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Antidifferentiation: The \"Int\" a nd \"int\" commands, Integral Curves. " }}{PARA 203 "" 0 "" {TEXT 208 101 "The command for antidifferentiation (Indefinite Integration) is \+ \"int(function, variable)\". Note: " }{TEXT 209 56 "Maple does not put the \"+C\" in the answer of integration" }{TEXT 208 79 ". We sta rt with an easy one. What is an antiderivative of the function f(x) = \+ " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 210 1 " " }{TEXT 210 15 "? ie. What is " }{XPPEDIT 18 0 "Int(x^2,x);" "6#-%$IntG6$*$%\"xG \"\"#F'" }{TEXT 210 1 " " }{TEXT 210 4 " ? " }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 12 "Int(x^2, x);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#-I$IntG6$I*protectedGF&I(_syslibG6\"6$*$I\"xGF(\"\"#F+" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 294 "This has done us litt le good. It just rewrote the integral in mathematical notation. The \+ only use I can think of for the \"Int\" with a capital \"I\" is that i t produces mathematical notation good for cutting and pasting into tex t. You can actually evalute this integral with the command \"value\"" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 9 "value(%);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#,$*$I\"xG6\"\"\"$#\"\"\"F'" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 104 "Note: This is \"an\" anti derivative. Specifically, the one where the constant of integration eq uals zero." }}{PARA 203 "" 0 "" {TEXT 208 71 "The \"int\" command with a small \"i\" replaces this sequence of commands. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 11 "int(x^2,x);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#,$*$I\"xG6\"\"\"$#\"\"\"F'" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 86 "Notice, you must put in the independen t variable. Here's what happens if you don't. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 9 "int(x^2);" }}{PARA 205 "" 1 "" {TEXT 212 51 "Error, (in int) wrong number (or type) of arguments" }{TEXT 212 1 "\n" }}}{EXCHG {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 208 87 "How \+ about a more difficult problem. What is the antiderivative of the func tion f(x) = " }{XPPEDIT 18 0 "(exp(x)+exp(-x))/(exp(x)-exp(-x));" "6# *&,&-%$expG6#%\"xG\"\"\"-F&6#,$F(!\"\"F)F),&F%F)F*F-F-" }{TEXT 210 1 " " }{TEXT 210 22 " ? If you let u = " }{XPPEDIT 18 0 "exp(x)-exp(- x);" "6#,&-%$expG6#%\"xG\"\"\"-F%6#,$F'!\"\"F," }{TEXT 210 1 " " } {TEXT 210 57 " you will notice that you have an integral of the form \+ " }{XPPEDIT 18 0 "Int(1/u,u);" "6#-%$IntG6$*&\"\"\"F'%\"uG!\"\"F(" } {TEXT 210 1 " " }{TEXT 210 73 " which you should recognize as ln(|u|) + C. Can Maple figure this out? " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 78 "f := x -> (exp(x) + exp(-x))/(exp(x) - exp(-x)); # this defines the integrand" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"fG 6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%*&,&-I$expGF%6#9$\"\"\"-F/6 #,$F1!\"\"F2F2,&F.F2F3F6F6F%F%F%" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 46 "int(f(x),x); # this finds the antideriv ative" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&-%$expG6#%\"xG\"\" \"-F(6#,$F*!\"\"F/" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 99 "Maple got it. Again, the arbitrary constant is not included in the answer. The actual answer is " }{XPPEDIT 18 0 "ln(exp(x)-exp(- x));" "6#-%#lnG6#,&-%$expG6#%\"xG\"\"\"-F(6#,$F*!\"\"F/" }{TEXT 210 1 " " }{TEXT 210 7 " + C. " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 209 16 "I ntegral Curves:" }{TEXT 208 215 " Suppose we want to plot some integra l curves. This amounts to plotting the antideritive of f(x) for vario us values of the arbitrary constant C. Here I define a function (the \+ antiderivative of f(x)) of x and C. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 39 "F := (x,C) -> ln(exp(x) - exp(-x)) + C;" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"FG6\"f*6$I\"xGF%I\"CGF%F%6$I)operatorGF %I&arrowGF%F%,&-I#lnGF%6#,&-I$expGF%6#9$\"\"\"-F36#,$F5!\"\"F:F69%F6F% F%F%" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 77 "and pl ot this function over a range of x values for different values of C by " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 60 "plot([F(x,-1),F(x, 0),F(x,1)],x=0..1,color=[red,green,blue]);" }}{PARA 206 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6&7^o7$$\"3`*****\\n5; \"o!#@$!3e8xt0Xc)f(!#<7$$\"3#******\\8ABO\"!#?$!3*4Z(=$4bF- 7$$\"3S+++v1h6oF1$!3y**)yqvrfH&F-7$$\"3Y******4G$R<)F1$!35[[x(zYO6&F-7 $$\"3N*****\\%\\DO&*F1$!3Qj_!o4#\\f\\F-7$$\"3%******zqd)*3\"!#>$!3-r\" 4W1cf#[F-7$$\"33+++N@Ki8F^o$!3>V$ywP,Gg%F-7$$\"3*)*****>c'yM;F^o$!3W(4 )*4@m/U%F-7$$\"3()******))4D2>F^o$!31(GboW*HmUF-7$$\"3')*****fT:(z@F^o $!3j4P`&\\\\F8%F-7$$\"3#*******zZ*z7$F^o$!3AK-M%yn9x$F-7$$\"33+++XTFwS F^o$!3JF'4eril]$F-7$$\"3=+++qMrU^F^o$!35j@'**R,SF$F-7$$\"3&******4z_\" 4iF^o$!3%*GkFogN&3$F-7$$\"3y*****\\;hEG(F^o$!3zYH.pGkDHF-7$$\"3o****** R&phN)F^o$!3#R#\\e$Qfyy#F-7$$\"3++++*=)H\\5!#=$!3#3@\"p\"*>[fDF-7$$\"3 %******z/3uC\"Ffr$!3CQhDT!ydQ#F-7$$\"35+++J$RDX\"Ffr$!3'RSYQk5EB#F-7$$ \"37+++)R'ok;Ffr$!3w\"o%[!y'=&4#F-7$$\"3-+++1J:w=Ffr$!3WbS1OZNu>F-7$$ \"33+++3En$4#Ffr$!3Q$)R9IIAj=F-7$$\"3-+++/RE&G#Ffr$!3a98(3XnUx\"F-7$$ \"3\")*****\\K]4]#Ffr$!3[M*QP=kBo\"F-7$$\"3$******\\PAvr#Ffr$!3Cyf#GGR uf\"F-7$$\"3)******\\nHi#HFfr$!3\\os'eE#\\@:F-7$$\"3*)*****p*ev:JFfr$! 3w\"*o5s#QoX\"F-7$$\"3$)*****z!47TLFfr$!3CaO/c^f%Q\"F-7$$\"3?+++LY.KNF fr$!3dR'HP\\doK\"F-7$$\"3u*****4o7Tv$Ffr$!3_gBs!R1KE\"F-7$$\"31+++$Q*o ]RFfr$!3*)R;=*eo'47F-7$$\"3?+++\"=lj;%Ffr$!3=@i<(\\GO:\"F-7$$\"3++++V& Rrl+\"F-7$$\"3!)*****H5k]*\\Ffr$!3Of$>V*pV(f*Ffr7$$ \"3A+++(RQb@&Ffr$!3y?d2yIjG\"*Ffr7$$\"3`+++=>Y2aFfr$!3S))=(RZ+Rt)Ffr7$ $\"3\\+++yXu9cFfr$!3OF\\#R%*3/K)Ffr7$$\"3d******[y))GeFfr$!3[/mn)fHh!z Ffr7$$\"3.+++i_QQgFfr$!3!)G@&otID^(Ffr7$$\"3A+++!y%3TiFfr$!3Ayk!*f'z=9 (Ffr7$$\"35+++O![hY'Ffr$!3N()RkWU=TnFfr7$$\"3I+++#Qx$omFfr$!3gGCUEwe_`Ffr7$$\"3#******H,M^ \\(Ffr$!3/cS]:v[K]Ffr7$$\"3S+++0#=bq(Ffr$!35%[ZP$pI/ZFfr7$$\"3Y*****p? 27\"zFfr$!3cg['o5\"R*Q%Ffr7$$\"3a+++IXaE\")Ffr$!3#z_d_(RglSFfr7$$\"37+ ++l*RRL)Ffr$!3!*>+QKg6fPFfr7$$\"3i*****HvJga)Ffr$!3mI`=&*[z]MFfr7$$\"3 s*****HJnjv)Ffr$!33&*o>Q]#)\\JFfr7$$\"3k******[Qk\\*)Ffr$!3qin)H9Ps(GF fr7$$\"3w******o0;r\"*Ffr$!3-(*H$RHK#pDFfr7$$\"3[*****\\w(Gp$*Ffr$!39G %Q(o0_(H#Ffr7$$\"37+++!oK0e*Ffr$!3#ol]v'>_6?Ffr7$$\"33+++<5s#y*Ffr$!3# \\Z,>yr6u\"Ffr7$$\"\"\"\"\"!$!3u\"f)oyX8a9Ffr7^o7$F($!3e8xt0Xc)f'F-7$F /$!3*4Z(=$4XF -7$FS$!3y**)yqvrfH%F-7$FX$!35[[x(zYO6%F-7$Fgn$!3#HE0o4#\\fRF-7$F\\o$!3 Zr\"4W1cf#QF-7$Fbo$!3uU$ywP,Gg$F-7$Fgo$!3W(4)*4@m/U$F-7$F\\p$!3i'GboW* HmKF-7$Fap$!3j4P`&\\\\F8$F-7$Ffp$!3AK-M%yn9x#F-7$F[q$!3JF'4eril]#F-7$F `q$!35j@'**R,SF#F-7$Feq$!3%*GkFogN&3#F-7$Fjq$!3zYH.pGkD>F-7$F_r$!39C \\e$Qfyy\"F-7$Fdr$!3/67p\"*>[f:F-7$Fjr$!3YQhDT!ydQ\"F-7$F_s$!3'RSYQk5E B\"F-7$Fds$!3w\"o%[!y'=&4\"F-7$Fis$!3:`0kgtaV(*Ffr7$F^t$!3sK)R9IIAj)Ff r7$Fct$!3bYJr3XnUxFfr7$Fht$!3#[M*QP=kBoFfr7$F]u$!3e$yf#GGRufFfr7$Fbu$! 3!\\os'eE#\\@&Ffr7$Fgu$!3==*o5s#QoXFfr7$F\\v$!3'HaO/c^f%QFfr7$Fav$!3E' R'HP\\doKFfr7$Ffv$!3G1OA2R1KEFfr7$F[w$!3S*R;=*eo'4#Ffr7$F`w$!3z6Awr\\G O:Ffr7$Few$!3+_HN#3Ri-\"Ffr7$Fjw$!3K@#4*z3Qh^F^o7$F_x$!3YV()Hms>rlF17$ Fdx$\"3V,k!o0Ic-%F^o7$Fix$\"34)yU#>#pOr)F^o7$F^y$\"3K6\"Gg_*4m7Ffr7$Fc y$\"3Os]2c5fz;Ffr7$Fhy$\"3_&RB8SqQ4#Ffr7$F]z$\"3[ry9j#pu[#Ffr7$Fbz$\"3 y@N4S.7eGFfr7$Fgz$\"3k7gNbd\")eKFfr7$F\\[l$\"3TrvdtB#)4OFfr7$Fa[l$\"3y m$fd>Le(RFfr7$Ff[l$\"3\\4U$\\>,-I%Ffr7$F[\\l$\"3U:E]0!=uk%Ffr7$F`\\l$ \"3'R%f\\%[7v'\\Ffr7$Fe\\l$\"3!f^_i1$p&H&Ffr7$Fj\\l$\"3WR^8$*)31h&Ffr7 $F_]l$\"32sCuCgRMfFfr7$Fd]l$\"35!)*>w'R)3C'Ffr7$Fi]l$\"3MpY\"[50#\\lFf r7$F^^l$\"3#\\5.='\\<]oFfr7$Fc^l$\"3IPK,dGwArFfr7$Fh^l$\"3*H+ngqn2V(Ff r7$F]_l$\"3'=dh7VzCq(Ffr7$Fb_l$\"3=V$\\C.y%))zFfr7$Fg_l$\"33D&)4=#G)e# )Ffr7$F\\`l$\"3C39J@a'ea)Ffr7^o7$F($!3e8xt0Xc)f&F-7$F/$!3*4Z(=$4NF-7$FS$!3y**)yqvrf H$F-7$FX$!35[[x(zYO6$F-7$Fgn$!3#HE0o4#\\fHF-7$F\\o$!3Zr\"4W1cf#GF-7$Fb o$!3uU$ywP,Gg#F-7$Fgo$!3W(4)*4@m/U#F-7$F\\p$!3i'GboW*HmAF-7$Fap$!3j4P` &\\\\F8#F-7$Ffp$!3AK-M%yn9x\"F-7$F[q$!3JF'4eril]\"F-7$F`q$!35j@'**R,SF \"F-7$Feq$!3%*GkFogN&3\"F-7$Fjq$!3#zYH.pGkD*Ffr7$F_r$!3KT#\\e$QfyyFfr7 $Fdr$!3W5@\"p\"*>[f&Ffr7$Fjr$!3g%QhDT!ydQFfr7$F_s$!3`RSYQk5EBFfr7$Fds$ !3cw\"o%[!y'=&*F^o7$Fis$\"3WoWf$REXc#F^o7$F^t$\"3Gn,c)ppxO\"Ffr7$Fct$ \"3W`oG\"\\DtD#Ffr7$Fht$\"3=b1hi\"ej<$Ffr7$F]u$\"3V;-urrgDSFfr7$Fbu$\" 35:tKTt2&y%Ffr7$Fgu$\"3P#3J*yshJaFfr7$F\\v$\"3gdMcR%[S:'Ffr7$Fav$\"3I/ Oqi]UJnFfr7$Ffv$\"3s$RwF4OzO(Ffr7$F[w$\"39,O=3TJ.zFfr7$F`w$\"3?)yP#G]r j%)Ffr7$Few$\"3'y/Zw\"4wt*)Ffr7$Fjw$\"3My!4?\">'Q[*Ffr7$F_x$\"3m7qLF!) GM**Ffr7$Fdx$\"31k!o0Ic-/\"F-7$Fix$\"3#zU#>#pOr3\"F-7$F^y$\"306Gg_*4m7 \"F-7$Fcy$\"392vg0\"fz;\"F-7$Fhy$\"3mRB8SqQ47F-7$F]z$\"3C(y9j#pu[7F-7$ Fbz$\"3=_$4S.7eG\"F-7$Fgz$\"3F,c`v:)eK\"F-7$F\\[l$\"39dxNPA)4O\"F-7$Fa [l$\"3iOfd>Le(R\"F-7$Ff[l$\"3+@M\\>,-I9F-7$F[\\l$\"3]h-b+=uk9F-7$F`\\l $\"3S%f\\%[7v'\\\"F-7$Fe\\l$\"3f^_i1$p&H:F-7$Fj\\l$\"3%Q^8$*)31h:F-7$F _]l$\"3?ZUZ-'RMf\"F-7$Fd]l$\"3!z*>w'R)3C;F-7$Fi]l$\"3/n9[50#\\l\"F-7$F ^^l$\"3Q5.='\\<]o\"F-7$Fc^l$\"3iB8q&GwAr\"F-7$Fh^l$\"3I+ngqn2V " 0 "" {MPLTEXT 1 211 28 "C1 := fsolve(F(1,C) = \+ 0,C); " }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I#C1G6\"$!+?a'ea)!#5" } {TEXT 20 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 23 "plot(F (x,C1),x=0..1.5);" }}{PARA 206 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"3/+++,;u@5!#?$!3oN:@VeoZq!#<7$$\"33 +++-K[V?F*$!3!4ut19QXN'F-7$$\"3#******R![AlIF*$!31nkFi@2\\fF-7$$\"3s** ***\\Smp3%F*$!3_Z(e>())QhcF-7$$\"3z******4'\\/8'F*$!3ccgf$G?fD&F-7$$\" 3Y******4G$R<)F*$!3%z%[xRLBo\\F-7$$\"3'******>#**3E7!#>$!3G*fZ>JaFc%F- 7$$\"3*)*****>c'yM;FJ$!3G(4)*Hv_]F%F-7$$\"3#******R%)z@X#FJ$!3!**)fWo> `pQF-7$$\"3')*****\\7t&pKFJ$!3=+'G<&>x\"e$F-7$$\"3/+++s@*>p%FJ$!3[&fb( )R&Q?KF-7$$\"3B+++>7T9hFJ$!39cv&[/Q`&HF-7$$\"3d*****\\?qSr(FJ$!30_7.-? dAFF-7$$\"30+++)=HPJ*FJ$!3o%e9xjuO`#F-7$$\"3/+++v\"*R#4\"!#=$!35mEJ'** fOP#F-7$$\"35+++JaU`7F]p$!3NalE276(=F]p$!3L.hN.KmJ=F-7$$\"3#******p**3)y@F]p$!3q,2 cElMx;F-7$$\"3/+++(fHq\\#F]p$!3>ah\\v@bQ:F-7$$\"3))******e'HU\"GF]p$!3 ^fc$GwSF] p$!3fw;z!3(HJ5F-7$$\"3?+++7XM*Q%F]p$!3ib80\"3x$H&*F]p7$$\"3'******p%Qj tYF]p$!3)>0o][U%f))F]p7$$\"3.+++i8o6]F]p$!3h\"=&QMSO2\")F]p7$$\"31+++] >0)H&F]p$!3;mi?i2L.vF]p7$$\"3\"******>-p6j&F]p$!3O4kB_#\\S$oF]p7$$\"3d *****\\2Mg#fF]p$!35HGSur4oiF]p7$$\"3M+++sxa\\iF]p$!3;AK@2lZscF]p7$$\"3 U+++;$4wb'F]p$!3K0A'*e5GF^F]p7$$\"3[+++>#R!zoF]p$!3EBBR'yZ)yXF]p7$$\"3 Y+++4A@urF]p$!3i\\6%p*pu\"4%F]p7$$\"3%)*****f:'f#\\(F]p$!3C9e6\")F]p$!3!ee( 3&zgVj#F]p7$$\"3M+++po6A%)F]p$!3'\\O<)z!)=w@F]p7$$\"3Z*****\\xJLu)F]p$ !3TL#))G`4Ur\"F]p7$$\"3e*****R*ydd!*F]p$!3)eH%=e\\Xs7F]p7$$\"3%******> v3/*[UFJ7$$\" 3'******\\!Q*>1\"F-$\"3A![9\"*ok!3!)FJ7$$\"35+++R(zS4\"F-$\"3XWLFSAr07 F]p7$$\"36+++-,FC6F-$\"33mOZ&4P8e\"F]p7$$\"3-+++Jx#e:\"F-$\"3o7[LBV$)o >F]p7$$\"3!******43\"o'=\"F-$\"3*pvYWG\"F- $\"3gT<+(o/=Z$F]p7$$\"32+++(4bMJ\"F-$\"3:vxtu/?QQF]p7$$\"36+++ylWU8F-$ \"3Q8*Qx(Q#><%F]p7$$\"33+++'3ucP\"F-$\"3#y(G;`\"o6b%F]p7$$\"3++++lJR09 F-$\"3o)G7;rMw)[F]p7$$\"3-+++-*zqV\"F-$\"3c_*H>nIPC&F]p7$$\"31+++`\"3u Y\"F-$\"3e&\\+2sq@e&F]p7$$\"3++++++++:F-$\"34$)HdqFWVfF]p-%&COLORG6&%$ RGBG$\"#5!\"\"$\"\"!Fh^lFi^l-%+AXESLABELSG6$Q\"x6\"Q!F__l-%%FONTG6$%*H ELVETICAGFg^l-%%VIEWG6$;Fi^l$\"#:Fh^l;$!2/\"phXk_+s!#;$\"2?vEYz[=Z(F-" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 108 "It sure looks like this goes through the po int (1,0). How can you be sure? Evaluate F at x = 1 and C = C1. " }}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 8 "F(1,C1);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#,&-I#lnG6$I*protectedGF'I(_syslibG6\"6#,&-I$expGF& 6#\"\"\"F/-F-6#!\"\"F2F/$!+?a'ea)!#5F/" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 66 "Is this zero? Have Maple evaluate thi s as a floating point number" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 15 "evalf(F(1,C1));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#$\"\"!F$ " }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 7 "Bingo. " }} }{EXCHG {PARA 203 "" 0 "" {TEXT 209 10 "Summary: " }}{PARA 203 "" 0 " " {TEXT 208 91 "1) Use the \"int(function, variable)\" to evaluate ind efinite integrals (antidifferentiate). " }}{PARA 203 "" 0 "" {TEXT 208 88 "2) If you want to see the integral in mathematical notation us e \"Int(function,variable)\"" }}{PARA 203 "" 0 "" {TEXT 208 78 "3) Rem ember, that the constant of integration is excluded from Maple's answe r." }}{PARA 203 "" 0 "" {TEXT 208 76 "4) You can solve for the constan t of integration using Maple if necessary. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 21 "# end of this section" }}}}{SECT 1 {PARA 202 "" 0 "" {TEXT 207 44 "2. Area Approximations, Summation Techniques" }} {EXCHG {PARA 203 "" 0 "" {TEXT 208 442 "In this section you will learn how to use Maple to visualize the approximate area under a positive f unction using boxes (rectangles really). There is another command to find the summed area of these boxes (again rectangles). This command will be used with the limit command to find the area under the curve. The maple commands that draw boxes and compute the areas are part of the \"student\" package which you must load in order to access." }}} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 39 "restart: # this clear s all variables" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 48 "with (student): # this loads the student library" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 39 "To approximate the area under the cuve " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 210 1 " " } {TEXT 210 102 " over the interval from 1 to 3, several boxes can be us ed. The following command draws the graph of " }{XPPEDIT 18 0 "f(x) \+ = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 210 1 " " }{TEXT 210 41 " a nd six boxes using the left end points." }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 25 "leftbox(x^2, x = 1..3,6);" }}{PARA 206 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%)POLYGONSG6)7&7$$\"\"\"\"\"!$F* F*7$F(F(7$$\"+LLLL8!\"*F(7$F.F+7&F17$F.$\"+yxxxq!#;$\"1#=t:!zg>!*FjnFhnFhnF [oFhnFhnF[oFhnFhnF[oFhnFhnF[oFhnFhnF[oFhn-%'CURVESG6&7SF,7$$\"3ALLL3Vf V5!#<$\"3EGdQ!3*3*3\"Fdo7$$\"3smm\"H[D:3\"Fdo$\"3CH\")>qtpp6Fdo7$$\"3X LL$e0$=C6Fdo$\"3krF-Vvyj7Fdo7$$\"3QLL$3RBr;\"Fdo$\"3%yHI%4qLLezs$HL\"Fdo$\"3)o,.8z@n x\"Fdo7$$\"31++D@1Bv8Fdo$\"3?hOih#f7*=Fdo7$$\"3pmmm@Xt=9Fdo$\"3PX(oHk2 G,#Fdo7$$\"3MLL$3y_qX\"Fdo$\"3r\\T81G+B@Fdo7$$\"3'******\\1!>+:Fdo$\"3 +/Z7J-d]AFdo7$$\"3*******\\Z/Na\"Fdo$\"3yD]MkgS#Q#Fdo7$$\"35+++NfC&e\" Fdo$\"3kCSVuY+8DFdo7$$\"3LLLez6:B;Fdo$\"3`mFy^(>Yj#Fdo7$$\"3_mmm\"=C#o ;Fdo$\"3->uH?>(Hy#Fdo7$$\"3gmmmEpS1Fdo$\"3e5Su!*envOFdo7$$\"3mmm\"zihl&>Fdo$\"3%p\"f$QS L\"GQFdo7$$\"3KLL$3#G,**>Fdo$\"3CSg&yD_g*RFdo7$$\"3 " 0 "" {MPLTEXT 1 211 22 "lef tsum(x^2,x=1..3,6);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#,$-I$SumGI(_sy slibG6\"6$*$,&\"\"\"F+I\"iGF'#F+\"\"$\"\"#/F,;\"\"!\"\"&F-" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 9 "evalf(%);" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#$\"+p.Pqt!\"*" }{TEXT 20 1 " " }}} {EXCHG {PARA 203 "" 0 "" {TEXT 208 114 "From the graph, you may assume that this value is smaller than the actual area under the curve. Usi ng rightboxes:" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 23 "right box(x^2,x=1..3,6);" }}{PARA 206 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6&7S7$$\"\"\"\"\"!F(7$$\"3ALLL3VfV5!#<$\"3EGd Q!3*3*3\"F.7$$\"3smm\"H[D:3\"F.$\"3CH\")>qtpp6F.7$$\"3XLL$e0$=C6F.$\"3 krF-Vvyj7F.7$$\"3QLL$3RBr;\"F.$\"3%yHI%4qLLezs$HL\"F.$\"3)o,.8z@nx\"F.7$$\"31++D@1Bv8F.$\"3 ?hOih#f7*=F.7$$\"3pmmm@Xt=9F.$\"3PX(oHk2G,#F.7$$\"3MLL$3y_qX\"F.$\"3r \\T81G+B@F.7$$\"3'******\\1!>+:F.$\"3+/Z7J-d]AF.7$$\"3*******\\Z/Na\"F .$\"3yD]MkgS#Q#F.7$$\"35+++NfC&e\"F.$\"3kCSVuY+8DF.7$$\"3LLLez6:B;F.$ \"3`mFy^(>Yj#F.7$$\"3_mmm\"=C#o;F.$\"3->uH?>(Hy#F.7$$\"3gmmmEpS1F.$\"3e5Su!*envOF.7$$\"3mmm\"zihl&> F.$\"3%p\"f$QSL\"GQF.7$$\"3KLL$3#G,**>F.$\"3CSg&yD_g*RF.7$$\"3q!#; $\"1#=t:!zg>!*Fj^lFh^lFh^lF[_lFh^lFh^lF[_lFh^lFh^lF[_lFh^lFh^lF[_lFh^l Fh^lF[_lFh^l-%+AXESLABELSG6$Q\"x6\"Q!Fa_l-%%FONTG6$%*HELVETICAGFc[l-%% VIEWG6$;Fb[l$\"#IFd[l;$!#=!\"#$\"$=*F``l" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 30 "evalf(rightsum(x^2,x=1..3,6));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#$\"+/Pq.5!\")" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 168 "and we may safely assume this value is larger than the actual area under the curve. Therefore, it is safe to assume tha t the actual area is between 7.37 and 10.04. " }}{PARA 203 "" 0 "" {TEXT 208 41 "Question: How do we find the exact area ?" }}{PARA 203 " " 0 "" {TEXT 208 94 "Answer: By taking the limit of either of the abov e as the number of boxes goes to infinity. " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 66 "First lets try 50 boxes evaluating f at the right-h and endpoints. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 25 "righ tbox(x^2, x=1..3,50);" }}{PARA 206 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6&7S7$$\"\"\"\"\"!F(7$$\"3ALLL3VfV5!#<$\"3EGd Q!3*3*3\"F.7$$\"3smm\"H[D:3\"F.$\"3CH\")>qtpp6F.7$$\"3XLL$e0$=C6F.$\"3 krF-Vvyj7F.7$$\"3QLL$3RBr;\"F.$\"3%yHI%4qLLezs$HL\"F.$\"3)o,.8z@nx\"F.7$$\"31++D@1Bv8F.$\"3 ?hOih#f7*=F.7$$\"3pmmm@Xt=9F.$\"3PX(oHk2G,#F.7$$\"3MLL$3y_qX\"F.$\"3r \\T81G+B@F.7$$\"3'******\\1!>+:F.$\"3+/Z7J-d]AF.7$$\"3*******\\Z/Na\"F .$\"3yD]MkgS#Q#F.7$$\"35+++NfC&e\"F.$\"3kCSVuY+8DF.7$$\"3LLLez6:B;F.$ \"3`mFy^(>Yj#F.7$$\"3_mmm\"=C#o;F.$\"3->uH?>(Hy#F.7$$\"3gmmmEpS1F.$\"3e5Su!*envOF.7$$\"3mmm\"zihl&> F.$\"3%p\"f$QSL\"GQF.7$$\"3KLL$3#G,**>F.$\"3CSg&yD_g*RF.7$$\"3F_\\l7$$\"+++++9F_\\lFf`l7$Fi`lF[\\l7&F[al7$Fi`l$\"+++gt?F_\\l 7$F^^lF^al7$F^^lF[\\l7&Faal7$F^^l$\"+++S!>#F_\\l7$$\"++++![\"F_\\lFdal 7$FgalF[\\l7&Fial7$Fgal$\"+++S5BF_\\l7$$\"++++?:F_\\lF\\bl7$F_blF[\\l7 &Fabl7$F_bl$\"+++gLCF_\\l7$$\"++++g:F_\\lFdbl7$FgblF[\\l7&Fibl7$Fgbl$ \"++++gDF_\\l7$$\"+++++;F_\\lF\\cl7$F_clF[\\l7&Facl7$F_cl$\"+++g*o#F_ \\l7$$\"++++S;F_\\lFdcl7$FgclF[\\l7&Ficl7$Fgcl$\"+++SAGF_\\l7$$\"++++! o\"F_\\lF\\dl7$F_dlF[\\l7&Fadl7$F_dl$\"+++SeHF_\\l7$$\"++++?F_\\lF\\gl7$F_glF[\\l7&Fagl7$F_gl$\"+++gTQF_\\l7$Ff`lFdgl7$Ff`lF[ \\l7&Fggl7$Ff`l$\"\"%F*7$$F][lF*Fjgl7$F]hlF[\\l7&F^hl7$F]hl$\"+++ghTF_ \\l7$$\"++++S?F_\\lFahl7$FdhlF[\\l7&Ffhl7$Fdhl$\"+++SEVF_\\l7$$\"++++! 3#F_\\lFihl7$F\\ilF[\\l7&F^il7$F\\il$\"+++S%\\%F_\\l7$$\"++++?@F_\\lFa il7$FdilF[\\l7&Ffil7$Fdil$\"+++glYF_\\l7$$\"++++g@F_\\lFiil7$F\\jlF[ \\l7&F^jl7$F\\jl$\"++++S[F_\\l7$$\"+++++AF_\\lFajl7$FdjlF[\\l7&Ffjl7$F djl$\"+++g<]F_\\l7$$\"++++SAF_\\lFijl7$F\\[mF[\\l7&F^[m7$F\\[m$\"+++S) >&F_\\l7$$\"++++!G#F_\\lFa[m7$Fd[mF[\\l7&Ff[m7$Fd[m$\"+++S#Q&F_\\l7$$ \"++++?BF_\\lFi[m7$F\\\\mF[\\l7&F^\\m7$F\\\\m$\"+++gpbF_\\l7$$\"++++gB F_\\lFa\\m7$Fd\\mF[\\l7&Ff\\m7$Fd\\m$\"++++gdF_\\l7$$\"+++++CF_\\lFi \\m7$F\\]mF[\\l7&F^]m7$F\\]m$\"+++g`fF_\\l7$$\"++++SCF_\\lFa]m7$Fd]mF[ \\l7&Ff]m7$Fd]m$\"+++S]hF_\\l7$$\"++++![#F_\\lFi]m7$F\\^mF[\\l7&F^^m7$ F\\^m$\"+++S]jF_\\l7$$\"++++?DF_\\lFa^m7$Fd^mF[\\l7&Ff^m7$Fd^m$\"+++g` lF_\\l7$F\\clFi^m7$F\\clF[\\l7&F\\_m7$F\\cl$\"++++gnF_\\l7$$\"+++++EF_ \\lF__m7$Fb_mF[\\l7&Fd_m7$Fb_m$\"+++gppF_\\l7$$\"++++SEF_\\lFg_m7$Fj_m F[\\l7&F\\`m7$Fj_m$\"+++S#=(F_\\l7$$\"++++!o#F_\\lF_`m7$Fb`mF[\\l7&Fd` m7$Fb`m$\"+++S)R(F_\\l7$$\"++++?FF_\\lFg`m7$Fj`mF[\\l7&F\\am7$Fj`m$\"+ ++gq!#;$\"1#=t:!zg>!*FddmFbdmFbdmFed mFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmF edmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbd mFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmF bdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbd mFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmF bdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFed mFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmF edmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbdmFedmFbdmFbd mFedmFbdmFbdmFedmFbdm-%+AXESLABELSG6$Q\"x6\"Q!F[em-%%FONTG6$%*HELVETIC AGFc[l-%%VIEWG6$;Fb[l$\"#IFd[l;$!#=!\"#$\"$=*Fjem" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 31 "evalf(rightsum(x^2,x=1..3,50));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#$\"+++?F))!\"*" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 157 "It seems as though this is getting closer t o the exact area by inspection of the graph, but is still too big. We create a function of the number of boxes by" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 40 "boxarea := n -> rightsum(x^2, x=1..3,n);" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#>I(boxareaG6\"f*6#I\"nGF%F%6$I)operat orGF%I&arrowGF%F%-I)rightsumGF%6%*$I\"xGF%\"\"#/F0;\"\"\"\"\"$9$F%F%F% " }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 20 "eva lf(boxarea(100));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#$\"+++!ou)!\"*" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 98 "This should \+ be closer still. To get the limit as \"n\" goes to infinity we use th e \"limit\" command:" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 29 "limit(boxarea(n),n=infinity);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6##\" #E\"\"$" }{TEXT 20 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 81 "Thi s is the exact area which we can check by the fundemental theorem of c alculus." }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 47 "We can see that ou r error in using 100 boxes is" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 51 "evalf(boxarea(100) - Limit(boxarea(n),n=infinity));" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#$\")LL8!)!\"*" }{TEXT 20 1 " " }}} {EXCHG {PARA 203 "" 0 "" {TEXT 208 262 "Not bad. Notice that using rig ht boxes to approximate the area under an increasing function will alw ays be greater than the exact area. If the function is decreasing, do \+ you expect the approximation with right boxes to be larger or smaller \+ than the exact area? " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 211 21 "# end of this section" }}}}{SECT 1 {PARA 202 "" 0 "" {TEXT 207 10 "Assignment" }}{EXCHG {PARA 207 "" 0 "" {TEXT 213 51 "Problem # 1: Ant iddifferentiation, Integral Curves." }}{PARA 203 "" 0 "" {TEXT 208 68 "Generate a graph of at least three integral curves of the function " }{XPPEDIT 18 0 "f(x) = (1+x)*sin(x);" "6#/-%\"fG6#%\"xG*&,&\"\"\"F*F' F*F*-%$sinGF&F*" }{TEXT 210 1 " " }{TEXT 210 21 " over the interval ( " }{XPPEDIT 18 0 "-Pi/2,Pi/2;" "6$,$*&%#PiG\"\"\"\"\"#!\"\"F(F$" } {TEXT 210 1 " " }{TEXT 210 3 "). " }}{PARA 203 "" 0 "" {TEXT 208 50 "F ind the solution to the initial value problem: " }{XPPEDIT 18 0 "dy/ dx = f(x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6#%\"xG" }{TEXT 210 1 " " }{TEXT 210 18 ", y(0) = 3.2" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 214 31 "Problem #2: Area Approximations" }}{PARA 203 "" 0 "" {TEXT 208 5 "Let " }{XPPEDIT 18 0 "f(x) = exp(-x^2);" "6#/-%\"fG6#%\"xG-%$e xpG6#,$*$F'\"\"#!\"\"" }{TEXT 210 1 " " }{TEXT 210 91 " . Approxima te the area under this curve defined by y = f(x) over the interval [0, 3] by " }}{PARA 203 "" 0 "" {TEXT 208 110 "a) using 20 rectangles with the height determined by f evaluated at the right hand endpoint of ea ch interval. " }}{PARA 203 "" 0 "" {TEXT 208 104 "b) using 20 rectangl es with height determined by f evaluated at the left hand endpoint of \+ each interval." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 208 163 " c) Give an upper bound and a lower bound of the actual area under the \+ curve based on the function being increasing or decreasing and the res ults from (a) and (b). " }}{PARA 203 "" 0 "" {TEXT 208 203 "d) Using a limiting process to find the actual area under the curve. Does this l ie within the bounds from part (c). You will need to use the command \+ \"evalf\" on the limit answer to get an actual number. " }}{PARA 203 " " 0 "" }}}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" }{PARA 208 "" 0 "" }} {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }