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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#riemann" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#Duane A. Cooper, Dept. of Mathemat
ics, Maple Workshop, 10/8/02" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#Enter the rule of f and the values
 of a, b, and n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#
\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 5 "b:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"
#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=4;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"nG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "#Next we compute the left- and right-handed sums." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "lhsum:=sum((b-a)/n*f(a+(i-1)/n*(b-a)),i=1..n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&lhsumG#\"\"(\"\"%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "rhsum:=sum((b-a)/n*f(a+i/n*(b-a)),i=1..n)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rhsumG#\"#:\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#Now let's evaluate the actual inte
gral." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "int(f(x),x=a..b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\")\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "#Now let's get a visual look at our sums." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "w
ith(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "leftbox(f(
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\\lF[\\l7$F_\\lF(-%&COLORG6&F[[l$\"\"(!\"\"$\"\"*Fg\\lFe\\l-Fg[l6$7&Fa
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!\"*7$$\"+++++:Fh]lFf]l7$Fj]lF(Fb\\l-Fg[l6$7&F\\^l7$Fj]lFfzFcz7$FdzF(F
b\\l-%+AXESLABELSG6$Q\"x6\"Q!Ff^l-%%VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C
urve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
64 "#Of course, in the midst of discovering the leftbox and rightbox" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "#commands, I also found t
he leftsum and rightsum commands, below," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "#which would have saved the need to define those left
- and right-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "#handed sum
s above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "leftsum(f(x),x=a..b,n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%$SumG6$,$*&\"\"%!\"\"%\"
iGF'F&/F/;\"\"!\"\"$F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"(\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"++++]<!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "rightsum(f(x),x=a..b,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
#\"\"\"\"\"#F&-%$SumG6$,$*&\"\"%!\"\"%\"iGF'F&/F/;F&F-F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"#:\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 
"evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]P!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "#The end." }}}}{MARK "34" 0 }
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