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{SECT 0 {PARA 258 "" 0 "" {TEXT -1 12 "RIEMANN SUMS" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT -1 19 "Calculus I Project " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 256 11 "Objectives:" }}
{PARA 0 "" 0 "" {TEXT -1 43 "To calculate Riemann sums for a function.
  " }}{PARA 0 "" 0 "" {TEXT -1 80 "To visualize the Riemann sums as a \+
series of rectangles, and evaluate the limit " }}{PARA 0 "" 0 "" 
{TEXT -1 56 "of the sums for  the definite integral for the function.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "This \+
project uses the student package in Maple V to define, evaluate and " 
}}{PARA 0 "" 0 "" {TEXT -1 82 "visualize the Riemann right and left su
ms, and compute algebraically the limit of " }}{PARA 0 "" 0 "" {TEXT 
-1 57 "the sums as the number of subdivisions goes to infinity. " }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 257 16 "Solved Example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 6 "Let f(" }{TEXT 270 1 "x" }{TEXT 269 4 ") = " }{XPPEDIT 
18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 268 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT 258 1 " " }{TEXT -1 34 "a)   Calculate the left sum for f(" }
{TEXT 271 1 "x" }{TEXT -1 34 ") on [1, 2] using 10 subintervals." }}
{PARA 0 "" 0 "" {TEXT -1 68 " b)   Construct the rectangles that repre
sents left sum  for n = 10." }}{PARA 0 "" 0 "" {TEXT -1 36 " c)   Calc
ulate the right sum for f(" }{TEXT 272 1 "x" }{TEXT -1 36 ") on [1, 2]
 using 10  subintervals. " }}{PARA 0 "" 0 "" {TEXT -1 71 " d)   Constr
uct the rectangles that represents right sum  for n = 10.  " }}{PARA 
0 "" 0 "" {TEXT -1 35 " e)   Calculate the left sum for f(" }{TEXT 
273 1 "x" }{TEXT -1 33 ") on [1, 2] using n subintervals." }}{PARA 0 "
" 0 "" {TEXT -1 52 " f)   Calculate the limit of  the left sum as n ->
  " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 36 " g)   Calculate the right sum for f(" }
{TEXT 274 1 "x" }{TEXT -1 37 ") on [1, 2] using n subintervals..   " }
}{PARA 0 "" 0 "" {TEXT -1 46 " h)   Calculate the limit of right sum a
s n ->" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 38 " i)  Using the above results evaluate " }
{XPPEDIT 18 0 "int(x^2,x = 1 .. 2);" "6#-%$intG6$*$%\"xG\"\"#/F';\"\"
\"F(" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 10 " Solution:" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 0 "" }{TEXT 275 17 " Define function." }}{PARA 0 "" 0 "" 
{TEXT -1 23 "Define the function f ." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "f:= x -> x^2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT 276 24 "a) Compute the left sum." }}
{PARA 0 "" 0 "" {TEXT -1 73 "Activate the student calculus package tha
t contains the estimates of the " }}{PARA 0 "" 0 "" {TEXT -1 11 "integ
ral.. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(student);\n" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#7@%\"DG%%DiffG%*DoubleintG%$IntG%&Li
mitG%(LineintG%(ProductG%$SumG%*TripleintG%*changevarG%/completesquare
G%)distanceG%'equateG%*integrandG%*interceptG%)intpartsG%(leftboxG%(le
ftsumG%)makeprocG%*middleboxG%*middlesumG%)midpointG%(powsubsG%)rightb
oxG%)rightsumG%,showtangentG%(simpsonG%&slopeG%(summandG%*trapezoidG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "leftsum(f(x), x = 1..2, 1
0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#5F&-%$SumG6$*$)
,&F&F&*&F'!\"\"%\"iGF&F&\"\"#F&/F0;\"\"!\"\"*F&F&" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6##\"$P%\"$+#" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 277 
29 "b) Graph the left rectangles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 278 2 "  " }{TEXT -1 51 "Represent the left Riemann Sum by left \+
rectangles. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "leftbox(f(x)
, x = 1..2, 10);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 286 168 168 
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0" "Curve 11" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 279 
25 "c) Compute the right sum." }}{PARA 0 "" 0 "" {TEXT -1 39 " Define \+
the right Riemann Sum for f(x)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "rightsum(f(x), x = 1..2,10);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&#\"\"\"\"#5F&-%$SumG6$*$),&F&F&*&F'!\"\"%\"iGF&F&\"\"#F&/F0;
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1 0 10 "value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$(\\\"$+#" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 
"" }{TEXT 280 31 "d)  Graph the right rectangles." }}{PARA 0 "" 0 "" 
{TEXT -1 39 "Represent the right sum by right boxes." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 " rightbox(f(x), x = 1..2,10);\n" }}{PARA 
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{PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 281 62 " e) Define and evaluate t
he left sum for n equal subintervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "leftsum(f(x), x = 1..2, \+
n);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"nG!\"\"-%$SumG6$*$),&\"
\"\"F,*&%\"iGF,F$F%F,\"\"#F,/F.;\"\"!,&F$F,F,F%F," }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&%\"nG!\"\",(*(\"\"(\"\"\"\"\"$F%F$F)F)#F*\"\"#F%*&F)F)*&\"\"'F)F$
F)F%F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(%);\n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"'!\"\",(*&\"#9\"\"\")%\"nG
\"\"#F*F**&\"\"*F*F,F*F&F*F*F*F,!\"#F*" }}}}{PARA 0 "" 0 "" {TEXT 259 
0 "" }{TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 
282 58 "f) Compute the limit of left sum as n approaches infinity." }}
{PARA 0 "" 0 "" {TEXT -1 79 "Note how the limit of the left sum is cal
culated as n approaches to infinity.  " }}{PARA 0 "" 0 "" {TEXT -1 70 
"Maple uses the limit command and as is done in the earlier labs takes
 " }}{PARA 0 "" 0 "" {TEXT -1 74 "n = infinity.  If you are calculatin
g the limit manually you should write " }}{PARA 0 "" 0 "" {TEXT -1 77 
"n -> infinity since infinity is not a number.  It is another way of s
aying n " }}{PARA 0 "" 0 "" {TEXT -1 23 "increases indefinitely." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "limit(%, n = infinity);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"
(\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 69 "This means that as n increases
 indefinitely, the left sum approaches " }{XPPEDIT 18 0 "7/3;" "6#*&\"
\"(\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 0 "" }{TEXT 283 57 "g) Define and evaluate the right sum with n sub
intervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rightsum(f(x),
 x = 1..2, n);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"nG!\"\"-%$Sum
G6$*$),&\"\"\"F,*&%\"iGF,F$F%F,\"\"#F,/F.;F,F$F," }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*&%\"nG!\"\",.F$\"\"\"*&F$F%,&F$F'F'F'\"\"#F'*&F)F'F$F%F%*(\"\"$F%F$
!\"#F)F-F'*(F*F%F$F.F)F*F%*(\"\"'F%F$F.F)F'F'F'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "simplify(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*(\"\"'!\"\"%\"nG!\"#,(*&\"#9\"\"\")F'\"\"#F,F,*&\"\"*F,F'F,F,
F,F,F,F," }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 284 63 "h) \+
Compute the limit of the right sum as n approaches infinity." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "\nlimit(%, n = infinity);\n
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"(\"\"$" }}}{PARA 0 "" 0 "" 
{TEXT -1 69 "This means that as n increases indefinitely the right sum
 approaches " }{XPPEDIT 18 0 "7/3;" "6#*&\"\"(\"\"\"\"\"$!\"\"" }
{TEXT -1 2 " ." }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 1 " " }{TEXT 285 13 "i) Conclusion" }}{PARA 0 "" 0 "" 
{TEXT -1 72 " This means that as n increases indefinitely, the right  \+
sum approaches " }{XPPEDIT 18 0 "7/3;" "6#*&\"\"(\"\"\"\"\"$!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Since both the left and
 right  limits approach " }{XPPEDIT 18 0 "7/3;" "6#*&\"\"(\"\"\"\"\"$!
\"\"" }{TEXT -1 17 "  as n increases " }}{PARA 0 "" 0 "" {TEXT -1 31 "
indefinitely, we conclude that " }{XPPEDIT 18 0 "int(x^2,x = 1 .. 2);
" "6#-%$intG6$*$%\"xG\"\"#/F';\"\"\"F(" }{TEXT -1 5 "  is " }{XPPEDIT 
18 0 "7/3" "6#*&\"\"(\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 49 "Also using Maple V we can evaluate the instegral." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int(x^2, x = 1..2);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"(\"\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 259 "" 1 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 67 "_________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 260 8 "Problem:" }}
{PARA 0 "" 0 "" {TEXT -1 8 "  Let f(" }{TEXT 261 1 "x" }{TEXT -1 6 ") \+
=   " }{XPPEDIT 18 0 "(6+7*x-x^3)/4;" "6#*&,(\"\"'\"\"\"*&\"\"(F&%\"xG
F&F&*$F)\"\"$!\"\"F&\"\"%F," }{TEXT -1 5 " .   " }}{PARA 0 "" 0 "" 
{TEXT -1 41 "      (a)   Calculate the left sum for f(" }{TEXT 262 1 "
x" }{TEXT -1 35 ") on [-1, 1] using 20 subintervals." }}{PARA 0 "" 0 "
" {TEXT -1 74 "      (b)   Construct the rectangles that represents le
ft sum  for n = 20." }}{PARA 0 "" 0 "" {TEXT -1 42 "      (c)   Calcul
ate the right sum for f(" }{TEXT 263 1 "x" }{TEXT -1 37 ") on [-1, 1] \+
using 20  subintervals. " }}{PARA 0 "" 0 "" {TEXT -1 77 "      (d)   C
onstruct the rectangles that represents right sum  for n = 20.  " }}
{PARA 0 "" 0 "" {TEXT -1 41 "      (e)   Calculate the left sum for f(
" }{TEXT 264 1 "x" }{TEXT -1 34 ") on [-1, 1] using n subintervals." }
}{PARA 0 "" 0 "" {TEXT -1 68 "      (f)   Calculate the limit of  the \+
left sum as n ->  infinity. " }}{PARA 0 "" 0 "" {TEXT -1 42 "      (g)
   Calculate the right sum for f(" }{TEXT 265 1 "x" }{TEXT -1 37 ") on
 [-1, 1] using n subintervals.   " }}{PARA 0 "" 0 "" {TEXT -1 66 "    \+
  (h)   Calculate the limit of the right sum as n -> infinity." }}
{PARA 0 "" 0 "" {TEXT -1 85 "      (i)    Using the above results what
 is the value of the definite integral of f(" }{TEXT 266 1 "x" }{TEXT 
-1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 29 "             with respect to \+
" }{TEXT 267 1 "x" }{TEXT -1 14 " from -1 to 1." }}}{PARA 0 "" 0 "" 
{TEXT -1 66 "_________________________________________________________
_________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
99 "MSIP Grant #P120A80089-98:  \"Three Urban Calculus Reform programs
: Adopting the Best\" 1998-2001    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{MARK "0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }