm;aFE$\"3-+5T0 \\sb7F27$$\"3+++++++DcFE$\"37+!=a,eUG\"F27$$\"3%)**RLLLLLeFE$\"3/+]e)f #e88F27$$\"3;+gmmmmTgFE$\"3&***Re*>'[V8F27$$\"3+++++++]iFE$\"3%****)\\ Wbvt8F27$$\"3_**HLLLLekFE$\"38+]RM2=/9F27$$\"3o**zmmmmmmFE$\"3&***f@\" ))eXV\"F27$$\"3+++++++voFE$\"3))**>M]rpk9F27$$\"3#)**RLLLL$3(FE$\"31+I A1_T%\\\"F27$$\"3[+qmmmm\"H(FE$\"35+!G83ZN_\"F27$$\"3]**>+++++vFE$\"35 +g]0D%>b\"F27$$\"3`**HLLLL3xFE$\"3!****yPun%z:F27$$\"3Z+qmmmm;zFE$\"36 +g/(*f+1;F27$$\"3+++++++D\")FE$\"33+]2BuXJ;F27$$\"3:+]LLLLL$)FE$\"3-++ cN)Qdl\"F27$$\"3;+gmmmmT&)FE$\"3++!eu8$yy;F27$$\"3+++++++]()FE$\"39+!o >aQ0q\"F27$$\"3_**HLLLLe*)FE$\"3&***HiIw'4s\"F27$$\"3o**zmmmmm\"*FE$\" 36+!QNHY+u\"F27$$\"3+++++++v$*FE$\"35+S'>fixv\"F27$$\"3_**HLLLL$e*FE$ \"3'***pGyb6uE=F27$$\"3'***HLLLL$3\"F2$\"35+?(p>SY%= F27$$\"3+++++++D6F2$\"3!***4V(4b%e=F27$$\"3-+qmmmmm6F2$\"37+?>FD-o=F27 $$\"31+SLLLL37F2$\"3%***4\"=vZP(=F27$$FJF2$\"3%)***\\f1Sg(=F27$$\"3-+q mmmm\"H\"F2$\"3'***\\6z)*Hv=F27$$\"31+SLLLLL8F2$\"3%)**\\:h&4>(=F27$$ \"3+++++++v8F2$\"3%***zr*QFi'=F27$$\"3%***fmmmm;9F2$\"3))**f0lRee=F27$ $\"31+SLLLLe9F2$\"37+S'*>0G\\=F27$$\"3++++++++:F2$\"3%***z]L')eQ=F27$$ \"3.+qmmmmT:F2$\"3&)**RuU4vE=F27$$\"3'***HLLLL$e\"F2$\"36+?1GD)R\"=F27 $$\"3+++++++D;F2$\"3++IV?FZ+=F27$$\"3-+qmmmmm;F2$\"3/+]uSrQ'y\"F27$$\" 31+SLLLL3F2$\"3-+![^.*f'p\"F27$$\" 31+SLLLLe>F2$\"3-+!=d(Qc\"o\"F27$$\"\"#F)Fb_l-%'COLOURG6&%$RGBG$\")=TH v!\")F_blF_bl-%&STYLEG6#%%LINEG-F$6%F&-%&COLORG6&F^blF($\"\"&!\"\"$F+F ]cl-Fcbl6#%&POINTG-%&TITLEG6#%2evaluation~pointsG-%'SYMBOLG6#%'CIRCLEG -%+AXESLABELSG6$Q!6\"F]dl-%%VIEWG6$%(DEFAULTGFbdl" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#t" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+a'pk4$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, Maple's Ne wton-Cotes integration can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(sqrt(1+D(f)(x)^2),x=0. .2,method=_NCrule);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I ntG6%*$,&\"\"\"F(*(\"\"*F(%\"xG\"\"%,&*$)F+\"\"$F(F(F(F(!\"#F(#F(\"\"# /F+;\"\"!F3/%'methodG%(_NCruleG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +a'pk4$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 12 " }{TEXT 295 53 ".. disecting a region into two pieces with equal \+ area" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 " f(x)=exp(-x^2)" "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f := x -> exp(-x^2);\nplot(f(x),x=0..2.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$ expG6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 254 171 171 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G \\a!#>$\"3k'fx'*H\\.(**!#=7$$\"3PL$ek`o!>5F2$\"39KHz6to'*)*F27$$\"3omm \"z>)G_:F2$\"3Gl)G,/?>w*F27$$\"3-nmT&QU!*3#F2$\"33fj?Re(Hd*F27$$\"3HL$ eRZXKi#F2$\"3.Z)H+Q,]L*F27$$\"3xm;z>,_=JF2$\"3u+9gNiFt!*F27$$\"3v**\\7 G$[8j$F2$\"3'zjr'yjdk()F27$$\"35n;z%*frhTF2$\"3w&3!GJTq4%)F27$$\"3A+]i lFQ!p%F2$\"35jsd=G@D!)F27$$\"3@ML$3_\"=M_F2$\"3Mf&Gs*\\d.wF27$$\"3HnmT g(fJr&F2$\"33e0?,7::sF27$$\"3k++]7eP_iF2$\"3[V!=U$*GVw'F27$$\"3Q++]Pf! Qz'F2$\"3S3NOp(3II'F27$$\"3@++](=ubJ(F2$\"3czgr42mbeF27$$\"37n;zW(*Q*y (F2$\"339ECH&=7X&F27$$\"3#QLL3F-GN)F2$\"3#)4;K)>At(\\F27$$\"3=MLL$e'3I ))F2$\"3!zzTF'eT&e%F27$$\"3?+]7.\"Fjq$\"3ouApA#*z$R#F27$$\"3km;/Egw[7Fjq$\"3iv;=Hqe-@F27$$\"3zm\" z%*f%)QI\"Fjq$\"3?`)*[x]iE=F27$$\"3/+voza'=N\"Fjq$\"3C#)pzmi33;F27$$\" 3(om\"zWho.9Fjq$\"3GbNR[<5%R\"F27$$\"3-++]i>Ad9Fjq$\"3G?V!y[Fh>\"F27$$ \"32+]i:jf4:Fjq$\"3gVK.Js)R-\"F27$$\"39+DJ&>r-c\"Fjq$\"3UVI'f=FYw)F/7$ $\"3++]P4q`;;Fjq$\"3een9&G?+L(F/7$$\"3;LL$eM%4n;Fjq$\"3H7Ds8Pz3iF/7$$ \"37++v$4v5s\"Fjq$\"3)y`G.'Q-r^F/7$$\"3cm\"zWn*)*pFjq$\"31_q!*zpXXCF/7$$\"3CLL3-=!y(>Fjq$\"3))QBdyhk+?F /7$$\"3))*\\7G8O;.#Fjq$\"3d.28t/C7;F/7$$\"3!pmm;*\\[$3#Fjq$\"3h\\\\&HT nCI\"F/7$$\"3*pmT&Qz]O@Fjq$\"3l*[t,\\N8/\"F/7$$\"3iLekG=4*=#Fjq$\"3'o$ HEq%G[H)!#?7$$\"3F++]i4TPAFjq$\"3AwzI:!)e(p'F[y7$$\"3qL$3F9!z#H#Fjq$\" 3lF8k&y/:@&F[y7$$\"3'pmm;%>KUBFjq$\"3Wi26.EVUTF[y7$$\"3/+DJqJ8&R#Fjq$ \"3'oTms=9bA$F[y7$$\"3G+voa-oXCFjq$\"3\\(Rz6')4a_#F[y7$$\"3++++++++DFj q$\"3_4xAOTXI>F[y-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6 \"Q!Fe[l-%%VIEWG6$;F($\"#DF`[l%(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 13 "The function " }{XPPEDIT 18 0 " p(x)=Int(exp(-t^2),t=0..x)" "6#/-%\"pG6#%\"xG-%$IntG6$-%$expG6#,$*$%\" tG\"\"#!\"\"/F0;\"\"!F'" }{TEXT -1 50 ", gives the area under this gra ph starting at the " }{TEXT 282 1 "y" }{TEXT -1 56 " axis and going as far as the vertical line situated at " }{TEXT 283 1 "x" }{TEXT -1 2 " . 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "p_ NC := x -> NCint(exp(-t^2),t=0..x);\nplot('p_NC(x)',x=0..2.5,color=blu e); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%p_NCGf*6#%\"xG6\"6$%)operat orG%&arrowGF(-%&NCintG6$-%$expG6#,$*$)%\"tG\"\"#\"\"\"!\"\"/F5;\"\"!9$ F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 304 191 191 {PLOTDATA 2 "6&-%'CUR VESG6#7S7$$\"\"!F)F(7$$\"+T&)G\\a!#6$\"+,&**QW&F-7$$\"+O&o!>5!#5$\"+@= b:5F37$$\"+)>)G_:F3$\"+S)4*R:F37$$\"+&QU!*3#F3$\"+lo/f?F37$$\"+uaCBEF3 $\"+[aHkDF37$$\"+?,_=JF3$\"+(y3.-$F37$$\"+G$[8j$F3$\"+K8&yZ$F37$$\"+&* frhTF3$\"+,UVLRF37$$\"+lFQ!p%F3$\"+Jq)zO%F37$$\"+?:=M_F3$\"+C\"RIz%F37 $$\"+g(fJr&F3$\"+&*e)z9&F37$$\"+7eP_iF3$\"+Ce$\\_&F37$$\"+Pf!Qz'F3$\"+ FTryeF37$$\"+(=ubJ(F3$\"+;_\"f>'F37$$\"+W(*Q*y(F3$\"+)*)pPY'F37$$\"+qA !GN)F3$\"+R9]dnF37$$\"+$e'3I))F3$\"+mum&)pF37$$\"+.@QzxF37$$\"+Ol]Y6Fhq$\"+ZVMKzF37$$\"+N?q&>\"Fhq$\"+r=@d!)F37$$ \"+Egw[7Fhq$\"+*)))Rw\")F37$$\"+*f%)QI\"Fhq$\"+E;c%G)F37$$\"+![l=N\"Fh q$\"+A@)oO)F37$$\"+Xho.9Fhq$\"+eEdW%)F37$$\"+i>Ad9Fhq$\"+1b!Q^)F37$$\" +;jf4:Fhq$\"+qQ&=d)F37$$\"+&>r-c\"Fhq$\"+&4I*>')F37$$\"+4q`;;Fhq$\"+oE 6l')F37$$\"+YV4n;Fhq$\"+XJF*p)F37$$\"+%4v5s\"Fhq$\"+Os\"*H()F37$$\"+u' *)*pFhq$\"+.Z;0))F37$$\"+-=!y(>Fhq$\"+7Hc;))F37$$\"+LhjJ?Fhq $\"+v[DE))F37$$\"+#*\\[$3#Fhq$\"+CfyL))F37$$\"+Qz]O@Fhq$\"+#zw*R))F37$ $\"+H=4*=#Fhq$\"+'pw[%))F37$$\"+i4TPAFhq$\"+dk[[))F37$$\"+V,z#H#Fhq$\" +B&o<&))F37$$\"+U>KUBFhq$\"+If2a))F37$$\"+qJ8&R#Fhq$\"+'H7g&))F37$$\"+ b-oXCFhq$\"+l\"fu&))F37$$\"#D!\"\"$\"+OFme))F3-%'COLOURG6&%$RGBGF(F($ \"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fc[l-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Now consi der the problem of solving the equation " }{XPPEDIT 18 0 "p(x) = 2/3; " "6#/-%\"pG6#%\"xG*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". 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In this case the procdure " } {TEXT 0 5 "NCint" }{TEXT -1 71 " will be used for the function evaluat ions needed by the secant method." }}{PARA 257 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "secant('p_NC(x)'=2/3,x=0.82 ..0.83);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+9C,t\")!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Now consider th e probem of finding the value " }{XPPEDIT 18 0 "x[mid]" "6#&%\"xG6#%$m idG" }{TEXT -1 29 " such that the vertical line " }{XPPEDIT 18 0 "x=x[ mid]" "6#/%\"xG&F$6#%$midG" }{TEXT -1 36 " divides the region under th e graph " }{XPPEDIT 18 0 "y=exp(-x^2)" "6#/%\"yG-%$expG6#,$*$%\"xG\"\" #!\"\"" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 44 " \+ into two regions which have the same area. 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F\\\\l7$$\"+,D./7Fi`m$\"+d,UYBF\\\\l7$$\"+-cTO7Fi`m$\"+GC7o@F\\\\l7$$ \"+UD0q7Fi`m$\"+<=#G*>F\\\\l7$$\"+)fL$*H\"Fi`m$\"+fUR[=F\\\\l7$$\"+Sz& 4L\"Fi`m$\"+&[j3q\"F\\\\l7$$\"+!)*GOO\"Fi`m$\"+*yHvb\"F\\\\l7$$\"+>7f& R\"Fi`m$\"+^E0E9F\\\\l7$$\"+Gj^E9Fi`m$\"+!proI\"F\\\\l7$$\"+!R`3Y\"Fi` m$\"+7$>N=\"F\\\\l7$$\"+mnq\"\\\"Fi`m$\"+v)p/3\"F\\\\l7$$\"+K$\\Y_\"Fi `m$\"+=5p#y*Fi[l7$$\"+#>+Xb\"Fi`m$\"+&)*)fB*)Fi[l7$$\"+n]8(e\"Fi`m$\"+ 7!HS0)Fi[l7$$\"+T>%yh\"Fi`m$\"+t![\"*H(Fi[l7$$\"+t%R*\\;Fi`m$\"+J*RBd' Fi[l7$$\"+w1K\"o\"Fi`m$\"+jv'*>fFi[l7$$\"+))Q<9o^MUFi[l7$$\"+,OE5= Fi`m$\"+jh#Rx$Fi[l7$$\"+\"3^(R=Fi`m$\"+1[$))Q$Fi[l7$$\"+gqat=Fi`m$\"+q v:*)HFi[l7$$\"+)euP!>Fi`m$\"+3?imEFi[l7$$\"+OM+O>Fi`m$\"+#GDiN#Fi[l7$$ \"+c0&o'>Fi`m$\"+Cd'*)3#Fi[l7$$\"\"#F)$\"+*)QcJ=Fi[l7$FcjmF(-Fjjl6&F][ l$\"#&*F^[m$\"\"*F`[lF\\[nF_[m-F$6$7$7$$\"3A+++hIPVZF2F(7$Fb[n$\"3!*** ***p3&>&)zF2-F[[l6&F][lF)F)F)-F$6$7$Fgjm7$Fcjm$\"3A=M()))QcJ=F/Fg[n-%% TEXTG6&7$$\"#ZF^[m$!\"&F^[mQ\"x6\"Fg[n-%%FONTG6$%*HELVETICAGF_[l-F`\\n 6&7$$\"\"$F`[l$\"\"%F`[lQ%p(x)Fh\\nFg[nFi\\n-F`\\n6&7$$\"\"(F`[lFb]nQ% q(x)Fh\\nFg[nFi\\n-F`\\n6&7$$\"$v&!\"$$!\"(F^[mQ$midFh\\nFg[n-Fj\\n6$F \\]n\"\")-%*AXESTICKSG6$FdjmFdjm-%+AXESLABELSG6%%!GF]_n-Fj\\n6#%(DEFAU LTG-%%VIEWG6$;F($\"#DF`[lF`_n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "This equation can be solved usi ng " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "q := x -> Int(exp(-t^ 2),t=x..2);\nfsolve(p(x)=q(x),x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$-%$expG6#,$*$) %\"tG\"\"#\"\"\"!\"\"/F5;9$F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+hIPVZ!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Alternati vely, we can calculate the total area under the graph " }{XPPEDIT 18 0 "y=exp(-x^2)" "6#/%\"yG-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 6 " f rom " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A=Int(exp(-x^2),x=0..2)" "6#/%\"AG-% $IntG6$-%$expG6#,$*$%\"xG\"\"#!\"\"/F-;\"\"!F." }{XPPEDIT 18 0 "`` = p (2)" "6#/%!G-%\"pG6#\"\"#" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "and then solve the equation " }{XPPEDIT 18 0 "p(x)=A/2" "6#/-%\"pG 6#%\"xG*&%\"AG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "For variety, let's do this using " }{TEXT 278 7 "p_NC(x) " }{TEXT -1 24 " and the secant method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "p_NC := x -> NCint(ex p(-t^2),t=0..x):\nA := p_NC(2);\nsecant('p_NC'(x)=A/2,x=0..2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"+3R\"3#))!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hIPVZ!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "Using the bisection method along with num erical integration via " }{TEXT 0 5 "NCint" }{TEXT -1 162 "causes a no ticable delay in Maple supplying the solution. This is because two int egrals have to be computed for each of 39 steps required by the bisect ion method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 "q_NC := x -> NCint(exp(-t^2),t=x..2);\nbisect('p_NC (x)=q_NC(x)',x=0..2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% q_NCGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%&NCintG6$-%$expG6#,$*$)%\"tG \"\"#\"\"\"!\"\"/F5;9$F6F(F(F(" }}{PARA 6 "" 1 "" {TEXT -1 69 " start [-] 0.000000000000 2.000000000000 [+]" }} {PARA 6 "" 1 "" {TEXT -1 69 " step 1 [-] 0.000000000000 \+ 1.000000000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 2 \+ [-] 0.000000000000 .5000000000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 3 [-]-> .2500000000000 \+ .5000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 4 [-] -> .3750000000000 .5000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 5 [-]-> .4375000000000 .5000 000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 6 [-]-> \+ .4687500000000 .5000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 7 [-] .4687500000000 .484375000 0000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 8 [-] . 4687500000000 .4765625000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 9 [-]-> .4726562500000 .4765625000000 \+ [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 10 [-] .472656 2500000 .4746093750000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 11 [-]-> .4736328125000 .4746093750000 [+] " }}{PARA 6 "" 1 "" {TEXT -1 69 " step 12 [-]-> .474121093750 0 .4746093750000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " ste p 13 [-] .4741210937500 .4743652343750 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 69 " step 14 [-]-> .4742431640625 \+ .4743652343750 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 15 [-]-> .4743041992187 .4743652343750 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 16 [-]-> .4743347167969 \+ .4743652343750 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 17 [-] .4743347167969 .4743499755859 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 18 [-] .4743347167969 .4743 423461914 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 19 [-] \+ .4743347167969 .4743385314941 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 20 [-]-> .4743366241454 .474338531 4941 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 21 [-] . 4743366241454 .4743375778197 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 22 [-]-> .4743371009825 .4743375778197 \+ [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 23 [-] .474337 1009825 .4743373394010 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 24 [-]-> .4743372201917 .4743373394010 [+] " }}{PARA 6 "" 1 "" {TEXT -1 69 " step 25 [-]-> .474337279796 3 .4743373394010 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " ste p 26 [-] .4743372797963 .4743373095987 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 69 " step 27 [-]-> .4743372946976 \+ .4743373095987 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 28 [-]-> .4743373021482 .4743373095987 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 29 [-]-> .4743373058735 \+ .4743373095987 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 30 [-] .4743373058735 .4743373077362 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 31 [-] .4743373058735 .4743 373068049 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 32 [-] \+ .4743373058735 .4743373063392 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 33 [-] .4743373058735 .474337306 1064 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 34 [-]-> . 4743373059900 .4743373061064 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 35 [-]-> .4743373060482 .4743373061064 \+ [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 36 [-]-> .474337 3060773 .4743373061064 [+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " step 37 [-] .4743373060773 .4743373060918 <-[+] " }}{PARA 6 "" 1 "" {TEXT -1 69 " step 38 [-] .474337306077 3 .4743373060845 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 69 " ste p 39 [-] .4743373060773 .4743373060808 <-[+]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hIPVZ!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Another observation is th at Newton's method can be used to solve the equation " }{XPPEDIT 18 0 "Int(exp(-t^2),t = 0 .. x)=Int(exp(-t^2),t=x..2)" "6#/-%$IntG6$-%$expG 6#,$*$%\"tG\"\"#!\"\"/F,;\"\"!%\"xG-F%6$-F(6#,$*$F,F-F./F,;F2F-" } {TEXT -1 84 ", because Maple \"knows\" how to obtain analytical expres sions for the derivatives of " }{XPPEDIT 18 0 "p(x)=Int(exp(-t^2),t = \+ 0 .. x)" "6#/-%\"pG6#%\"xG-%$IntG6$-%$expG6#,$*$%\"tG\"\"#!\"\"/F0;\" \"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)=Int(exp(-t^2),t = x .. 2)" "6#/-%\"qG6#%\"xG-%$IntG6$-%$expG6#,$*$%\"tG\"\"#!\"\"/F0;F'F1" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 119 "p := x -> Int(exp(-t^2),t=0..x):\nDiff(p(x),x )=diff(p(x),x);\nq := x -> Int(exp(-t^2),t=x..2):\nDiff(q(x),x)=diff(q (x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$IntG6$-%$expG 6#,$*$)%\"tG\"\"#\"\"\"!\"\"/F0;\"\"!%\"xGF7-F+6#,$*$)F7F1F2F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$IntG6$-%$expG6#,$*$)%\"t G\"\"#\"\"\"!\"\"/F0;%\"xGF1F6,$-F+6#,$*$)F6F1F2F3F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "newton(p( x)=q(x),x=0.5,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxi mation~1~~->~~~G$\".Q>E4,u%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~2~~->~~~G$\".zdbsLu%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%7approximation~3~~->~~~G$\".x21tLu%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".x21tLu%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hIPVZ!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 11 "Example 13 " }{TEXT 304 40 ".. solving equations involv ing integrals" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 303 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 35 "Find the solution of the equation " }{XPPEDIT 18 0 "Int(1/(t+exp(t)),t = 0 .. x) = 2-x^2; " "6#/-%$IntG6$*&\"\"\"F(,&%\"tGF(-%$expG6#F*F(!\"\"/F*;\"\"!%\"xG,&\" \"#F(*$F2F4F." }{TEXT -1 24 " correct to 10 digits. " }}{PARA 258 "" 0 "" {TEXT 302 8 "Solution" }{TEXT 305 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = Int(1/(t+exp(t)),t = 0 .. x);" "6# /-%\"fG6#%\"xG-%$IntG6$*&\"\"\"F,,&%\"tGF,-%$expG6#F.F,!\"\"/F.;\"\"!F '" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "g(x)=2-x^2" "6#/-%\"gG6#%\"xG ,&\"\"#\"\"\"*$F'F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "We can plot the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"f G6#%\"xG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = g(x)" "6#/%\"yG-%\" gG6#%\"xG" }{TEXT -1 82 " in the same picture showing the single point of intersection of the two curves. " }}{PARA 0 "" 0 "" {TEXT -1 8 "M aple's " }{TEXT 0 4 "plot" }{TEXT -1 72 " command will automatically u se numerical integration to plot the curve " }{XPPEDIT 18 0 "y = f(x); " "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> Int(1/(t+ex p(t)),t=0..x);\ng := x -> 2-x^2;\nplot([f(x),g(x)],x=0..1.6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%$IntG6$*&\"\"\"F0,&%\"tGF0-%$expG6#F2F0!\"\"/F2;\"\"!9$F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&\"\"#\"\"\"*$)9$F-F.!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 353 255 255 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"+mYa([$!#6$ \"+kkkqLF-7$$\"+L'Q?_'F-$\"+K,[EhF-7$$\"+mWkM**F-$\"+Kq))[!*F-7$$\"+Fr )pL\"!#5$\"+Jr'>=\"F=7$$\"+.r()y;F=$\"+cTIU9F=7$$\"+wG&e*>F=$\"+\\c8r; F=7$$\"+IH1CBF=$\"+\"zCk*=F=7$$\"+O#)\\jEF=$\"+F\"Rz6#F=7$$\"+q\\%=+$F =$\"+F6.GBF=7$$\"+th()\\LF=$\"+l&HQ`#F=7$$\"+YAUcOF=$\"+Tg02FF=7$$\"+? 0_,SF=$\"+;$zO*GF=7$$\"++e.[VF=$\"+aUssIF=7$$\"+![n>o%F=$\"+p'yyB$F=7$ $\"+O%4_)\\F=$\"+T$e>Q$F=7$$\"+`MzX`F=$\"+*=>ka$F=7$$\"+8aD^cF=$\"+auB !o$F=7$$\"+*G!e1gF=$\"+8.#*HQF=7$$\"+8I5@jF=$\"+.'Rt&RF=7$$\"+*G%=mmF= $\"+R`&>4%F=7$$\"+pKy%*pF=$\"+0;O:UF=7$$\"+L=kPtF=$\"+?1TRVF=7$$\"+BI \\_wF=$\"+dfH\\WF=7$$\"+mD5#*zF=$\"+TiqjXF=7$$\"+O9'[M)F=$\"+d(R#yYF=7 $$\"+p!R>l)F=$\"+WY`uZF=7$$\"+E8f$)*)F=$\"+?W8v[F=7$$\"+f0AE$*F=$\"+fh ^v\\F=7$$\"+>kTh'*F=$\"+*\\%Qq]F=7$$\"+\\ct&)**F=$\"+@'o\"f^F=7$$\"+'o $eM5!\"*$\"+OBVa_F=7$$\"+\"QSp1\"F_u$\"+1z\"F_u$\"+Nza\\cF=7$$\"+8H)GB\"F_u$\"+1**)Gs&F=7$$\"+`Jzl7F_u$\"+QEO#z &F=7$$\"+DrC+8F_u$\"+9l\"G'eF=7$$\"+&RIML\"F_u$\"+o#\\&GfF=7$$\"+\"3lt O\"F_u$\"+vTp$*fF=7$$\"+q(=5S\"F_u$\"+y$*HcgF=7$$\"+;I%>V\"F_u$\"+f387 hF=7$$\"+\"p&Qn9F_u$\"+J0>uhF=7$$\"+Vg3*\\\"F_u$\"+T%4!GiF=7$$\"+H_)G` \"F_u$\"+@_p$G'F=7$$\"+j`Bl:F_u$\"+!)pSNjF=7$$\"#;!\"\"$\"+)e0$*Q'F=-% 'COLOURG6&%$RGBG$\"#5FezF(F(-F$6$7S7$F($\"\"#F)7$$\"3QmmmmYa([$!#>$\"3 e+)>Kq$y)*>!#<7$$\"3%QLLLjQ?_'Fg[l$\"3%3`17IYd*>Fj[l7$$\"3BnmmmWkM**Fg [l$\"3!H4KRGI,*>Fj[l7$$\"3kmmmEr)pL\"!#=$\"3/w7BaY7#)>Fj[l7$$\"3GLLL.r ()y;Fh\\l$\"3%3.>nr8=(>Fj[l7$$\"3mmmmwG&e*>Fh\\l$\"3!Q,ZHrl,'>Fj[l7$$ \"3')******HH1CBFh\\l$\"37)*R(\\J()f%>Fj[l7$$\"3_mmmO#)\\jEFh\\l$\"3\\ NFVrx0H>Fj[l7$$\"3&*******p\\%=+$Fh\\l$\"3*p&3wn#*))4>Fj[l7$$\"3wLLLth ()\\LFh\\l$\"3IOLB'H$y()=Fj[l7$$\"3#ommmCAkl$Fh\\l$\"3!4\"3ajdIm=Fj[l7 $$\"3O+++?0_,SFh\\l$\"3H*=!GN$y)R=Fj[l7$$\"3o*******zN![VFh\\l$\"3p$=> o%e%4\"=Fj[l7$$\"3'*******zu'>o%Fh\\l$\"3YCA;0=z!y\"Fj[l7$$\"3gmmmO%4_ )\\Fh\\l$\"33'pD(ooZ^K *4#R;Fj[l7$$\"3NKLL8I5@jFh\\l$\"3t#G[pcO/g\"Fj[l7$$\"37+++!H%=mmFh\\l$ \"3'=d<,()>cb\"Fj[l7$$\"35+++qKy%*pFh\\l$\"3*4Gd+2I2^\"Fj[l7$$\"3vLLLL =kPtFh\\l$\"3UmrDB,fh9Fj[l7$$\"3ELLLBI\\_wFh\\l$\"3tY$y_]$R99Fj[l7$$\" 3smmmmD5#*zFh\\l$\"3!4!)Qc'HEh8Fj[l7$$\"3CmmmO9'[M)Fh\\l$\"3bN$Gg(Gj.8 Fj[l7$$\"3%)******p!R>l)Fh\\l$\"3Ov+H.&R9D\"Fj[l7$$\"3smmmE8f$)*)Fh\\l $\"3O%Ra(o3&H>\"Fj[l7$$\"3A******f0AE$*Fh\\l$\"3-LBm+h@I6Fj[l7$$\"3o** ****>kTh'*Fh\\l$\"3'Qa$fF.dm5Fj[l7$$\"3S******\\ct&)**Fh\\l$\"3'3>$GN3 &G+\"Fj[l7$$\"3&******fo$eM5Fj[l$\"39MlimfO'H*Fh\\l7$$\"39LLL\"QSp1\"F j[l$\"3u;G!oA#Q;')Fh\\l7$$\"35+++g!)[,6Fj[l$\"3_iVxO0CnyFh\\l7$$\"3gmm m\"R$zK6Fj[l$\"3E,L'zJ\"znrFh\\l7$$\"3-+++)Q=q;\"Fj[l$\"3;9))p?3o!Q'Fh \\l7$$\"3;LLLU9A*>\"Fj[l$\"3c+'zCKz'=cFh\\l7$$\"33+++8H)GB\"Fj[l$\"3cU jKGs***z%Fh\\l7$$\"3KLLL`Jzl7Fj[l$\"3mkXuHpnxRFh\\l7$$\"3%)*****\\7Z-I \"Fj[l$\"3qZBHRTd$4$Fh\\l7$$\"3zmmm%RIML\"Fj[l$\"3-s4zDQj>AFh\\l7$$\"3 immm!3ltO\"Fj[l$\"3\">SMV\"Fj[l$!37Bw62,3Y]Fg[l7$$\"3ULLL\"p&Qn9Fj[l$!3-Z !)Hrw?K:Fh\\l7$$\"3cmmmUg3*\\\"Fj[l$!3wm+=L'*esCFh\\l7$$\"38+++H_)G`\" Fj[l$!3)y#Q'GDrt\\$Fh\\l7$$\"3++++j`Bl:Fj[l$!3e " 0 "" {MPLTEXT 1 0 98 "f := x -> Int(1/(t+ex p(t)),t=0..x);\ng := x -> 2-x^2;\nx1 := secant(f(x)=g(x),x=1.2..1.4,in fo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(-%$IntG6$*&\"\"\"F0,&%\"tGF0-%$expG6#F2F0!\"\"/F2;\"\" !9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,&\"\"#\"\"\"*$)9$F-F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".[f#*p\")>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".y&y40)>\"!#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".lbqU!)> \"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\". 1_qU!)>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~ ~~G$\".1_qU!)>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+0F/)> \"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The answer can be checked as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(f(x1));\ng(x 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ln$pk&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*xOpk&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 51 ": We can force the nu merical integration needed by " }{TEXT 0 6 "secant" }{TEXT -1 34 " to \+ be performed by the procedure " }{TEXT 0 5 "NCint" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "fn := x -> NCint(1/(t+exp(t)),t=0..x);\ng := x -> 2-x^2;\nsecant(' fn(x)'=g(x),x=1.2..1.4,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#fnGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%&NCintG6$*&\"\"\"F0,&%\"tGF 0-%$expG6#F2F0!\"\"/F2;\"\"!9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"#\"\"\"*$)9$F-F.!\"\" F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\" .[f#*p\")>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~ ->~~~G$\".y&y40)>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximat ion~3~~->~~~G$\".lbqU!)>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~4~~->~~~G$\".1_qU!)>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~5~~->~~~G$\".1_qU!)>\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0F/)>\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 4 "Note " }{TEXT -1 78 ": Questions 4 to 9 are general questions which involve numerical integration. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Find an interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6 #-%\"pG6#%\"xG" }{TEXT -1 55 " of degree 10 whose values match those o f the function " }{XPPEDIT 18 0 "f(x) = 4/(1+exp(x^2+1))" "6#/-%\"fG6# %\"xG*&\"\"%\"\"\",&F*F*-%$expG6#,&*$F'\"\"#F*F*F*F*!\"\"" }{TEXT -1 44 " at 11 equally spaced nodes bewteen 0 and 2." }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 113 ": You may want to use 15 digit preci sion for the calculation of the coefficients of the interpolating poly nomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "(b) Calculate " }{XPPEDIT 18 0 "Int(p(x),x=0..2)" "6#-%$IntG6$-%\" pG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 36 " to obtain an approximate value for " }{XPPEDIT 18 0 "Int(f(x),x=0..2)" "6#-%$IntG6$-%\"fG6#%\"xG/F); \"\"!\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "(c) Calcul ate an approximate value for " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2)" " 6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 24 " by using the pr ocedure " }{TEXT 0 5 "NCint" }{TEXT -1 19 " with the options \"" } {TEXT 278 14 "adaptive=false" }{TEXT -1 7 "\" and \"" }{TEXT 278 12 "n umpoints=11" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 41 "(d) Comp are the results from (b) and (c)." }}{PARA 0 "" 0 "" {TEXT -1 14 "(e) \+ Calculate " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2)" "6#-%$IntG6$-%\"fG6# %\"xG/F);\"\"!\"\"#" }{TEXT -1 30 " by using using the procedure " } {TEXT 0 5 "NCint" }{TEXT -1 19 " with the options \"" }{TEXT 278 13 "a daptive=true" }{TEXT -1 7 "\" and \"" }{TEXT 278 12 "numpoints=11" } {TEXT -1 91 "\" and compare the result with that obtained by using Map le's numerical integration through " }{TEXT 0 5 "evalf" }{TEXT -1 5 " \+ and " }{TEXT 0 3 "Int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 34 "(a) Find an approximate value for " } {XPPEDIT 18 0 "Int(cos(x^2),x = 0 .. 1);" "6#-%$IntG6$-%$cosG6#*$%\"xG \"\"#/F*;\"\"!\"\"\"" }{TEXT -1 67 " which is correct to 10 digits by using a Newton-Cotes method via " }{MPLTEXT 1 0 5 "NCint" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "( b) Try various choices for the option \"" }{TEXT 278 9 "numpoints" } {TEXT -1 91 "\" in order to discover the minimum number of function ev aluations with which the procedure " }{MPLTEXT 1 0 5 "NCint" }{TEXT -1 73 " can evaluate the integral in part (a) correct to 10 digits, wh en in the " }{TEXT 261 8 "adaptive" }{TEXT -1 6 " mode." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "(c) Find the minimu m number of function evaluations needed to calculate the integral in ( a) correct to 10 digits when using " }{TEXT 0 5 "NCint" }{TEXT -1 8 " \+ in the " }{TEXT 261 12 "non-adaptive" }{TEXT -1 6 " mode." }}{PARA 0 " " 0 "" {TEXT -1 36 "____________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "__ __________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 35 "(a) Plot the graph of the function " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 10 " given by " } {XPPEDIT 18 0 "h(x) = 1/((30*x-11*sqrt(2))^2+1/10)" "6#/-%\"hG6#%\"xG* &\"\"\"F),&*$,&*&\"#IF)F'F)F)*&\"#6F)-%%sqrtG6#\"\"#F)!\"\"F4F)*&F)F) \"#5F5F)F5" }{TEXT -1 31 " over the interval from 0 to 1." }}{PARA 0 " " 0 "" {TEXT -1 21 "(b) For the function " }{XPPEDIT 18 0 "h(x)" "6#-% \"hG6#%\"xG" }{TEXT -1 44 " in part (a), find an approximate value for " }{XPPEDIT 18 0 "Int(h(x),x = 0 .. 1);" "6#-%$IntG6$-%\"hG6#%\"xG/F) ;\"\"!\"\"\"" }{TEXT -1 67 " which is correct to 10 digits by using a Newton-Cotes method via " }{MPLTEXT 1 0 5 "NCint" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "(c) Try v arious choices for the option \"" }{TEXT 278 9 "numpoints" }{TEXT -1 91 "\" in order to discover the minimum number of function evaluations with which the procedure " }{MPLTEXT 1 0 5 "NCint" }{TEXT -1 60 " can evaluate the integral in part (b) correct to 10 digits." }}{PARA 0 " " 0 "" {TEXT -1 35 "___________________________________" }}{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 36 "__ __________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }{TEXT 334 66 " .. calculating the area of a region enclosed between two curv es " }}{PARA 0 "" 0 "" {TEXT -1 85 "Calculate (correct to 10 digits) \+ the area of the region enclosed between the curves " }{XPPEDIT 18 0 " y=16*exp(-x)*arctan(x)" "6#/%\"yG*(\"#;\"\"\"-%$expG6#,$%\"xG!\"\"F'-% 'arctanG6#F,F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=2-4/(1+exp(x^2+1 ))" "6#/%\"yG,&\"\"#\"\"\"*&\"\"%F',&F'F'-%$expG6#,&*$%\"xGF&F'F'F'F'! \"\"F1" }{TEXT -1 43 " between their two points of intersection. " }} {PARA 0 "" 0 "" {TEXT -1 70 "Perform any necessary numerical integrati on by a Newton-Cotes method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {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 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 "Q" } {TEXT 328 85 "5 .. calculating the area of a region enclosed between t wo curves (random functions) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "Execute the code below to obtain two Mapl e functions " }{TEXT 278 1 "f" }{TEXT -1 5 " and " }{TEXT 278 1 "g" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 56 " is a quadratic function with rational coefficients and \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 28 " is a sine wav e of the form " }{XPPEDIT 18 0 "g(x) = p+q*sin(r*x);" "6#/-%\"gG6#%\"x G,&%\"pG\"\"\"*&%\"qGF*-%$sinG6#*&%\"rGF*F'F*F*F*" }{TEXT -1 8 ", wher e " }{TEXT 335 1 "p" }{TEXT -1 2 ", " }{TEXT 336 1 "q" }{TEXT -1 5 " a nd " }{TEXT 337 1 "r" }{TEXT -1 23 " are rational numbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Calculate (corr ect to 10 digits) the area of the region enclosed between the curves \+ " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 42 " b etween their two points of intersection." }}{PARA 0 "" 0 "" {TEXT -1 70 "Perform any necessary numerical integration by a Newton-Cotes meth od. " }}{PARA 0 "" 0 "" {TEXT -1 35 "_________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "randomize():\nrand(64..96)()/16-rand(8..16)()/16*x^2 :\nf := unapply(%,x):\n'f(x)'=f(x);\nrand(28..36)()/16-rand(12..20)()/ 16*sin(rand(18..26)()/16*x):\ng := unapply(%,x):\n'g(x)'=g(x);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 4 "Note" } {TEXT -1 60 ": It is advisable to transfer a copy of the expressions f or " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 69 " to the next comm and line so that you can recover the definitions of " }{TEXT 278 1 "f " }{TEXT -1 5 " and " }{TEXT 278 1 "g" }{TEXT -1 39 " if you return to this question later. " }}{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 36 "_______________________________ _____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 "Q" }{TEXT 330 44 "6 .. calculating arc len gth along a curve " }}{PARA 0 "" 0 "" {TEXT -1 64 "Calculate (correc t to 10 digits) the arc length along the curve " }{XPPEDIT 18 0 "y=(4- x^2)*x^2" "6#/%\"yG*&,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"F(*$F*F+F(" }{TEXT -1 22 " from the point where " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\" \"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "Perform any neces sary numerical integration by a Newton-Cotes method." }}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________ ________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 "Q" }{TEXT 329 44 "7 .. calcu lating arc length along a curve " }}{PARA 0 "" 0 "" {TEXT -1 64 "Cal culate (correct to 10 digits) the arc length along the curve " } {XPPEDIT 18 0 "y = x*(Pi-x)*sqrt(sin(x));" "6#/%\"yG*(%\"xG\"\"\",&%#P iGF'F&!\"\"F'-%%sqrtG6#-%$sinG6#F&F'" }{TEXT -1 22 " from the point wh ere " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 20 " to the point where " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 69 "Perform any necessary numerical integrati on by a Newton-Cotes method." }}{PARA 0 "" 0 "" {TEXT -1 35 "_________ __________________________" }}{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 36 "____________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }{TEXT 331 53 ".. disecting a region into two piec es with equal area" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Find the area of the region under the graph " }{XPPEDIT 18 0 "y = ln (arctan(x)+1);" "6#/%\"yG-%#lnG6#,&-%'arctanG6#%\"xG\"\"\"F-F-" } {TEXT -1 14 " between the " }{TEXT 285 1 "y" }{TEXT -1 19 " axis and \+ the line " }{XPPEDIT 18 0 "x = 4" "6#/%\"xG\"\"%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 22 "(b) Find the value of " }{TEXT 286 1 "a" }{TEXT -1 20 " such that the line " }{XPPEDIT 18 0 "x = a" "6#/%\"xG% \"aG" }{TEXT -1 69 " divides the region described in (a) into two piec es with equal area." }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________ ________________" }}{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 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q9 " }{TEXT 332 55 ".. disecting a region into three piece s with equal area" }{TEXT -1 1 " " }}{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 82 ", find (corr ect to 10 digits) the area of the region enclosed beween the graph of \+ " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 9 " and th e " }{TEXT 307 1 "x" }{TEXT -1 18 " axis between the " }{TEXT 308 1 "y " }{TEXT -1 64 " axis and the first positive intersection of the graph with the " }{TEXT 309 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "(b) Find (correct to 10 d igits) 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 havi ng 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 whic h 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 4 "Q10 " }{TEXT 333 50 ".. solving an equation which involves an integral " }}{PARA 0 "" 0 "" {TEXT -1 4 "L et " }{XPPEDIT 18 0 "f(x) = Int(t^3/(t-arctan(t)),t=0..x)" "6#/-%\"fG6 #%\"xG-%$IntG6$*&%\"tG\"\"$,&F,\"\"\"-%'arctanG6#F,!\"\"F3/F,;\"\"!F' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "(a) Plot the graphs \+ of " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "y = exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 21 " in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 66 "(b) Find (co rrect to 10 digits) 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 " t o " }{XPPEDIT 18 0 "x = 4" "6#/%\"xG\"\"%" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 39 "_______________________________________" }}{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 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 0 "" 0 "" {TEXT -1 18 "code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 48 "Pictures for Simpson's 3/8 rule and Bode's rule " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1159 "inter p([0,1,2,3,4],[2,2.4,2.2,2,1.8],x):\nf := unapply(%,x):\ninterp([0,1,2 ,3],[2,2.4,2.2,2],x):\ng := unapply(%,x):\np1 := plot([f(x),g(x)],x=-. 5..3.5,color=[red,blue],thickness=1):\np2 := plot([[[0,0],[0,2]],[[1,0 ],[1,2.4]],[[2,0],[2,2.2]],\n [[3,0],[3,2]],[[-.5,0],[3.5,0]]],color =black):\np3 := plot([[[0,2],[1,2.4],[2,2.2],[3,2]]$3],\n style=poi nt,symbol=[circle,diamond,cross],color=black):\np4 := plot(g(x),x=0..3 ,color=COLOR(RGB,.9,.85,1),\n adaptive=false,numpoints=20,fi lled=true):\nt1 := plots[textplot]([[3.5,-0.1,`x`],\n [-.24,2.1,`(x \+ ,y )`],[.95,2.53,`(x ,y )`],\n [2.18,2.33,`(x ,y )`],[3,2.15,`(x ,y )`],\n [0,-0.08,`x`],[1,-0.08,`x`],[2,-0.08,`x`],\n [3,-0.08, `x`]],font=[HELVETICA,10]):\nt2 := plots[textplot]([[-.19,2.05,`0 0 `],[.98,2.48,`1 1`],\n [2.22,2.28,`2 2`],[3.04,2.1,`3 3`],\n [0.08,-0.13,`0`],[1.08,-0.13,`1`],[2.08,-0.13,`2`],\n [3.08,-0.13,` 3`]],font=[HELVETICA,8]):\nt3 := plots[textplot]([3.5,1.85,`y = f(x)`] ,color=red,font=[HELVETICA,10]):\nt4 := plots[textplot]([-.5,1.82,`y = p(x)`],color=blue,font=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4, t1,t2,t3,t4],\n view=[-.5..3.5,-.2..2.6],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1267 "in terp([0,1,2,3,4,5],[2,2.4,2.2,2,2.2,2.2],x):\nf := unapply(%,x):\ninte rp([0,1,2,3,4],[2,2.4,2.2,2,2.2],x):\ng := unapply(%,x):\np1 := plot([ f(x),g(x)],x=-.5..4.5,color=[red,blue],thickness=1):\np2 := plot([[[0, 0],[0,2]],[[1,0],[1,2.4]],[[2,0],[2,2.2]],\n [[3,0],[3,2]],[[4,0],[4 ,2.2]],[[-.5,0],[4.5,0]]],color=black):\np3 := plot([[[0,2],[1,2.4],[2 ,2.2],[3,2],[4,2.2]]$3],\n style=point,symbol=[circle,diamond,cross ],color=black):\np4 := plot(g(x),x=0..4,color=COLOR(RGB,.9,.85,1),\n \+ adaptive=false,numpoints=20,filled=true):\nt1 := plots[textpl ot]([[4.5,-0.1,`x`],\n [-.3,2.1,`(x ,y )`],[.95,2.55,`(x ,y )`], \n [2.2,2.35,`(x ,y )`],[3,2.15,`(x ,y )`],\n [3.77,2.32,`(x ,y )`],\n [0,-0.08,`x`],[1,-0.08,`x`],[2,-0.08,`x`],\n [3,-0.08,`x`] ,[4,-0.08,`x`]],font=[HELVETICA,10]):\nt2 := plots[textplot]([[-.25,2. 05,`0 0`],[1,2.5,`1 1`],\n [2.24,2.3,`2 2`],[3.04,2.1,`3 \+ 3`],[3.81,2.27,`4 4`],\n [0.08,-0.13,`0`],[1.08,-0.13,`1`],[2.08,- 0.13,`2`],\n [3.08,-0.13,`3`],[4.08,-0.13,`4`]],font=[HELVETICA,8]): \nt3 := plots[textplot]([4.5,2.2,`y = f(x)`],color=red,font=[HELVETICA ,10]):\nt4 := plots[textplot]([-.5,1.25,`y = p(x)`],color=blue,font=[H ELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3,t4],\n view= [-.5..4.5,-.2..2.7],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 22 "Pictures for examples " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "f := x -> exp(-x^2):\np 1 := plot(f(x),x=0..2.5):\np2 := plots[polygonplot]([[0,0],op(op(1,op( 1,plot(f(x),x=0..0.9)))),\n [.9,0]],style=patchnogrid,color=COLOR( RGB,.9,.9,.9)):\np3 := plot([[.9,0],[.9,f(.9)]],color=black):\nt1 := p lots[textplot]([[.9,-.05,`x`],[.45,.5,`p(x)`]],color=black):\nplots[di splay]([p1,p2,p3,t1],labels=[``,``],\n tickmarks=[2,2]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 754 "f := x -> exp(-x^2):\nxmid := .4743373061:\np1 := pl ot(f(x),x=0..2.5):\np2 := plots[polygonplot]([[0,0],op(op(1,op(1,plot( f(x),x=0..xmid)))),\n [xmid,0]],style=patchnogrid,color=COLOR(RGB, .93,.93,.93)):\np3 := plots[polygonplot]([[xmid,0],\n \+ op(op(1,op(1,plot(f(x),x=xmid..2)))),\n [2,0]],style=patchnogrid, color=COLOR(RGB,.95,.9,.9)):\np4 := plot([[[xmid,0],[xmid,f(xmid)]],\n [[2,0],[2,f(2)]]],color=black):\nt1 := plots[textplo t]([[.47,-.05,`x`],[.3,.4,`p(x)`],\n [.7,.4,`q(x)`]],color=bl ack,\n font=[HELVETICA,10]):\nt2 := plots[textplot]( [.575,-.07,`mid`],color=black,\n font=[HELVETICA,8 ]):\nplots[display]([p1,p2,p3,p4,t1,t2],labels=[``,``],\n \+ tickmarks=[2,2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 51 " The area of the region enclosed between two graphs " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 905 "g := x -> 2+sin(x/2)-cos(5/2*x)/4:\nf := x -> 5+cos(x/3)-sin(3*x)/5:\na := .7: \+ b := 3.4:\np1 := plot([f(x),g(x)],x=.5..3.6,y=0..4,color=[red,blue],th ickness=2):\np2 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]]],\n \+ color=COLOR(RGB,.4,.4,.4)):\np3 := plot([[[a,1],[a,g(a)] ],[[b,1],[b,g(b)]]],\n color=COLOR(RGB,.5,.5,.5),linest yle=3):\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,numpoints=25): \ngpts := op(1,op(1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-1],fp ts[i],gpts[i],gpts[i-1]],i=2..25)],\n style=patchnogrid,colo r=COLOR(RGB,.85,.85,.85)):\nt1 := plots[textplot]([2.4,6.1,`y = f(x)`] ,color=red):\nt2 := plots[textplot]([2.4,2.45,`y = g(x)`],color=blue): \nt3 := plots[textplot]([[a,.8,`x = a`],[b,.8,`x = b`]],color=black): \nplots[display]([p1,p2,p3,p4,t1,t2,t3],view=[0..4.03,.8..6.1],axes=no ne);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }