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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 83 "Lab # 6 Antidifferentiati
on, Integral Curves, Direction Fields, Area Appoximations " }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 72 "1. Antidifferentiation: The  \"Int\" an
d \"int\" commands, Integral Curves. " }}{PARA 0 "" 0 "" {TEXT -1 101 
"The command for antidifferentiation (Indefinite Integration) is \"int
(function, variable)\".    Note:  " }{TEXT 261 56 "Maple does not put \+
the \"+C\" in the answer of integration" }{TEXT -1 79 ".  We start wit
h an easy one. What is an antiderivative of the function f(x) = " }
{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 15 "?  ie. What is " }
{XPPEDIT 18 0 "Int(x^2,x);" "6#-%$IntG6$*$%\"xG\"\"#F'" }{TEXT -1 4 " \+
 ? " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Int(x^2" }{TEXT -1 0 "
" }{MPLTEXT 1 0 5 ", x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*
$)%\"xG\"\"#\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 294 "This has
 done us little good.  It just rewrote the integral in mathematical no
tation.  The only use I can think of for the \"Int\" with a capital \"
I\" is that it produces mathematical notation good for cutting and pas
ting into text.  You can actually evalute this integral with the comma
nd \"value\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%\"xGF%\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Note:  This is \"an\" antiderivat
ive. Specifically, the one where the constant of integration equals ze
ro." }}{PARA 0 "" 0 "" {TEXT -1 71 "The \"int\" command with a small \+
\"i\" replaces this sequence of commands. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "int(x^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
\"\"$!\"\"%\"xGF%\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Notice
, you must put in the independent variable.  Here's what happens if yo
u don't.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(x^2);" }}
{PARA 8 "" 1 "" {TEXT -1 52 "Error, (in int) wrong number (or type) of
 arguments\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 87 "How about a more difficult problem. What is the antider
ivative of the function f(x) =  " }{XPPEDIT 18 0 "(exp(x)+exp(-x))/(ex
p(x)-exp(-x));" "6#*&,&-%$expG6#%\"xG\"\"\"-F&6#,$F(!\"\"F)F),&-F&6#F(
F)-F&6#,$F(F-F-F-" }{TEXT -1 22 "   ?  If you let u =  " }{XPPEDIT 18 
0 "exp(x)-exp(-x);" "6#,&-%$expG6#%\"xG\"\"\"-F%6#,$F'!\"\"F," }{TEXT 
-1 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 -1 73 "  which you should recognize as ln(|u|) + C.  Can Maple f
igure this out? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x \+
-> (exp(x) + exp(-x))/(exp(x) - exp(-x));  # this defines the integran
d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&
arrowGF(*&,&-%$expG6#9$\"\"\"-F/6#,$F1!\"\"F2F2,&F.F2F3F6F6F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "int(f(x),x);   # this finds \+
the antiderivative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&-%$exp
G6#%\"xG\"\"\"-F(6#,$F*!\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 
"Maple got it. Again, the arbitrary constant is not included in the an
swer.    The actual answer is " }{XPPEDIT 18 0 "ln(exp(x)-exp(-x));" "
6#-%#lnG6#,&-%$expG6#%\"xG\"\"\"-F(6#,$F*!\"\"F/" }{TEXT -1 7 "  + C. \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 16 "Integral Curves:" }{TEXT -1 
215 " Suppose we want to plot some integral curves.  This amounts to p
lotting the antideritive of f(x) for various values of the arbitrary c
onstant C.  Here I define a function (the antiderivative of f(x))  of \+
x and C.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F := (x,C) ->
 ln(exp(x) - exp(-x)) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*
6$%\"xG%\"CG6\"6$%)operatorG%&arrowGF),&-%#lnG6#,&-%$expG6#9$\"\"\"-F3
6#,$F5!\"\"F:F69%F6F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "and \+
plot this function over a range of x values for different values of C \+
by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([F(x,-1),F(x,0)
,F(x,1)],x=0..1,color=[red,green,blue]);" }}{PARA 13 "" 1 "" 
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****z[D9v$F]p$!3a)z(3#HU&=6F-7$$\"3')*****>c$GwSF]p$!3fw;z!3(HJ5F-7$$
\"3?+++7XM*Q%F]p$!3ib80\"3x$H&*F]p7$$\"3'******p%QjtYF]p$!3)>0o][U%f))
F]p7$$\"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$$\"3d*****\\2Mg#fF]p$!35
HGSur4oiF]p7$$\"3M+++sxa\\iF]p$!3;AK@2lZscF]p7$$\"3U+++;$4wb'F]p$!3K0A
'*e5GF^F]p7$$\"3[+++>#R!zoF]p$!3EBBR'yZ)yXF]p7$$\"3Y+++4A@urF]p$!3i\\6
%p*pu\"4%F]p7$$\"3%)*****f:'f#\\(F]p$!3C9e<G4N#e$F]p7$$\"3*)*****pf2L#
yF]p$!3e[C+*oQ\"pIF]p7$$\"3x*****z(G>6\")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$$\"3%******><F;O*F]p$!3.vXb9,GQ
&)FJ7$$\"3w*****f0A#*p*F]p$!3)z,4=$f&H)RFJ7$$\"3%******zgc-+\"F-$\"3#o
14:c;\"pL!#@7$$\"3,+++c]kK5F-$\"3#p')>v3/*[UFJ7$$\"3'******\\!Q*>1\"F-
$\"3A![9\"*ok!3!)FJ7$$\"35+++R(zS4\"F-$\"3XWLFSAr07F]p7$$\"36+++-,FC6F
-$\"33mOZ&4P8e\"F]p7$$\"3-+++Jx#e:\"F-$\"3o7[LBV$)o>F]p7$$\"3!******43
\"o'=\"F-$\"3*pvYW<wHM#F]p7$$\"3'*******z;)*=7F-$\"36a6HH'[+t#F]p7$$\"
35+++&*44]7F-$\"3M%R'*=U-()4$F]p7$$\"3++++jZ!>G\"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++++lJR09F-$\"3o)G7;rMw)[F]p
7$$\"3-+++-*zqV\"F-$\"3c_*H>nIPC&F]p7$$\"31+++`\"3uY\"F-$\"3e&\\+2sq@e
&F]p7$$\"3++++++++:F-$\"34$)HdqFWVfF]p-%'COLOURG6&%$RGBG$\"#5!\"\"$\"
\"!Fj^lFi^l-%+AXESLABELSG6$Q\"x6\"Q!F__l-%%VIEWG6$;Fi^l$\"#:Fh^l%(DEFA
ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "It sure looks like this goe
s through the point (1,0).  How can you be sure? Evaluate F at x = 1 a
nd C = C1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F(1,C1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&-%$expG6#\"\"\"F+-F)6#!\"
\"F.F+$\"+?a'ea)!#5F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Is this \+
zero?  Have Maple evaluate this as a floating point number" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(F(1,C1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Bingo
. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 10 "Summary:  \+
" }}{PARA 0 "" 0 "" {TEXT -1 91 "1) Use the \"int(function, variable)
\" to evaluate indefinite integrals (antidifferentiate). " }}{PARA 0 "
" 0 "" {TEXT -1 88 "2) If you want to see the integral in mathematical
 notation use \"Int(function,variable)\"" }}{PARA 0 "" 0 "" {TEXT -1 
78 "3) Remember, that the constant of integration is excluded from Map
le's answer." }}{PARA 0 "" 0 "" {TEXT -1 76 "4) You can solve for the \+
constant of integration using Maple if necessary.  " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this section" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 44 "2. Area Approximations, Summation Techniques" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 442 "In this section you will learn h
ow to use Maple to visualize the approximate area under a positive fun
ction using boxes (rectangles really).   There is another command to f
ind the summed area of these boxes (again rectangles).  This command w
ill 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 t
he \"student\" package which you must load in order to access." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "restart:    # this clears al
l variables" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with(student
):  # this loads the student library" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 39 "To approximate the area under the cuve " }{XPPEDIT 18 0 "f(x) =
 x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 102 " over the interval f
rom 1 to 3, several boxes can be used.   The following command draws t
he graph of " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#
" }{TEXT -1 41 " and six boxes using the left end points." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "leftbox(x^2, x = 1..3,6);" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%)POLYGONSG6$7&7$$\"
\"\"\"\"!$F*F*7$F(F(7$$\"+LLLL8!\"*F(7$F.F+-%&COLORG6&%$RGBG$\"\"(!\"
\"$\"\"*F8F6-F$6$7&F17$F.$\"+yxxx<F07$$\"+nmmm;F0F?7$FBF+F2-F$6$7&FD7$
FB$\"+yxxxFF07$$\"\"#F*FI7$FLF+F2-F$6$7&FN7$FL$\"\"%F*7$$\"+LLLLBF0FS7
$FVF+F2-F$6$7&FX7$FV$\"+WWWWaF07$$\"+nmmmEF0Fgn7$FjnF+F2-F$6$7&F\\o7$F
jn$\"+6666rF07$$\"\"$F*Fao7$FdoF+F2-%'CURVESG6&7SF,7$$\"3ALLL3VfV5!#<$
\"3EGdQ!3*3*3\"F^p7$$\"3smm\"H[D:3\"F^p$\"3CH\")>qtpp6F^p7$$\"3XLL$e0$
=C6F^p$\"3krF-Vvyj7F^p7$$\"3QLL$3RBr;\"F^p$\"3%yHI%4q<i8F^p7$$\"3imm\"
zjf)47F^p$\"3iZ)fMMgPY\"F^p7$$\"3WLLe4;[\\7F^p$\"3gdpo#H/7c\"F^p7$$\"3
-++Dmy]!H\"F^p$\"3+GJ&Gb5am\"F^p7$$\"3>LLezs$HL\"F^p$\"3)o,.8z@nx\"F^p
7$$\"31++D@1Bv8F^p$\"3?hOih#f7*=F^p7$$\"3pmmm@Xt=9F^p$\"3PX(oHk2G,#F^p
7$$\"3MLL$3y_qX\"F^p$\"3r\\T81G+B@F^p7$$\"3'******\\1!>+:F^p$\"3+/Z7J-
d]AF^p7$$\"3*******\\Z/Na\"F^p$\"3yD]MkgS#Q#F^p7$$\"35+++NfC&e\"F^p$\"
3kCSVuY+8DF^p7$$\"3LLLez6:B;F^p$\"3`mFy^(>Yj#F^p7$$\"3_mmm\"=C#o;F^p$
\"3->uH?>(Hy#F^p7$$\"3gmmmEpS1<F^p$\"3PyfP*fC=\"HF^p7$$\"3%)***\\i`A3v
\"F^p$\"33#)3W`&z`1$F^p7$$\"3Ymmmwy8!z\"F^p$\"3&4kwuh$f/KF^p7$$\"3/++D
OIFL=F^p$\"3!RHTa-!*3O$F^p7$$\"3!****\\(3zMu=F^p$\"3y(\\NI3!=8NF^p7$$
\"3emm;H_?<>F^p$\"3e5Su!*envOF^p7$$\"3mmm\"zihl&>F^p$\"3%p\"f$QSL\"GQF
^p7$$\"3KLL$3#G,**>F^p$\"3CSg&yD_g*RF^p7$$\"3<LLezw5V?F^p$\"3'=Rs.**)G
uTF^p7$$\"3o***\\PQ#\\\"3#F^p$\"37c#4Oa5EL%F^p7$$\"3BLL$e\"*[H7#F^p$\"
3CDzB*47p]%F^p7$$\"3#*******pvxl@F^p$\"3'[5:F[#f!p%F^p7$$\"3z****\\_qn
2AF^p$\"3u'3N\"oz$Q([F^p7$$\"3%)***\\i&p@[AF^p$\"3)Q,qB[zW0&F^p7$$\"3#
)****\\2'HKH#F^p$\"3W.YrK?!*e_F^p7$$\"3_mmmwanLBF^p$\"3c$R&RI7/YaF^p7$
$\"3u*****\\2goP#F^p$\"3G/!Hh\"QY\\cF^p7$$\"3CLLeR<*fT#F^p$\"3F+\\t&3;
q$eF^p7$$\"3'******\\)HxeCF^p$\"3I5ew\"fkb/'F^p7$$\"3Cmm\"H!o-*\\#F^p$
\"3y$*eph\\8XiF^p7$$\"3))***\\7k.6a#F^p$\"3U3Sd:x?dkF^p7$$\"3emmmT9C#e
#F^p$\"3/X21j3(zm'F^p7$$\"3\"****\\i!*3`i#F^p$\"3=rbB`oC#*oF^p7$$\"3;L
LL$*zymEF^p$\"3YDoQ,#e<6(F^p7$$\"30LL$3N1#4FF^p$\"3zkcR^!*zRtF^p7$$\"3
kmm\"HYt7v#F^p$\"3wm%\\xm0&pvF^p7$$\"3%*******p(G**y#F^p$\"3!Hrt;a-Py(
F^p7$$\"3Umm;9@BMGF^p$\"3M_O(pnrG.)F^p7$$\"3/LLL`v&Q(GF^p$\"35^5&oBd!f
#)F^p7$$\"30++DOl5;HF^p$\"3,s*f2LxO])F^p7$$\"3/++v.UacHF^p$\"3<@xsGO:T
()F^p7$Fdo$F:F*-%'COLOURG6&F5$\"*++++\"!\")F+F+-%&STYLEG6#%%LINEG-%*TH
ICKNESSG6#FM-%+AXESLABELSG6$Q\"x6\"Q!Fj_l-%%VIEWG6$;F(Fdo%(DEFAULTG" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 69 "To find the combined area of these boxes \+
we use the \"leftsum\" command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "leftsum(x^2,x=1..3,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
#\"\"\"\"\"$F&-%$SumG6$*$),&F&F&*&F'!\"\"%\"iGF&F&\"\"#F&/F0;\"\"!\"\"
&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+p.Pqt!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 114 "From the graph, you may assume that this value is smalle
r than the actual area under the curve.  Using rightboxes:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rightbox(x^2,x=1..3,6);" }}{PARA 
13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6&7S7$$\"\"
\"\"\"!F(7$$\"3ALLL3VfV5!#<$\"3EGdQ!3*3*3\"F.7$$\"3smm\"H[D:3\"F.$\"3C
H\")>qtpp6F.7$$\"3XLL$e0$=C6F.$\"3krF-Vvyj7F.7$$\"3QLL$3RBr;\"F.$\"3%y
HI%4q<i8F.7$$\"3imm\"zjf)47F.$\"3iZ)fMMgPY\"F.7$$\"3WLLe4;[\\7F.$\"3gd
po#H/7c\"F.7$$\"3-++Dmy]!H\"F.$\"3+GJ&Gb5am\"F.7$$\"3>LLezs$HL\"F.$\"3
)o,.8z@nx\"F.7$$\"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$$\"3_mmm\"=C#o;F.$\"3
->uH?>(Hy#F.7$$\"3gmmmEpS1<F.$\"3PyfP*fC=\"HF.7$$\"3%)***\\i`A3v\"F.$
\"33#)3W`&z`1$F.7$$\"3Ymmmwy8!z\"F.$\"3&4kwuh$f/KF.7$$\"3/++DOIFL=F.$
\"3!RHTa-!*3O$F.7$$\"3!****\\(3zMu=F.$\"3y(\\NI3!=8NF.7$$\"3emm;H_?<>F
.$\"3e5Su!*envOF.7$$\"3mmm\"zihl&>F.$\"3%p\"f$QSL\"GQF.7$$\"3KLL$3#G,*
*>F.$\"3CSg&yD_g*RF.7$$\"3<LLezw5V?F.$\"3'=Rs.**)GuTF.7$$\"3o***\\PQ#
\\\"3#F.$\"37c#4Oa5EL%F.7$$\"3BLL$e\"*[H7#F.$\"3CDzB*47p]%F.7$$\"3#***
****pvxl@F.$\"3'[5:F[#f!p%F.7$$\"3z****\\_qn2AF.$\"3u'3N\"oz$Q([F.7$$
\"3%)***\\i&p@[AF.$\"3)Q,qB[zW0&F.7$$\"3#)****\\2'HKH#F.$\"3W.YrK?!*e_
F.7$$\"3_mmmwanLBF.$\"3c$R&RI7/YaF.7$$\"3u*****\\2goP#F.$\"3G/!Hh\"QY
\\cF.7$$\"3CLLeR<*fT#F.$\"3F+\\t&3;q$eF.7$$\"3'******\\)HxeCF.$\"3I5ew
\"fkb/'F.7$$\"3Cmm\"H!o-*\\#F.$\"3y$*eph\\8XiF.7$$\"3))***\\7k.6a#F.$
\"3U3Sd:x?dkF.7$$\"3emmmT9C#e#F.$\"3/X21j3(zm'F.7$$\"3\"****\\i!*3`i#F
.$\"3=rbB`oC#*oF.7$$\"3;LLL$*zymEF.$\"3YDoQ,#e<6(F.7$$\"30LL$3N1#4FF.$
\"3zkcR^!*zRtF.7$$\"3kmm\"HYt7v#F.$\"3wm%\\xm0&pvF.7$$\"3%*******p(G**
y#F.$\"3!Hrt;a-Py(F.7$$\"3Umm;9@BMGF.$\"3M_O(pnrG.)F.7$$\"3/LLL`v&Q(GF
.$\"35^5&oBd!f#)F.7$$\"30++DOl5;HF.$\"3,s*f2LxO])F.7$$\"3/++v.UacHF.$
\"3<@xsGO:T()F.7$$\"\"$F*$\"\"*F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*
F][l-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"#-%)POLYGONSG6$7&7$F(F][l7$F(
$\"+yxxx<!\"*7$$\"+LLLL8F^\\lF\\\\l7$F`\\lF][l-%&COLORG6&Fiz$\"\"(!\"
\"$FezFh\\lFf\\l-Fg[l6$7&Fb\\l7$F`\\l$\"+yxxxFF^\\l7$$\"+nmmm;F^\\lF^]
l7$Fa]lF][lFc\\l-Fg[l6$7&Fc]l7$Fa]l$\"\"%F*7$$Fe[lF*Fh]l7$F[^lF][lFc\\
l-Fg[l6$7&F\\^l7$F[^l$\"+WWWWaF^\\l7$$\"+LLLLBF^\\lFa^l7$Fd^lF][lFc\\l
-Fg[l6$7&Ff^l7$Fd^l$\"+6666rF^\\l7$$\"+nmmmEF^\\lF[_l7$F^_lF][lFc\\l-F
g[l6$7&F`_l7$F^_lFdzFaz7$FbzF][lFc\\l-%+AXESLABELSG6$Q\"x6\"Q!Fj_l-%%V
IEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf(righ
tsum(x^2,x=1..3,6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/Pq.5!\")
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "and we may safely assume thi
s value is larger than the actual area under the curve.  Therefore, it
 is safe to assume that the actual area is between 7.37 and 10.04.    \+
" }}{PARA 0 "" 0 "" {TEXT -1 41 "Question: How do we find the exact ar
ea ?" }}{PARA 0 "" 0 "" {TEXT -1 94 "Answer: By taking the limit of ei
ther of the above as the number of boxes goes to infinity.   " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "First lets try 50 boxes evaluating
 f at the right-hand endpoints. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "rightbox(x^2, x=1..3,50);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6W-%'CURVESG6&7S7$$\"\"\"\"\"!F(7$$
\"3ALLL3VfV5!#<$\"3EGdQ!3*3*3\"F.7$$\"3smm\"H[D:3\"F.$\"3CH\")>qtpp6F.
7$$\"3XLL$e0$=C6F.$\"3krF-Vvyj7F.7$$\"3QLL$3RBr;\"F.$\"3%yHI%4q<i8F.7$
$\"3imm\"zjf)47F.$\"3iZ)fMMgPY\"F.7$$\"3WLLe4;[\\7F.$\"3gdpo#H/7c\"F.7
$$\"3-++Dmy]!H\"F.$\"3+GJ&Gb5am\"F.7$$\"3>LLezs$HL\"F.$\"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+8D
F.7$$\"3LLLez6:B;F.$\"3`mFy^(>Yj#F.7$$\"3_mmm\"=C#o;F.$\"3->uH?>(Hy#F.
7$$\"3gmmmEpS1<F.$\"3PyfP*fC=\"HF.7$$\"3%)***\\i`A3v\"F.$\"33#)3W`&z`1
$F.7$$\"3Ymmmwy8!z\"F.$\"3&4kwuh$f/KF.7$$\"3/++DOIFL=F.$\"3!RHTa-!*3O$
F.7$$\"3!****\\(3zMu=F.$\"3y(\\NI3!=8NF.7$$\"3emm;H_?<>F.$\"3e5Su!*env
OF.7$$\"3mmm\"zihl&>F.$\"3%p\"f$QSL\"GQF.7$$\"3KLL$3#G,**>F.$\"3CSg&yD
_g*RF.7$$\"3<LLezw5V?F.$\"3'=Rs.**)GuTF.7$$\"3o***\\PQ#\\\"3#F.$\"37c#
4Oa5EL%F.7$$\"3BLL$e\"*[H7#F.$\"3CDzB*47p]%F.7$$\"3#*******pvxl@F.$\"3
'[5:F[#f!p%F.7$$\"3z****\\_qn2AF.$\"3u'3N\"oz$Q([F.7$$\"3%)***\\i&p@[A
F.$\"3)Q,qB[zW0&F.7$$\"3#)****\\2'HKH#F.$\"3W.YrK?!*e_F.7$$\"3_mmmwanL
BF.$\"3c$R&RI7/YaF.7$$\"3u*****\\2goP#F.$\"3G/!Hh\"QY\\cF.7$$\"3CLLeR<
*fT#F.$\"3F+\\t&3;q$eF.7$$\"3'******\\)HxeCF.$\"3I5ew\"fkb/'F.7$$\"3Cm
m\"H!o-*\\#F.$\"3y$*eph\\8XiF.7$$\"3))***\\7k.6a#F.$\"3U3Sd:x?dkF.7$$
\"3emmmT9C#e#F.$\"3/X21j3(zm'F.7$$\"3\"****\\i!*3`i#F.$\"3=rbB`oC#*oF.
7$$\"3;LLL$*zymEF.$\"3YDoQ,#e<6(F.7$$\"30LL$3N1#4FF.$\"3zkcR^!*zRtF.7$
$\"3kmm\"HYt7v#F.$\"3wm%\\xm0&pvF.7$$\"3%*******p(G**y#F.$\"3!Hrt;a-Py
(F.7$$\"3Umm;9@BMGF.$\"3M_O(pnrG.)F.7$$\"3/LLL`v&Q(GF.$\"35^5&oBd!f#)F
.7$$\"30++DOl5;HF.$\"3,s*f2LxO])F.7$$\"3/++v.UacHF.$\"3<@xsGO:T()F.7$$
\"\"$F*$\"\"*F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F][l-%&STYLEG6#%%L
INEG-%*THICKNESSG6#\"\"#-%)POLYGONSG6$7&7$F(F][l7$F($\"+++g\"3\"!\"*7$
$\"++++S5F^\\lF\\\\l7$F`\\lF][l-%&COLORG6&Fiz$\"\"(!\"\"$FezFh\\lFf\\l
-Fg[l6$7&Fb\\l7$F`\\l$\"+++Sm6F^\\l7$$\"++++!3\"F^\\lF^]l7$Fa]lF][lFc
\\l-Fg[l6$7&Fc]l7$Fa]l$\"+++Sa7F^\\l7$$\"++++?6F^\\lFh]l7$F[^lF][lFc\\
l-Fg[l6$7&F]^l7$F[^l$\"+++gX8F^\\l7$$\"++++g6F^\\lFb^l7$Fe^lF][lFc\\l-
Fg[l6$7&Fg^l7$Fe^l$\"++++S9F^\\l7$$\"+++++7F^\\lF\\_l7$F__lF][lFc\\l-F
g[l6$7&Fa_l7$F__l$\"+++gP:F^\\l7$$\"++++S7F^\\lFf_l7$Fi_lF][lFc\\l-Fg[
l6$7&F[`l7$Fi_l$\"+++SQ;F^\\l7$$\"++++!G\"F^\\lF``l7$Fc`lF][lFc\\l-Fg[
l6$7&Fe`l7$Fc`l$\"+++SU<F^\\l7$$\"++++?8F^\\lFj`l7$F]alF][lFc\\l-Fg[l6
$7&F_al7$F]al$\"+++g\\=F^\\l7$$\"++++g8F^\\lFdal7$FgalF][lFc\\l-Fg[l6$
7&Fial7$Fgal$\"++++g>F^\\l7$$\"+++++9F^\\lF^bl7$FablF][lFc\\l-Fg[l6$7&
Fcbl7$Fabl$\"+++gt?F^\\l7$F\\_lFhbl7$F\\_lF][lFc\\l-Fg[l6$7&F[cl7$F\\_
l$\"+++S!>#F^\\l7$$\"++++![\"F^\\lF`cl7$FcclF][lFc\\l-Fg[l6$7&Fecl7$Fc
cl$\"+++S5BF^\\l7$$\"++++?:F^\\lFjcl7$F]dlF][lFc\\l-Fg[l6$7&F_dl7$F]dl
$\"+++gLCF^\\l7$$\"++++g:F^\\lFddl7$FgdlF][lFc\\l-Fg[l6$7&Fidl7$Fgdl$
\"++++gDF^\\l7$$\"+++++;F^\\lF^el7$FaelF][lFc\\l-Fg[l6$7&Fcel7$Fael$\"
+++g*o#F^\\l7$$\"++++S;F^\\lFhel7$F[flF][lFc\\l-Fg[l6$7&F]fl7$F[fl$\"+
++SAGF^\\l7$$\"++++!o\"F^\\lFbfl7$FeflF][lFc\\l-Fg[l6$7&Fgfl7$Fefl$\"+
++SeHF^\\l7$$\"++++?<F^\\lF\\gl7$F_glF][lFc\\l-Fg[l6$7&Fagl7$F_gl$\"++
+g(4$F^\\l7$$\"++++g<F^\\lFfgl7$FiglF][lFc\\l-Fg[l6$7&F[hl7$Figl$\"+++
+SKF^\\l7$$\"+++++=F^\\lF`hl7$FchlF][lFc\\l-Fg[l6$7&Fehl7$Fchl$\"+++g&
Q$F^\\l7$$\"++++S=F^\\lFjhl7$F]ilF][lFc\\l-Fg[l6$7&F_il7$F]il$\"+++SMN
F^\\l7$$\"++++!)=F^\\lFdil7$FgilF][lFc\\l-Fg[l6$7&Fiil7$Fgil$\"+++S'o$
F^\\l7$$\"++++?>F^\\lF^jl7$FajlF][lFc\\l-Fg[l6$7&Fcjl7$Fajl$\"+++gTQF^
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Fc[mF][lFc\\l-Fg[l6$7&Fd[m7$Fc[m$\"+++ghTF^\\l7$$\"++++S?F^\\lFi[m7$F
\\\\mF][lFc\\l-Fg[l6$7&F^\\m7$F\\\\m$\"+++SEVF^\\l7$$\"++++!3#F^\\lFc
\\m7$Ff\\mF][lFc\\l-Fg[l6$7&Fh\\m7$Ff\\m$\"+++S%\\%F^\\l7$$\"++++?@F^
\\lF]]m7$F`]mF][lFc\\l-Fg[l6$7&Fb]m7$F`]m$\"+++glYF^\\l7$$\"++++g@F^\\
lFg]m7$Fj]mF][lFc\\l-Fg[l6$7&F\\^m7$Fj]m$\"++++S[F^\\l7$$\"+++++AF^\\l
Fa^m7$Fd^mF][lFc\\l-Fg[l6$7&Ff^m7$Fd^m$\"+++g<]F^\\l7$$\"++++SAF^\\lF[
_m7$F^_mF][lFc\\l-Fg[l6$7&F`_m7$F^_m$\"+++S)>&F^\\l7$$\"++++!G#F^\\lFe
_m7$Fh_mF][lFc\\l-Fg[l6$7&Fj_m7$Fh_m$\"+++S#Q&F^\\l7$$\"++++?BF^\\lF_`
m7$Fb`mF][lFc\\l-Fg[l6$7&Fd`m7$Fb`m$\"+++gpbF^\\l7$$\"++++gBF^\\lFi`m7
$F\\amF][lFc\\l-Fg[l6$7&F^am7$F\\am$\"++++gdF^\\l7$$\"+++++CF^\\lFcam7
$FfamF][lFc\\l-Fg[l6$7&Fham7$Ffam$\"+++g`fF^\\l7$$\"++++SCF^\\lF]bm7$F
`bmF][lFc\\l-Fg[l6$7&Fbbm7$F`bm$\"+++S]hF^\\l7$$\"++++![#F^\\lFgbm7$Fj
bmF][lFc\\l-Fg[l6$7&F\\cm7$Fjbm$\"+++S]jF^\\l7$$\"++++?DF^\\lFacm7$Fdc
mF][lFc\\l-Fg[l6$7&Ffcm7$Fdcm$\"+++g`lF^\\l7$F^elF[dm7$F^elF][lFc\\l-F
g[l6$7&F^dm7$F^el$\"++++gnF^\\l7$$\"+++++EF^\\lFcdm7$FfdmF][lFc\\l-Fg[
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$7&Fbem7$F`em$\"+++S#=(F^\\l7$$\"++++!o#F^\\lFgem7$FjemF][lFc\\l-Fg[l6
$7&F\\fm7$Fjem$\"+++S)R(F^\\l7$$\"++++?FF^\\lFafm7$FdfmF][lFc\\l-Fg[l6
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&F`gm7$F^gm$\"++++SyF^\\l7$$\"+++++GF^\\lFegm7$FhgmF][lFc\\l-Fg[l6$7&F
jgm7$Fhgm$\"+++gl!)F^\\l7$$\"++++SGF^\\lF_hm7$FbhmF][lFc\\l-Fg[l6$7&Fd
hm7$Fbhm$\"+++S%H)F^\\l7$$\"++++!)GF^\\lFihm7$F\\imF][lFc\\l-Fg[l6$7&F
^im7$F\\im$\"+++SE&)F^\\l7$$\"++++?HF^\\lFcim7$FfimF][lFc\\l-Fg[l6$7&F
him7$Ffim$\"+++gh()F^\\l7$$\"++++gHF^\\lF]jm7$F`jmF][lFc\\l-Fg[l6$7&Fb
jm7$F`jmFdzFaz7$FbzF][lFc\\l-%+AXESLABELSG6$Q\"x6\"Q!F\\[n-%%VIEWG6$;F
(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7
" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Cur
ve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 2
0" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "
Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curv
e 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39
" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "C
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(rightsum(x^2,x=1..3,50
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++?F))!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 157 "It seems as though this is getting close
r to 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 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "boxarea := n -> rightsum(x^2, x=1..3,n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(boxareaGf*6#%\"nG6\"6$%)operatorG%&
arrowGF(-%)rightsumG6%*$)%\"xG\"\"#\"\"\"/F1;F3\"\"$9$F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(boxarea(100));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++!ou)!\"*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 98 "This should be closer still.  To get the limit as \"
n\" goes to infinity we use the \"limit\" command:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "limit(boxarea(n),n=infinity);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6##\"#E\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
81 "This is the exact area which we can check by the fundemental theor
em of calculus." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We can see tha
t our error in using 100 boxes is" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "evalf(boxarea(100) - Limit(boxarea(n),n=infinity));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")LL8!)!\"*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 262 "Not bad. Notice that using right boxes to approxima
te the area under an increasing function will always be greater than t
he exact area. If the function is decreasing, do you expect the approx
imation with right boxes to be larger or smaller than the exact area? \+
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "# end of this section
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Assignment" }}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 51 "Problem # 1: Antiddifferentia
tion, Integral Curves." }}{PARA 0 "" 0 "" {TEXT -1 68 "Generate a grap
h of at least three integral curves of the function  " }{XPPEDIT 18 0 
"f(x) = (1+x)*sin(x);" "6#/-%\"fG6#%\"xG*&,&\"\"\"F*F'F*F*-%$sinG6#F'F
*" }{TEXT -1 21 "  over the interval (" }{XPPEDIT 18 0 "-Pi/2,Pi/2;" "
6$,$*&%#PiG\"\"\"\"\"#!\"\"F(*&F%F&F'F(" }{TEXT -1 3 "). " }}{PARA 0 "
" 0 "" {TEXT -1 48 "Find the solution to the initial value problem: " 
}{TEXT -1 2 "  " }{XPPEDIT 18 0 "dy/dx = f(x);" "6#/*&%#dyG\"\"\"%#dxG
!\"\"-%\"fG6#%\"xG" }{TEXT -1 18 ",       y(0) = 3.2" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "
" }{TEXT 260 31 "Problem #2: Area Approximations" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "Let  " }{XPPEDIT 18 0 "f(x) = exp(-x^2);" 
"6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 91 "  .   Approx
imate the area under this curve defined by y = f(x) over the interval \+
[0,3] by " }}{PARA 0 "" 0 "" {TEXT -1 110 "a) using 20 rectangles with
 the height determined by f evaluated at the right hand endpoint of ea
ch interval. " }}{PARA 0 "" 0 "" {TEXT -1 104 "b) using 20 rectangles \+
with height determined by f evaluated at the left hand endpoint of eac
h interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 163 "c) Give an upper bound and a lower bound of the actual area un
der the curve based on the function being increasing or decreasing and
 the results from (a) and (b). " }}{PARA 0 "" 0 "" {TEXT -1 203 "d) Us
ing a limiting process to find the actual area under the curve. Does t
his lie within the bounds from part (c).  You will need to use the com
mand \"evalf\" on the limit answer to get an actual number. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }