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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 24 "Integration the Easy Way" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 258 55 "Lab #1
: Project-based Learning Community (Calculus III)" }}{PARA 0 "" 0 "" 
{TEXT 259 24 "Date:  September 3, 2004" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 91 "The goals for this la
b are to do a little Maple work with integrals, proper and improper.  \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 226 "Just
 as in the first lab, there is a little background, with useful Maple \+
commands, in the next section.  The final section contains exercises t
hat you (and your partner(s) should work through.  Begin with two usef
ul commands:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{PARA 0 "" 
0 "" {TEXT -1 97 "If you start to get into trouble and nothing is maki
ing sense, come back here and re-execute the " }{TEXT 0 7 "restart" }
{TEXT -1 9 "; command" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Toolbox
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "There
 are two basic commands that you can use to evaluate integrals: " }
{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 6 "  and " }{HYPERLNK 17 "Int
" 2 "Int" "" }{TEXT -1 122 ".  The first actually evaluates an integra
l (definite or indefinite!).  The second just sets it up for later eva
luation.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "I1a := int(x^3
,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$I1aG,$*$)%\"xG\"\"%\"\"\"#F
*F)" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Notice that Maple does not add th
e arbitrary constant.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "I1b:= Int(x^3,x); " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$I1bG-%$IntG6$*$)%\"xG\"\"$\"\"\"F*" }}}{PARA 0 
"" 0 "" {TEXT -1 81 "This just \"set up\" the integral.  To actually c
ompute it, just ask for the value:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "value(I1b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%
\"xG\"\"%\"\"\"#F(F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 77 "If you want to do definite integrals, just specify the
 limits of integration:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "i
nt(x^3,x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "
" 0 "" {TEXT -1 83 "You can get nice formulas for reports by combining
 the two functions in one line.  " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "Int(x^3,x=0..2) = int(x^3,x=0..2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%$IntG6$*$)%\"xG\"\"$\"\"\"/F);\"\"!\"\"#\"\"%" }}}
{PARA 0 "" 0 "" {TEXT -1 54 "Finally, Maple has no trouble with improp
er integrals:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(1/x^(1/
3),x=0..2) = int(1/x^(1/3),x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%$IntG6$*&\"\"\"F(*$)%\"xG#F(\"\"$F(!\"\"/F+;\"\"!\"\"#,$*$)F2#F2F-
F(#F-F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "Int(x^(-2),x=1..infinity) = int(x^(-2),x=1..infinity)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)%\"xG\"\"#
F(!\"\"/F+;F(%)infinityGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "Int(x^(-2),x=0..2) = int(x^(-2),x=0..2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)%\"xG\"\"#F(!\"\"/F+;\"\"!F,%)in
finityG" }}}{PARA 0 "" 0 "" {TEXT -1 139 "If you want to actually see \+
how the integral is growing, you can do the truncation and then take a
 limit to get the value for the integral:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "Truncate1 := int(x^(-2),x=1..N);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*Truncate1G*&,&%\"NG\"\"\"F(!\"\"F(F'F)" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "TrueV
alue := limit(Truncate1,N=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%*TrueValueG\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 150 "One of the best ways to use Maple is to get some gu
idance in how to evaluate an integral by hand.  One of your homework p
roblems asked you to evaluate" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "Int(x^(1/3)/(x^(2/3)-9),x=0..27);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$IntG6$*&%\"xG#\"\"\"\"\"$,&*$)F'#\"\"#F*F)F)\"\"*!\"\"F1/F';
\"\"!\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 103 "The solution is a two-step
 substitution and a little tricky.  Maple can lead you to the right so
lution:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(x^(1/3)/(x^(2
/3)-9),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG#\"\"#\"\"$\"
\"\"#F)F(*&#\"#FF(F*-%#lnG6#,&F$F*\"\"*!\"\"F*F*" }}}{PARA 0 "" 0 "" 
{TEXT -1 122 "Study the solution formula.  The second term is a natura
l log;  it is telling you that part of the integral must really be" }}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"uG!\"\"F(
" }}}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"uG,&*$)%\"xG#\"\"#\"\"$\"\"\"F,\"\"*!\"\"" }}}
{PARA 0 "" 0 "" {TEXT -1 120 "This should lead you to try this substit
ution as a first step in evaluating the original integral... and it do
es work.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" {TEXT -1 128 "You and a par
tner should produce one report describing your solutions to the follow
ing problems.  The report is due on Wednesday" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "September 8 at 3:00pm.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 256 14 "Exercise #1:  " }}{PARA 0 "" 0 "" 
{TEXT -1 120 "You just handed in some homework in which you evaluated \+
improper integrals.  Now do the following integrals using Maple:" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 260 43 " Section 9.3:  26, 28;  \+
Section 9.4: 9, 18;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 257 12 "Exercise #2:" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 208 "You can make Maple do some strange things when evaluatin
g roots of negative numbers.  Here is one example in which you can do \+
the integral yourself pretty easily but Maple gives a completely diffe
rent answer:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(x^(-1/3)
,x=-1..2) = int(x^(-1/3),x=-1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%$IntG6$*&\"\"\"F(*$)%\"xG#F(\"\"$F(!\"\"/F+;F.\"\"#,(*$)F1#F1F-F(#F
-F1#F-\"\"%F(*&^##!\"$F8F(-%%sqrtG6#F-F(F(" }}}{PARA 0 "" 0 "" {TEXT 
-1 212 "Take a pencil and paper and recompute the  integral yourself. \+
   Your answer should not match Maples, but you are (probably) both ri
ght.  Explain what the heck is going on! (In Maple, I is the square ro
ot of -1.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
261 12 "Exercise #3:" }}{PARA 0 "" 0 "" {TEXT -1 50 "It is well known \+
to all Calculus III students that" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"/F);F(%)infinityGF-" }}}{PARA 0 "
" 0 "" {TEXT -1 30 " What value of N will give you" }}{EXCHG {PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"/F);F(%\"NG\"%+5
" }}}{PARA 0 "" 0 "" {TEXT -1 42 "(Hint:  You can use solve or fsolve \+
here.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "8" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }