{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "**** MAPLE FILE # 5 **** " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "This file explains how to use MAPLE to compute \+ Riemann sums and definite integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The commands described here come from t he MAPLE \"student package,\"" }}{PARA 0 "" 0 "" {TEXT -1 34 "so we mu st include them by typing:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Remember, gi ven a function f and an interval a..b, a Riemann sum" }}{PARA 0 "" 0 " " {TEXT -1 66 "depends on two things: our choice of a partition for a. .b, and our" }}{PARA 0 "" 0 "" {TEXT -1 34 "method of picking the poin ts c_i. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "MAPLE has built in commands for co mputing Riemann Sums." }}{PARA 0 "" 0 "" {TEXT -1 68 "They all assume \+ (as we did in the first couple of examples we did in" }}{PARA 0 "" 0 " " {TEXT -1 61 "class) that a UNIFORM partition is used (in other words , that" }}{PARA 0 "" 0 "" {TEXT -1 45 "(Delta x)_i = (b-a)/n for all i = 1,2,..,n). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "MAPLE allows us three choices for picking the \"c_i point \"" }}{PARA 0 "" 0 "" {TEXT -1 46 "on each subinterval: using the righ t endpoint," }}{PARA 0 "" 0 "" {TEXT -1 34 "the left endpoint, or the \+ middle.\n" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "M APLE can sketch the rectangles used in the Riemann Sums as well as" }} {PARA 0 "" 0 "" {TEXT -1 28 "compute the sums themselves:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f := x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$)%\"xG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rightbox (f, x=0..2, 5);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 426 426 {PLOTDATA 2 "6*-%)POLYGONSG6$7&7$$\"\"!F)F(7$F($ \"+++++;!#57$$\"+++++SF-F+7$F/F(-%&COLORG6&%$RGBG$\"\"(!\"\"$\"\"*F8F6 -F$6$7&F17$F/$\"+++++kF-7$$\"+++++!)F-F?7$FBF(F2-F$6$7&FD7$FB$\"++++S9 !\"*7$$\"+++++7FKFI7$FMF(F2-F$6$7&FO7$FM$\"++++gDFK7$$F,FKFT7$FWF(F2-F $6$7&FX7$FW$\"\"%F)7$$\"\"#F)Fgn7$FjnF(F2-%'CURVESG6&7SF'7$$\"39LLLL3V fV!#>$\"3'fth!>PY+>!#?7$$\"3&pmm;H[D:)Fdo$\"374dzkVSYmFgo7$$\"3LLLLe0$ =C\"!#=$\"3Ry/hNJ9U:Fdo7$$\"3HLLL3RBr;F`p$\"3Q3JOwF-$z#Fdo7$$\"3Ymm;zj f)4#F`p$\"3pU9lin5/WFdo7$$\"3=LL$e4;[\\#F`p$\"3!\\2H?N2TA'Fdo7$$\"3p** **\\i'y]!HF`p$\"3'py7`.#[R%)Fdo7$$\"3,LL$ezs$HLF`p$\"3V1NO@BZ36F`p7$$ \"3^****\\7iI_PF`p$\"364mB\">!)zS\"F`p7$$\"3#pmmm@Xt=%F`p$\"3^@TN'*fQ` +&F`p$\"3*=/ Z7,,>]#F`p7$$\"3')******\\Z/NaF`p$\"3Nc-XV6(R&HF`p7$$\"3&*******\\$fC& eF`p$\"34U-MW!G^U$F`p7$$\"3ELL$ez6:B'F`p$\"3m(*4;ER<$)QF`p7$$\"3Rmmm;= C#o'F`p$\"3*H'3kpbBlWF`p7$$\"3,mmmm#pS1(F`p$\"3!GXE/Y2,*\\F`p7$$\"3]** **\\i`A3vF`p$\"3?D)3%4[MPcF`p7$$\"3slmmm(y8!zF`p$\"3l$3L9kyJC'F`p7$$\" 3V++]i.tK$)F`p$\"3=RHTH&RM%pF`p7$$\"39++](3zMu)F`p$\"33$)\\NbE%[k(F`p7 $$\"3\"pmm;H_?<*F`p$\"3Fvn5CVl7%)F`p7$$\"3dmm;zihl&*F`p$\"3,Me-![,,:*F `p7$$\"39LLL3#G,***F`p$\"3;PP*=;m-)**F`p7$$\"3;LLezw5V5!#<$\"3vDd?JO2) 3\"F_w7$$\"3*)***\\PQ#\\\"3\"F_w$\"3*pD4hxD'p6F_w7$$\"3ALL$e\"*[H7\"F_ w$\"3ce7dnU,h7F_w7$$\"3#*******pvxl6F_w$\"3y/^rUt.f8F_w7$$\"3z****\\_q n27F_w$\"3:(3NJ'Q[e9F_w7$$\"3%)***\\i&p@[7F_w$\"3'R,q)pb/e:F_w7$$\"3#) ****\\2'HKH\"F_w$\"3c.YriD'eN\"3ZAF_w7$$\"3))***\\7k.6a\"F_w$\"3?3S2L/+vBF_w7$$\"3emmmT9C# e\"F_w$\"3W6uszz[.DF_w7$$\"3!****\\i!*3`i\"F_w$\"3#4dN2/H;k#F_w7$$\"3P LLL$*zym;F_w$\"3-g,s9A=yFF_w7$$\"3GLL$3N1#4uep1$F_w7$$\"3%*******p(G**y\"F_w$\"3c7Pn ,]%Q?$F_w7$$\"3kmm;9@BM=F_w$\"3[>.k[uSkLF_w7$$\"3DLLL`v&Q(=F_w$\"3!fQ% =I@M6NF_w7$$\"3/++DOl5;>F_w$\"3zs*f#eUYrOF_w7$$\"3/++v.Uac>F_w$\"3_@xA @_1GQF_wFin-%'COLOURG6&F5$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%*THICKN ESSG6#F[o-%+AXESLABELSG6$Q\"x6\"Q!Fa_l-%%VIEWG6$;F(Fjn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The rightbox command sketches the rectangles in a Riemann sum where the right" }}{PARA 0 "" 0 "" {TEXT -1 79 "sub-interval endp oints are used to compute rectangle heights. Its arguments are" }} {PARA 0 "" 0 "" {TEXT -1 68 "the function, the interval, and the numbe r of rectangles to be used." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "To com pute the value of the Riemann Sum (i.e., the total shaded area above), use" }}{PARA 0 "" 0 "" {TEXT -1 12 "the command:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rightsum(f,x=0..2,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"#\"\"&\"\"\"-%$SumG6$,$*(\"\"%F(\"#D!\"\"%\"iG F&F(/F1;F(F'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "This is the a ctual Riemann sum in \"sigma notation.\" If we want our answer in frac tion" }}{PARA 0 "" 0 "" {TEXT -1 18 "form, we can type:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#))\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "...or if we pref er a decimal, we can type:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eva lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++?N!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The commands leftbox, leftsum, middlebox \+ and middlesum are analogous to the above:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " leftbox(f,x=0..2,5);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 426 426 {PLOTDATA 2 "6*-%'CURVESG6&7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3'fth!>P Y+>!#?7$$\"3&pmm;H[D:)F-$\"374dzkVSYmF07$$\"3LLLLe0$=C\"!#=$\"3Ry/hNJ9 U:F-7$$\"3HLLL3RBr;F9$\"3Q3JOwF-$z#F-7$$\"3Ymm;zjf)4#F9$\"3pU9lin5/WF- 7$$\"3=LL$e4;[\\#F9$\"3!\\2H?N2TA'F-7$$\"3p****\\i'y]!HF9$\"3'py7`.#[R %)F-7$$\"3,LL$ezs$HLF9$\"3V1NO@BZ36F97$$\"3^****\\7iI_PF9$\"364mB\">!) zS\"F97$$\"3#pmmm@Xt=%F9$\"3^@TN'*fQ`+&F9$\"3*=/Z7,,>]#F97$$\"3')******\\Z/NaF9$\"3Nc -XV6(R&HF97$$\"3&*******\\$fC&eF9$\"34U-MW!G^U$F97$$\"3ELL$ez6:B'F9$\" 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"Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(middlesum(f,x= 0..2,5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++SE!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Notice above that the values obtained fro m rightsum (3.52) and leftsum (1.92) are" }}{PARA 0 "" 0 "" {TEXT -1 91 "an upper bound and a lower bound, respectively, on the exact value of the definite integral" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*$)%\"xG\"\"#\"\"\"/F(;\"\"!F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "In fact, MAPLE also has a command for evaluating definite integrals exactly:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int(f,x =0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\")\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmmE!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 ". ..notice that, as expected the exact value falls between 1.92 and 3.52 . Note also that the" }}{PARA 0 "" 0 "" {TEXT -1 88 "middlesum command yielded a much better approximation to the exact value of the integra l" }}{PARA 0 "" 0 "" {TEXT -1 90 "than either leftsum or rightsum. Loo k at the pictures and explain to yourself why this is." }{MPLTEXT 1 0 0 "" }}}}{MARK "31 2 1" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }