{VERSION 6 0 "IBM INTEL NT" "6.0" }
{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 }
{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 36 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 24 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 24 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 261 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
262 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 
0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }
{CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }
{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
282 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
287 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 291 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 14 17 
0 24 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
297 "" 1 14 116 134 184 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 1 
14 237 63 184 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 14 237 63 
184 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 1 14 237 63 184 0 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 1 14 0 0 15 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "" -1 302 "" 1 14 0 0 15 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" 
-1 303 "" 1 14 0 0 15 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 1 
14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 1 14 0 0 15 0 0 
0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 308 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 
14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 
-1 313 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 1 14 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 320 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
321 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 336 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "" 1 14 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 1 14 172 134 1 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "" -1 339 "" 1 14 172 134 1 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE 
"" -1 340 "" 1 14 172 134 1 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 341 "
" 1 14 172 134 1 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti
mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 
2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 
256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 
0 0 0 0 3 4 3 4 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T
imes" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 8 "MATH 149" }{TEXT 258 0 "
" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 259 23 "LABORATORY ASSIGNME
NT 7" }}{PARA 258 "" 0 "" {TEXT 261 21 "THE DEFINITE INTEGRAL" }}
{PARA 258 "" 0 "" {TEXT 260 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
262 0 "" }{TEXT 263 22 "In class we defined   " }{XPPEDIT 18 0 "Int(f(
x),x = a .. b);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 3 
"   " }{TEXT 264 2 "as" }{TEXT -1 3 "   " }{TEXT 265 3 "Lim" }
{XPPEDIT 18 0 "Sum(f(c[k])*(x[k+1]-x[k]),k = 0 .. n-1);" "6#-%$SumG6$*
&-%\"fG6#&%\"cG6#%\"kG\"\"\",&&%\"xG6#,&F-F.F.F.F.&F16#F-!\"\"F./F-;\"
\"!,&%\"nGF.F.F6" }{TEXT -1 0 "" }{TEXT 266 12 " , where  a=" }{TEXT 
-1 0 "" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 267 1 "<" }
{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 268 1 "
<" }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 0 "" }{TEXT 269 
5 "<...<" }{XPPEDIT 18 0 "x[n-1];" "6#&%\"xG6#,&%\"nG\"\"\"F(!\"\"" }
{TEXT -1 0 "" }{TEXT 270 1 "<" }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG
" }{TEXT -1 0 "" }{TEXT 271 35 "=b is a partition of [ a , b ] ,   " }
{XPPEDIT 18 0 "x[k];" "6#&%\"xG6#%\"kG" }{TEXT -1 0 "" }{TEXT 272 0 "
" }{TEXT 273 1 "<" }{XPPEDIT 18 0 "c[k];" "6#&%\"cG6#%\"kG" }{TEXT -1 
0 "" }{TEXT 274 0 "" }{TEXT 275 1 "<" }{XPPEDIT 18 0 "x[k+1];" "6#&%\"
xG6#,&%\"kG\"\"\"F(F(" }{TEXT -1 0 "" }{TEXT 276 28 ", and the limit i
s taken as " }{XPPEDIT 18 0 "proc (n) options operator, arrow; infinit
y end;" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }
{TEXT -1 1 " " }{TEXT 279 83 "and the lengths of the subintervals dete
rmined by the partition go to 0. If  f (x) " }{TEXT 280 1 ">" }{TEXT 
281 91 " 0 for all x in [ a , b ], then the integral represents the ar
ea of the region bounded by t" }{TEXT 277 0 "" }{TEXT 278 0 "" }{TEXT 
282 0 "" }{TEXT 283 74 "he curve  y = f (x) , the X-axis, and the line
s x = a and x = b. The sum  " }{XPPEDIT 18 0 "Sum(f(c[k])*(x[k+1]-x[k]
),k = 0 .. n-1)" "6#-%$SumG6$*&-%\"fG6#&%\"cG6#%\"kG\"\"\",&&%\"xG6#,&
F-F.F.F.F.&F16#F-!\"\"F./F-;\"\"!,&%\"nGF.F.F6" }{TEXT -1 2 "  " }
{TEXT 284 58 "represents the total area of the n rectangles with base \+
[ " }{XPPEDIT 18 0 "x[k];" "6#&%\"xG6#%\"kG" }{TEXT -1 0 "" }{TEXT 
285 3 " , " }{XPPEDIT 18 0 "x[k+1];" "6#&%\"xG6#,&%\"kG\"\"\"F(F(" }
{TEXT -1 1 " " }{TEXT 286 21 "] and height      f( " }{XPPEDIT 18 0 "c
[k];" "6#&%\"cG6#%\"kG" }{TEXT -1 1 " " }{TEXT 287 13 "), k=0...n-1." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 288 171 "There are commands in MAPLE's \"student\" package that \+
allow us to easily draw the approximating rectangles and calculate the
 corresponding sums for partitions in which the " }{XPPEDIT 18 0 "x[k]
;" "6#&%\"xG6#%\"kG" }{TEXT -1 0 "" }{TEXT 289 29 "'s are evenly space
d and the " }{XPPEDIT 18 0 "c[k];" "6#&%\"cG6#%\"kG" }{TEXT -1 0 "" }
{TEXT 290 96 "'s are either the left hand endpoints, the right endpoin
ts, or the midpoints of the intervals [ " }{XPPEDIT 18 0 "x[k];" "6#&%
\"xG6#%\"kG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x[k+1];" "6#&%\"xG6#,&%
\"kG\"\"\"F(F(" }{TEXT -1 0 "" }{TEXT 320 32 " ] determined by the par
tition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 291 58 "As an example, we will let f(x) =
 sin x, a = 0, and b = 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "A:=plot(sin,0..2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "B:=line([2,0],[2,sin(2)],color=red):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "C:=textplot([1.2,0.4,`REGION R`],co
lor=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "E:=textplot([0
.7,0.8,`y=sin x`]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "disp
lay(\{A,B,C,E\});" }}{PARA 13 "" 1 "" {GLPLOT2D 354 265 265 {PLOTDATA 
2 "6(-%'CURVESG6$7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3!)y==T,0eVF-7$$\"
3'pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLLLe0$=C\"!#=$\"3I#\\^x@T'Q7F87$$
\"3ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm;zjf)4#F8$\"3ctIO$>EK3#F87$$\"3=
LL$e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p****\\i'y]!HF8$\"3)z'**Hz%)QkGF87$$
\"3,LL$ezs$HLF8$\"3%f8@S</#oKF87$$\"3_****\\7iI_PF8$\"3+/tuT6([m$F87$$
\"3#pmmm@Xt=%F8$\"3uej'fKYg1%F87$$\"3QLLL3y_qXF8$\"3tO*fg>`IT%F87$$\"3
i******\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3()******\\Z/NaF8$\"3Uh8%*pkQr^F8
7$$\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF87$$\"3ELL$ez6:B'F8$\"3%o#4^'[pf
$eF87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F87$$\"3-mmmm#pS1(F8$\"3wN76%>Z5
\\'F87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C#oF87$$\"3slmmm(y8!zF8$\"3NBP6
&4.X5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S(F87$$\"39++](3zMu)F8$\"3;afM1N
DrwF87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF87$$\"3emm;zihl&*F8$\"3O`uvYv9s
\")F87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%)F87$$\"3<LLezw5V5!#<$\"3-2#QmI
K(R')F87$$\"3!****\\PQ#\\\"3\"Fas$\"3$[OLEF-m#))F87$$\"3BLL$e\"*[H7\"F
as$\"3#ylv\\<8Q,*F87$$\"3#*******pvxl6Fas$\"3-M-vk+&4>*F87$$\"3z****\\
_qn27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p@[7Fas$\"3uZ3fv(3U[*F87$$\"3#)
****\\2'HKH\"Fas$\"3<z/\"QV]sh*F87$$\"3_mmmwanL8Fas$\"3e9nS%H$=?(*F87$
$\"3'******\\2goP\"Fas$\"39AEdKB`7)*F87$$\"3CLLeR<*fT\"Fas$\"3atab(z;/
))*F87$$\"3'******\\)Hxe9Fas$\"3gCG+X%>t$**F87$$\"3Ymm\"H!o-*\\\"Fas$
\"3qbf1InDu**F87$$\"3))***\\7k.6a\"Fas$\"3_Q!)*f/#f&***F87$$\"3emmmT9C
#e\"Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i!*3`i\"Fas$\"3Iw,Vsb9&)**F87$$
\"3QLLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3GLL$3N1#4<Fas$\"3i:isYhO/**F87
$$\"3kmm\"HYt7v\"Fas$\"3yAkC$f\"eP)*F87$$\"3%*******p(G**y\"Fas$\"3kWG
U'3k3w*F87$$\"3lmm;9@BM=Fas$\"3;&yP#=+,b'*F87$$\"3ELLL`v&Q(=Fas$\"3Q44
i*ftUa*F87$$\"30++DOl5;>Fas$\"3jyU@b\\q4%*F87$$\"3/++v.Uac>Fas$\"37dI(
[,t^E*F87$$\"\"#F)$\"35<oDoU(H4*F8-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(
-F$6$7$7$FczF(7$Fcz$\"+oU(H4*!#5Fgz-%%TEXTG6%7$$\"#7!\"\"$\"\"%F\\\\l%
)REGION~RGFgz-Fg[l6$7$$\"\"(F\\\\l$\"\")F\\\\l%(y=sin~xG-%+AXESLABELSG
6$Q!6\"F[]l-%%VIEWG6$;F(Fcz%(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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
292 126 "We will initially approximate the area of the region using 5 \+
rectangles; the partition points will be equally spaced, and the " }
{XPPEDIT 18 0 "c[k];" "6#&%\"cG6#%\"kG" }{TEXT -1 0 "" }{TEXT 315 113 
"'s will be the midpoints of the subintervals; thus the partition poin
ts are 0, 0.4, 0.8, 1.2, 1.6 and 2, and the " }{XPPEDIT 18 0 "c[k];" "
6#&%\"cG6#%\"kG" }{TEXT -1 0 "" }{TEXT 321 34 "'s are 0.2, 0.6, 1.0, 1
.4 and 1.8." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "middlebox(sin(x),x=0.
.2,5);" }}{PARA 13 "" 1 "" {GLPLOT2D 328 235 235 {PLOTDATA 2 "6*-%)POL
YGONSG6$7&7$$\"\"!F)F(7$F($\"+3Lp')>!#57$$\"+++++SF-F+7$F/F(-%&COLORG6
&%$RGBG$\"\"(!\"\"$\"\"*F8F6-F$6$7&F17$F/$\"+MZUYcF-7$$\"+++++!)F-F?7$
FBF(F2-F$6$7&FD7$FB$\"+[)4ZT)F-7$$\"+++++7!\"*FI7$FLF(F2-F$6$7&FO7$FL$
\"++t\\a)*F-7$$\"+++++;FNFT7$FWF(F2-F$6$7&FY7$FW$\"+4jZQ(*F-7$$\"\"#F)
Fhn7$F[oF(F2-%'CURVESG6&7SF'7$$\"39LLLL3VfV!#>$\"3!)y==T,0eVFeo7$$\"3'
pmm;H[D:)Feo$\"3;VE!pV?N9)Feo7$$\"3LLLLe0$=C\"!#=$\"3I#\\^x@T'Q7F`p7$$
\"3ILLL3RBr;F`p$\"3=:p?3^Yj;F`p7$$\"3Ymm;zjf)4#F`p$\"3ctIO$>EK3#F`p7$$
\"3=LL$e4;[\\#F`p$\"3UFv?6l,pCF`p7$$\"3p****\\i'y]!HF`p$\"3)z'**Hz%)Qk
GF`p7$$\"3,LL$ezs$HLF`p$\"3%f8@S</#oKF`p7$$\"3_****\\7iI_PF`p$\"3+/tuT
6([m$F`p7$$\"3#pmmm@Xt=%F`p$\"3uej'fKYg1%F`p7$$\"3QLLL3y_qXF`p$\"3tO*f
g>`IT%F`p7$$\"3i******\\1!>+&F`p$\"3M:FtwK#fz%F`p7$$\"3()******\\Z/NaF
`p$\"3Uh8%*pkQr^F`p7$$\"3'*******\\$fC&eF`p$\"3#)HQ5YS/CbF`p7$$\"3ELL$
ez6:B'F`p$\"3%o#4^'[pf$eF`p7$$\"3Smmm;=C#o'F`p$\"3ux&[t$3$f>'F`p7$$\"3
-mmmm#pS1(F`p$\"3wN76%>Z5\\'F`p7$$\"3]****\\i`A3vF`p$\"3\"f)3()*p.C#oF
`p7$$\"3slmmm(y8!zF`p$\"3NBP6&4.X5(F`p7$$\"3V++]i.tK$)F`p$\"3!>,MN2j8S
(F`p7$$\"39++](3zMu)F`p$\"3;afM1NDrwF`p7$$\"3#pmm;H_?<*F`p$\"3f%f\"=mQ
0RzF`p7$$\"3emm;zihl&*F`p$\"3O`uvYv9s\")F`p7$$\"39LLL3#G,***F`p$\"3)y9
#)p*>P4%)F`p7$$\"3<LLezw5V5!#<$\"3-2#QmIK(R')F`p7$$\"3!****\\PQ#\\\"3
\"F_w$\"3$[OLEF-m#))F`p7$$\"3BLL$e\"*[H7\"F_w$\"3#ylv\\<8Q,*F`p7$$\"3#
*******pvxl6F_w$\"3-M-vk+&4>*F`p7$$\"3z****\\_qn27F_w$\"3#yNZUKMzM*F`p
7$$\"3%)***\\i&p@[7F_w$\"3uZ3fv(3U[*F`p7$$\"3#)****\\2'HKH\"F_w$\"3<z/
\"QV]sh*F`p7$$\"3_mmmwanL8F_w$\"3e9nS%H$=?(*F`p7$$\"3'******\\2goP\"F_
w$\"39AEdKB`7)*F`p7$$\"3CLLeR<*fT\"F_w$\"3atab(z;/))*F`p7$$\"3'******
\\)Hxe9F_w$\"3gCG+X%>t$**F`p7$$\"3Ymm\"H!o-*\\\"F_w$\"3qbf1InDu**F`p7$
$\"3))***\\7k.6a\"F_w$\"3_Q!)*f/#f&***F`p7$$\"3emmmT9C#e\"F_w$\"3]Ds(Q
0X$****F`p7$$\"3\"****\\i!*3`i\"F_w$\"3Iw,Vsb9&)**F`p7$$\"3QLLL$*zym;F
_w$\"3q\"4lgOjR&**F`p7$$\"3GLL$3N1#4<F_w$\"3i:isYhO/**F`p7$$\"3kmm\"HY
t7v\"F_w$\"3yAkC$f\"eP)*F`p7$$\"3%*******p(G**y\"F_w$\"3kWGU'3k3w*F`p7
$$\"3lmm;9@BM=F_w$\"3;&yP#=+,b'*F`p7$$\"3ELLL`v&Q(=F_w$\"3Q44i*ftUa*F`
p7$$\"30++DOl5;>F_w$\"3jyU@b\\q4%*F`p7$$\"3/++v.Uac>F_w$\"37dI([,t^E*F
`p7$F[o$\"35<oDoU(H4*F`p-%'COLOURG6&F5$\"*++++\"!\")F(F(-%&STYLEG6#%%L
INEG-%*THICKNESSG6#F\\o-%+AXESLABELSG6$%\"xGQ!6\"-%%VIEWG6$;F(F[o%(DEF
AULTG" 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 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 41 "We now
 increase the number of rectangles:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "middlebox(sin(x),x=0..2,10);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 354 265 265 {PLOTDATA 2 "6/-%'CURVESG6&7S7$$\"\"!F)F(7$$\"39
LLLL3VfV!#>$\"3!)y==T,0eVF-7$$\"3'pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLL
Le0$=C\"!#=$\"3I#\\^x@T'Q7F87$$\"3ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm;
zjf)4#F8$\"3ctIO$>EK3#F87$$\"3=LL$e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p****
\\i'y]!HF8$\"3)z'**Hz%)QkGF87$$\"3,LL$ezs$HLF8$\"3%f8@S</#oKF87$$\"3_*
***\\7iI_PF8$\"3+/tuT6([m$F87$$\"3#pmmm@Xt=%F8$\"3uej'fKYg1%F87$$\"3QL
LL3y_qXF8$\"3tO*fg>`IT%F87$$\"3i******\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3(
)******\\Z/NaF8$\"3Uh8%*pkQr^F87$$\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF8
7$$\"3ELL$ez6:B'F8$\"3%o#4^'[pf$eF87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F8
7$$\"3-mmmm#pS1(F8$\"3wN76%>Z5\\'F87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C
#oF87$$\"3slmmm(y8!zF8$\"3NBP6&4.X5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S(
F87$$\"39++](3zMu)F8$\"3;afM1NDrwF87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF8
7$$\"3emm;zihl&*F8$\"3O`uvYv9s\")F87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%)
F87$$\"3<LLezw5V5!#<$\"3-2#QmIK(R')F87$$\"3!****\\PQ#\\\"3\"Fas$\"3$[O
LEF-m#))F87$$\"3BLL$e\"*[H7\"Fas$\"3#ylv\\<8Q,*F87$$\"3#*******pvxl6Fa
s$\"3-M-vk+&4>*F87$$\"3z****\\_qn27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p
@[7Fas$\"3uZ3fv(3U[*F87$$\"3#)****\\2'HKH\"Fas$\"3<z/\"QV]sh*F87$$\"3_
mmmwanL8Fas$\"3e9nS%H$=?(*F87$$\"3'******\\2goP\"Fas$\"39AEdKB`7)*F87$
$\"3CLLeR<*fT\"Fas$\"3atab(z;/))*F87$$\"3'******\\)Hxe9Fas$\"3gCG+X%>t
$**F87$$\"3Ymm\"H!o-*\\\"Fas$\"3qbf1InDu**F87$$\"3))***\\7k.6a\"Fas$\"
3_Q!)*f/#f&***F87$$\"3emmmT9C#e\"Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i!
*3`i\"Fas$\"3Iw,Vsb9&)**F87$$\"3QLLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3G
LL$3N1#4<Fas$\"3i:isYhO/**F87$$\"3kmm\"HYt7v\"Fas$\"3yAkC$f\"eP)*F87$$
\"3%*******p(G**y\"Fas$\"3kWGU'3k3w*F87$$\"3lmm;9@BM=Fas$\"3;&yP#=+,b'
*F87$$\"3ELLL`v&Q(=Fas$\"3Q44i*ftUa*F87$$\"30++DOl5;>Fas$\"3jyU@b\\q4%
*F87$$\"3/++v.Uac>Fas$\"37dI([,t^E*F87$$\"\"#F)$\"35<oDoU(H4*F8-%'COLO
URG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%*THICKNESSG6#Fdz-%)PO
LYGONSG6$7&F'7$F($\"+l;M$)**!#67$$\"+++++?!#5Fj[l7$F^\\lF(-%&COLORG6&F
jz$\"\"(!\"\"$\"\"*Fg\\lFe\\l-Ff[l6$7&Fa\\l7$F^\\l$\"+n??bHF`\\l7$$\"+
++++SF`\\lF^]l7$Fa]lF(Fb\\l-Ff[l6$7&Fc]l7$Fa]l$\"+'QbUz%F`\\l7$$\"++++
+gF`\\lFh]l7$F[^lF(Fb\\l-Ff[l6$7&F]^l7$F[^l$\"+so<UkF`\\l7$$\"+++++!)F
`\\lFb^l7$Fe^lF(Fb\\l-Ff[l6$7&Fg^l7$Fe^l$\"+'4pK$yF`\\l7$$\"\"\"F)F\\_
l7$F__lF(Fb\\l-Ff[l6$7&Fa_l7$F__l$\"+,O27*)F`\\l7$$\"+++++7!\"*Ff_l7$F
i_lF(Fb\\l-Ff[l6$7&F\\`l7$Fi_l$\"+a=eN'*F`\\l7$$\"+++++9F[`lFa`l7$Fd`l
F(Fb\\l-Ff[l6$7&Ff`l7$Fd`l$\"+m)\\\\(**F`\\l7$$\"+++++;F[`lF[al7$F^alF
(Fb\\l-Ff[l6$7&F`al7$F^al$\"+0\"[m\"**F`\\l7$$\"+++++=F[`lFeal7$FhalF(
Fb\\l-Ff[l6$7&Fjal7$Fhal$\"+x3+j%*F`\\l7$FczF_bl7$FczF(Fb\\l-%+AXESLAB
ELSG6$%\"xGQ!6\"-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" }}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 294 13 "10 RECT
ANGLES" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "middlebox(sin(x),
x=0..2,20);" }}{PARA 258 "" 0 "" {TEXT 295 0 "" }}{PARA 13 "" 1 "" 
{GLPLOT2D 260 259 259 {PLOTDATA 2 "69-%'CURVESG6&7S7$$\"\"!F)F(7$$\"39
LLLL3VfV!#>$\"3!)y==T,0eVF-7$$\"3'pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLL
Le0$=C\"!#=$\"3I#\\^x@T'Q7F87$$\"3ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm;
zjf)4#F8$\"3ctIO$>EK3#F87$$\"3=LL$e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p****
\\i'y]!HF8$\"3)z'**Hz%)QkGF87$$\"3,LL$ezs$HLF8$\"3%f8@S</#oKF87$$\"3_*
***\\7iI_PF8$\"3+/tuT6([m$F87$$\"3#pmmm@Xt=%F8$\"3uej'fKYg1%F87$$\"3QL
LL3y_qXF8$\"3tO*fg>`IT%F87$$\"3i******\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3(
)******\\Z/NaF8$\"3Uh8%*pkQr^F87$$\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF8
7$$\"3ELL$ez6:B'F8$\"3%o#4^'[pf$eF87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F8
7$$\"3-mmmm#pS1(F8$\"3wN76%>Z5\\'F87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C
#oF87$$\"3slmmm(y8!zF8$\"3NBP6&4.X5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S(
F87$$\"39++](3zMu)F8$\"3;afM1NDrwF87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF8
7$$\"3emm;zihl&*F8$\"3O`uvYv9s\")F87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%)
F87$$\"3<LLezw5V5!#<$\"3-2#QmIK(R')F87$$\"3!****\\PQ#\\\"3\"Fas$\"3$[O
LEF-m#))F87$$\"3BLL$e\"*[H7\"Fas$\"3#ylv\\<8Q,*F87$$\"3#*******pvxl6Fa
s$\"3-M-vk+&4>*F87$$\"3z****\\_qn27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p
@[7Fas$\"3uZ3fv(3U[*F87$$\"3#)****\\2'HKH\"Fas$\"3<z/\"QV]sh*F87$$\"3_
mmmwanL8Fas$\"3e9nS%H$=?(*F87$$\"3'******\\2goP\"Fas$\"39AEdKB`7)*F87$
$\"3CLLeR<*fT\"Fas$\"3atab(z;/))*F87$$\"3'******\\)Hxe9Fas$\"3gCG+X%>t
$**F87$$\"3Ymm\"H!o-*\\\"Fas$\"3qbf1InDu**F87$$\"3))***\\7k.6a\"Fas$\"
3_Q!)*f/#f&***F87$$\"3emmmT9C#e\"Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i!
*3`i\"Fas$\"3Iw,Vsb9&)**F87$$\"3QLLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3G
LL$3N1#4<Fas$\"3i:isYhO/**F87$$\"3kmm\"HYt7v\"Fas$\"3yAkC$f\"eP)*F87$$
\"3%*******p(G**y\"Fas$\"3kWGU'3k3w*F87$$\"3lmm;9@BM=Fas$\"3;&yP#=+,b'
*F87$$\"3ELLL`v&Q(=Fas$\"3Q44i*ftUa*F87$$\"30++DOl5;>Fas$\"3jyU@b\\q4%
*F87$$\"3/++v.Uac>Fas$\"37dI([,t^E*F87$$\"\"#F)$\"35<oDoU(H4*F8-%'COLO
URG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%*THICKNESSG6#Fdz-%)PO
LYGONSG6$7&F'7$F($\"+Fp\"z*\\!#67$$\"+++++5!#5Fj[l7$F^\\lF(-%&COLORG6&
Fjz$\"\"(!\"\"$\"\"*Fg\\lFe\\l-Ff[l6$7&Fa\\l7$F^\\l$\"+D8Q%\\\"F`\\l7$
$\"+++++?F`\\lF^]l7$Fa]lF(Fb\\l-Ff[l6$7&Fc]l7$Fa]l$\"+$fRSZ#F`\\l7$$\"
+++++IF`\\lFh]l7$F[^lF(Fb\\l-Ff[l6$7&F]^l7$F[^l$\"+v!y*GMF`\\l7$$\"+++
++SF`\\lFb^l7$Fe^lF(Fb\\l-Ff[l6$7&Fg^l7$Fe^l$\"+T`l\\VF`\\l7$$\"+++++]
F`\\lF\\_l7$F__lF(Fb\\l-Ff[l6$7&Fa_l7$F__l$\"+*GsoA&F`\\l7$$\"+++++gF`
\\lFf_l7$Fi_lF(Fb\\l-Ff[l6$7&F[`l7$Fi_l$\"+dS'=0'F`\\l7$$\"+++++qF`\\l
F``l7$Fc`lF(Fb\\l-Ff[l6$7&Fe`l7$Fc`l$\"++wQ;oF`\\l7$$\"+++++!)F`\\lFj`
l7$F]alF(Fb\\l-Ff[l6$7&F_al7$F]al$\"+^S!G^(F`\\l7$$\"+++++!*F`\\lFdal7
$FgalF(Fb\\l-Ff[l6$7&Fial7$Fgal$\"+[]:M\")F`\\l7$$\"\"\"F)F^bl7$FablF(
Fb\\l-Ff[l6$7&Fcbl7$Fabl$\"+cABu')F`\\l7$$\"+++++6!\"*Fhbl7$F[clF(Fb\\
l-Ff[l6$7&F^cl7$F[cl$\"+.%Rw7*F`\\l7$$\"+++++7F]clFccl7$FfclF(Fb\\l-Ff
[l6$7&Fhcl7$Ffcl$\"+%>Y)*[*F`\\l7$$\"+++++8F]clF]dl7$F`dlF(Fb\\l-Ff[l6
$7&Fbdl7$F`dl$\"+yNBd(*F`\\l7$$\"+++++9F]clFgdl7$FjdlF(Fb\\l-Ff[l6$7&F
\\el7$Fjdl$\"+5*Hr#**F`\\l7$$\"+++++:F]clFael7$FdelF(Fb\\l-Ff[l6$7&Ffe
l7$Fdel$\"+Uw$y***F`\\l7$$\"+++++;F]clF[fl7$F^flF(Fb\\l-Ff[l6$7&F`fl7$
F^fl$\"+&G]'o**F`\\l7$$\"+++++<F]clFefl7$FhflF(Fb\\l-Ff[l6$7&Fjfl7$Fhf
l$\"+p%f)R)*F`\\l7$$\"+++++=F]clF_gl7$FbglF(Fb\\l-Ff[l6$7&Fdgl7$Fbgl$
\"+I?v7'*F`\\l7$$\"+++++>F]clFigl7$F\\hlF(Fb\\l-Ff[l6$7&F^hl7$F\\hl$\"
+]rf*G*F`\\l7$FczFchl7$FczF(Fb\\l-%+AXESLABELSG6$%\"xGQ!6\"-%%VIEWG6$;
F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve
 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "C
urve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve
 20" "Curve 21" }}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 296 
13 "20 RECTANGLES" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "middle
box(sin(x),x=0..2,50);" }}{PARA 13 "" 1 "" {GLPLOT2D 244 206 206 
{PLOTDATA 2 "6W-%'CURVESG6&7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3!)y==T,
0eVF-7$$\"3'pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLLLe0$=C\"!#=$\"3I#\\^x@
T'Q7F87$$\"3ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm;zjf)4#F8$\"3ctIO$>EK3#
F87$$\"3=LL$e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p****\\i'y]!HF8$\"3)z'**Hz%)
QkGF87$$\"3,LL$ezs$HLF8$\"3%f8@S</#oKF87$$\"3_****\\7iI_PF8$\"3+/tuT6(
[m$F87$$\"3#pmmm@Xt=%F8$\"3uej'fKYg1%F87$$\"3QLLL3y_qXF8$\"3tO*fg>`IT%
F87$$\"3i******\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3()******\\Z/NaF8$\"3Uh8%
*pkQr^F87$$\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF87$$\"3ELL$ez6:B'F8$\"3%
o#4^'[pf$eF87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F87$$\"3-mmmm#pS1(F8$\"3w
N76%>Z5\\'F87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C#oF87$$\"3slmmm(y8!zF8$
\"3NBP6&4.X5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S(F87$$\"39++](3zMu)F8$\"
3;afM1NDrwF87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF87$$\"3emm;zihl&*F8$\"3O
`uvYv9s\")F87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%)F87$$\"3<LLezw5V5!#<$\"
3-2#QmIK(R')F87$$\"3!****\\PQ#\\\"3\"Fas$\"3$[OLEF-m#))F87$$\"3BLL$e\"
*[H7\"Fas$\"3#ylv\\<8Q,*F87$$\"3#*******pvxl6Fas$\"3-M-vk+&4>*F87$$\"3
z****\\_qn27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p@[7Fas$\"3uZ3fv(3U[*F87
$$\"3#)****\\2'HKH\"Fas$\"3<z/\"QV]sh*F87$$\"3_mmmwanL8Fas$\"3e9nS%H$=
?(*F87$$\"3'******\\2goP\"Fas$\"39AEdKB`7)*F87$$\"3CLLeR<*fT\"Fas$\"3a
tab(z;/))*F87$$\"3'******\\)Hxe9Fas$\"3gCG+X%>t$**F87$$\"3Ymm\"H!o-*\\
\"Fas$\"3qbf1InDu**F87$$\"3))***\\7k.6a\"Fas$\"3_Q!)*f/#f&***F87$$\"3e
mmmT9C#e\"Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i!*3`i\"Fas$\"3Iw,Vsb9&)*
*F87$$\"3QLLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3GLL$3N1#4<Fas$\"3i:isYhO
/**F87$$\"3kmm\"HYt7v\"Fas$\"3yAkC$f\"eP)*F87$$\"3%*******p(G**y\"Fas$
\"3kWGU'3k3w*F87$$\"3lmm;9@BM=Fas$\"3;&yP#=+,b'*F87$$\"3ELLL`v&Q(=Fas$
\"3Q44i*ftUa*F87$$\"30++DOl5;>Fas$\"3jyU@b\\q4%*F87$$\"3/++v.Uac>Fas$
\"37dI([,t^E*F87$$\"\"#F)$\"35<oDoU(H4*F8-%'COLOURG6&%$RGBG$\"*++++\"!
\")F(F(-%&STYLEG6#%%LINEG-%*THICKNESSG6#Fdz-%)POLYGONSG6$7&F'7$F($\"+p
m')**>!#67$$\"+++++SF\\\\lFj[l7$F^\\lF(-%&COLORG6&Fjz$\"\"(!\"\"$\"\"*
Ff\\lFd\\l-Ff[l6$7&F`\\l7$F^\\l$\"+[1S'*fF\\\\l7$$\"+++++!)F\\\\lF]]l7
$F`]lF(Fa\\l-Ff[l6$7&Fb]l7$F`]l$\"+l;M$)**F\\\\l7$$\"+++++7!#5Fg]l7$Fj
]lF(Fa\\l-Ff[l6$7&F]^l7$Fj]l$\"+Y6V&R\"F\\^l7$$\"+++++;F\\^lFb^l7$Fe^l
F(Fa\\l-Ff[l6$7&Fg^l7$Fe^l$\"+MdH!z\"F\\^l7$$\"+++++?F\\^lF\\_l7$F__lF
(Fa\\l-Ff[l6$7&Fa_l7$F__l$\"+JiH#=#F\\^l7$$\"+++++CF\\^lFf_l7$Fi_lF(Fa
\\l-Ff[l6$7&F[`l7$Fi_l$\"+>b!3d#F\\^l7$$\"+++++GF\\^lF``l7$Fc`lF(Fa\\l
-Ff[l6$7&Fe`l7$Fc`l$\"+n??bHF\\^l7$$\"+++++KF\\^lFj`l7$F]alF(Fa\\l-Ff[
l6$7&F_al7$F]al$\"+@4([L$F\\^l7$$\"+++++OF\\^lFdal7$FgalF(Fa\\l-Ff[l6$
7&Fial7$Fgal$\"+%p/#4PF\\^l7$$F_\\lF\\^lF^bl7$FablF(Fa\\l-Ff[l6$7&Fbbl
7$Fabl$\"+JXgxSF\\^l7$$\"+++++WF\\^lFgbl7$FjblF(Fa\\l-Ff[l6$7&F\\cl7$F
jbl$\"+q5[RWF\\^l7$$\"+++++[F\\^lFacl7$FdclF(Fa\\l-Ff[l6$7&Ffcl7$Fdcl$
\"+'QbUz%F\\^l7$$\"+++++_F\\^lF[dl7$F^dlF(Fa\\l-Ff[l6$7&F`dl7$F^dl$\"+
<*f89&F\\^l7$$\"+++++cF\\^lFedl7$FhdlF(Fa\\l-Ff[l6$7&Fjdl7$Fhdl$\"+o$R
-[&F\\^l7$$\"+++++gF\\^lF_el7$FbelF(Fa\\l-Ff[l6$7&Fdel7$Fbel$\"+0;N5eF
\\^l7$$\"+++++kF\\^lFiel7$F\\flF(Fa\\l-Ff[l6$7&F^fl7$F\\fl$\"+?&o68'F
\\^l7$$\"+++++oF\\^lFcfl7$FfflF(Fa\\l-Ff[l6$7&Fhfl7$Fffl$\"+so<UkF\\^l
7$$\"+++++sF\\^lF]gl7$F`glF(Fa\\l-Ff[l6$7&Fbgl7$F`gl$\"+;\"zGu'F\\^l7$
$\"+++++wF\\^lFggl7$FjglF(Fa\\l-Ff[l6$7&F\\hl7$Fjgl$\"+#>%zKqF\\^l7$$F
a]lF\\^lFahl7$FdhlF(Fa\\l-Ff[l6$7&Fehl7$Fdhl$\"+(He9J(F\\^l7$$\"+++++%
)F\\^lFjhl7$F]ilF(Fa\\l-Ff[l6$7&F_il7$F]il$\"+HcUyvF\\^l7$$\"+++++))F
\\^lFdil7$FgilF(Fa\\l-Ff[l6$7&Fiil7$Fgil$\"+'4pK$yF\\^l7$$\"+++++#*F\\
^lF^jl7$FajlF(Fa\\l-Ff[l6$7&Fcjl7$Fajl$\"+/5ev!)F\\^l7$$\"+++++'*F\\^l
Fhjl7$F[[mF(Fa\\l-Ff[l6$7&F][m7$F[[m$\"+0P(\\I)F\\^l7$$\"\"\"F)Fb[m7$F
e[mF(Fa\\l-Ff[l6$7&Fg[m7$Fe[m$\"+>-3@&)F\\^l7$$\"++++S5!\"*F\\\\m7$F_
\\mF(Fa\\l-Ff[l6$7&Fb\\m7$F_\\m$\"+B[bB()F\\^l7$$\"++++!3\"Fa\\mFg\\m7
$Fj\\mF(Fa\\l-Ff[l6$7&F\\]m7$Fj\\m$\"+,O27*)F\\^l7$$\"++++?6Fa\\mFa]m7
$Fd]mF(Fa\\l-Ff[l6$7&Ff]m7$Fd]m$\"+h\\L'3*F\\^l7$$\"++++g6Fa\\mF[^m7$F
^^mF(Fa\\l-Ff[l6$7&F`^m7$F^^m$\"+C,1Y#*F\\^l7$$F[^lFa\\mFe^m7$Fh^mF(Fa
\\l-Ff[l6$7&Fi^m7$Fh^m$\"+jN*4R*F\\^l7$$\"++++S7Fa\\mF^_m7$Fa_mF(Fa\\l
-Ff[l6$7&Fc_m7$Fa_m$\"+;M!4_*F\\^l7$$\"++++!G\"Fa\\mFh_m7$F[`mF(Fa\\l-
Ff[l6$7&F]`m7$F[`m$\"+a=eN'*F\\^l7$$\"++++?8Fa\\mFb`m7$Fe`mF(Fa\\l-Ff[
l6$7&Fg`m7$Fe`m$\"+<a%[t*F\\^l7$$\"++++g8Fa\\mF\\am7$F_amF(Fa\\l-Ff[l6
$7&Faam7$F_am$\"+/``=)*F\\^l7$$\"+++++9Fa\\mFfam7$FiamF(Fa\\l-Ff[l6$7&
F[bm7$Fiam$\"+Hw^'))*F\\^l7$$\"++++S9Fa\\mF`bm7$FcbmF(Fa\\l-Ff[l6$7&Fe
bm7$Fcbm$\"+MOoQ**F\\^l7$$\"++++![\"Fa\\mFjbm7$F]cmF(Fa\\l-Ff[l6$7&F_c
m7$F]cm$\"+m)\\\\(**F\\^l7$$\"++++?:Fa\\mFdcm7$FgcmF(Fa\\l-Ff[l6$7&Fic
m7$Fgcm$\"+1$e_***F\\^l7$$\"++++g:Fa\\mF^dm7$FadmF(Fa\\l-Ff[l6$7&Fcdm7
$Fadm$\"+lkd****F\\^l7$$Ff^lFa\\mFhdm7$F[emF(Fa\\l-Ff[l6$7&F\\em7$F[em
$\"+Nu*y)**F\\^l7$$\"++++S;Fa\\mFaem7$FdemF(Fa\\l-Ff[l6$7&Ffem7$Fdem$
\"+**)R-'**F\\^l7$$\"++++!o\"Fa\\mF[fm7$F^fmF(Fa\\l-Ff[l6$7&F`fm7$F^fm
$\"+0\"[m\"**F\\^l7$$\"++++?<Fa\\mFefm7$FhfmF(Fa\\l-Ff[l6$7&Fjfm7$Fhfm
$\"+)y\">d)*F\\^l7$$\"++++g<Fa\\mF_gm7$FbgmF(Fa\\l-Ff[l6$7&Fdgm7$Fbgm$
\"+og'>y*F\\^l7$$\"+++++=Fa\\mFigm7$F\\hmF(Fa\\l-Ff[l6$7&F^hm7$F\\hm$
\"+*G\"4\"p*F\\^l7$$\"++++S=Fa\\mFchm7$FfhmF(Fa\\l-Ff[l6$7&Fhhm7$Ffhm$
\"+JGr%e*F\\^l7$$\"++++!)=Fa\\mF]im7$F`imF(Fa\\l-Ff[l6$7&Fbim7$F`im$\"
+x3+j%*F\\^l7$$\"++++?>Fa\\mFgim7$FjimF(Fa\\l-Ff[l6$7&F\\jm7$Fjim$\"+S
,:E$*F\\^l7$$\"++++g>Fa\\mFajm7$FdjmF(Fa\\l-Ff[l6$7&Ffjm7$Fdjm$\"+`&zV
<*F\\^l7$FczF[[n7$FczF(Fa\\l-%+AXESLABELSG6$%\"xGQ!6\"-%%VIEWG6$;F(Fcz
%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C
urve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 1
4" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "
Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curv
e 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33
" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "C
urve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve
 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" }}}}
{EXCHG {PARA 258 "" 0 "" {TEXT 297 13 "50 RECTANGLES" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 298 63 "The total area of the rectangles above can be \+
calculated using " }{TEXT 299 5 "Maple" }{TEXT 300 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf(value(middlesum(sin(x),x=0..2
,5)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g?jD9!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(value(middlesum(sin(x),x=0..2
,10)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q)4&=9!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(value(middlesum(sin(x),x=0..2
,20)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qqt;9!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(value(middlesum(sin(x),x=0..2
,50)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^7C;9!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 4 "The " }
{TEXT 302 5 "exact" }{TEXT 303 65 " value of the area of the region R \+
will be found by calculating  " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG
" }{TEXT -1 0 "" }{TEXT 305 71 " the total area of n rectangles constr
ucted as above, and then finding " }{TEXT -1 1 "." }{XPPEDIT 18 0 "lim
it(A[n],n=infinity" "6#-%&limitG6$&%\"AG6#%\"nG/F)%)infinityG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT 304 32 "As an example, we will show h
ow " }{TEXT -1 1 " " }{XPPEDIT 18 0 "A[10]" "6#&%\"AG6#\"#5" }{TEXT 
-1 1 " " }{TEXT 316 14 "is calculated." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "middlebox(sin(x
),x=0..2,10);  " }}{PARA 13 "" 1 "" {GLPLOT2D 354 265 265 {PLOTDATA 2 
"6/-%'CURVESG6&7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3!)y==T,0eVF-7$$\"3'
pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLLLe0$=C\"!#=$\"3I#\\^x@T'Q7F87$$\"3
ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm;zjf)4#F8$\"3ctIO$>EK3#F87$$\"3=LL$
e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p****\\i'y]!HF8$\"3)z'**Hz%)QkGF87$$\"3,
LL$ezs$HLF8$\"3%f8@S</#oKF87$$\"3_****\\7iI_PF8$\"3+/tuT6([m$F87$$\"3#
pmmm@Xt=%F8$\"3uej'fKYg1%F87$$\"3QLLL3y_qXF8$\"3tO*fg>`IT%F87$$\"3i***
***\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3()******\\Z/NaF8$\"3Uh8%*pkQr^F87$$
\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF87$$\"3ELL$ez6:B'F8$\"3%o#4^'[pf$eF
87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F87$$\"3-mmmm#pS1(F8$\"3wN76%>Z5\\'F
87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C#oF87$$\"3slmmm(y8!zF8$\"3NBP6&4.X
5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S(F87$$\"39++](3zMu)F8$\"3;afM1NDrwF
87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF87$$\"3emm;zihl&*F8$\"3O`uvYv9s\")F
87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%)F87$$\"3<LLezw5V5!#<$\"3-2#QmIK(R'
)F87$$\"3!****\\PQ#\\\"3\"Fas$\"3$[OLEF-m#))F87$$\"3BLL$e\"*[H7\"Fas$
\"3#ylv\\<8Q,*F87$$\"3#*******pvxl6Fas$\"3-M-vk+&4>*F87$$\"3z****\\_qn
27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p@[7Fas$\"3uZ3fv(3U[*F87$$\"3#)***
*\\2'HKH\"Fas$\"3<z/\"QV]sh*F87$$\"3_mmmwanL8Fas$\"3e9nS%H$=?(*F87$$\"
3'******\\2goP\"Fas$\"39AEdKB`7)*F87$$\"3CLLeR<*fT\"Fas$\"3atab(z;/))*
F87$$\"3'******\\)Hxe9Fas$\"3gCG+X%>t$**F87$$\"3Ymm\"H!o-*\\\"Fas$\"3q
bf1InDu**F87$$\"3))***\\7k.6a\"Fas$\"3_Q!)*f/#f&***F87$$\"3emmmT9C#e\"
Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i!*3`i\"Fas$\"3Iw,Vsb9&)**F87$$\"3Q
LLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3GLL$3N1#4<Fas$\"3i:isYhO/**F87$$\"
3kmm\"HYt7v\"Fas$\"3yAkC$f\"eP)*F87$$\"3%*******p(G**y\"Fas$\"3kWGU'3k
3w*F87$$\"3lmm;9@BM=Fas$\"3;&yP#=+,b'*F87$$\"3ELLL`v&Q(=Fas$\"3Q44i*ft
Ua*F87$$\"30++DOl5;>Fas$\"3jyU@b\\q4%*F87$$\"3/++v.Uac>Fas$\"37dI([,t^
E*F87$$\"\"#F)$\"35<oDoU(H4*F8-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&S
TYLEG6#%%LINEG-%*THICKNESSG6#Fdz-%)POLYGONSG6$7&F'7$F($\"+l;M$)**!#67$
$\"+++++?!#5Fj[l7$F^\\lF(-%&COLORG6&Fjz$\"\"(!\"\"$\"\"*Fg\\lFe\\l-Ff[
l6$7&Fa\\l7$F^\\l$\"+n??bHF`\\l7$$\"+++++SF`\\lF^]l7$Fa]lF(Fb\\l-Ff[l6
$7&Fc]l7$Fa]l$\"+'QbUz%F`\\l7$$\"+++++gF`\\lFh]l7$F[^lF(Fb\\l-Ff[l6$7&
F]^l7$F[^l$\"+so<UkF`\\l7$$\"+++++!)F`\\lFb^l7$Fe^lF(Fb\\l-Ff[l6$7&Fg^
l7$Fe^l$\"+'4pK$yF`\\l7$$\"\"\"F)F\\_l7$F__lF(Fb\\l-Ff[l6$7&Fa_l7$F__l
$\"+,O27*)F`\\l7$$\"+++++7!\"*Ff_l7$Fi_lF(Fb\\l-Ff[l6$7&F\\`l7$Fi_l$\"
+a=eN'*F`\\l7$$\"+++++9F[`lFa`l7$Fd`lF(Fb\\l-Ff[l6$7&Ff`l7$Fd`l$\"+m)
\\\\(**F`\\l7$$\"+++++;F[`lF[al7$F^alF(Fb\\l-Ff[l6$7&F`al7$F^al$\"+0\"
[m\"**F`\\l7$$\"+++++=F[`lFeal7$FhalF(Fb\\l-Ff[l6$7&Fjal7$Fhal$\"+x3+j
%*F`\\l7$FczF_bl7$FczF(Fb\\l-%+AXESLABELSG6$%\"xGQ!6\"-%%VIEWG6$;F(Fcz
%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C
urve 8" "Curve 9" "Curve 10" "Curve 11" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 306 218 "Each of the
 10 boxes has width 1/5 (2 units divided into 10 parts). The height of
 the first box is sin(1/10), the height of the second is sin(3/10), th
e third sin(5/10), etc.. Therefore, the total area of the boxes is:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "value(middlesum(sin(x),x=0
..2,10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*&#\"\"\"\"\"&F&-%$sinG
6##F&\"#5F&F&*&F%F&-F)6##\"\"$F,F&F&*&F%F&-F)6##F&\"\"#F&F&*&F%F&-F)6#
#\"\"(F,F&F&*&F%F&-F)6##\"\"*F,F&F&*&F%F&-F)6##\"#6F,F&F&*&F%F&-F)6##
\"#8F,F&F&*&F%F&-F)6##F1F6F&F&*&F%F&-F)6##\"#<F,F&F&*&F%F&-F)6##\"#>F,
F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 307 66 "Using the \"Sigma\" nota
tion, the last expression can be written as:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "A[10]:=middlesum(sin(x),x=0..2,10);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"#5,$*&#\"\"\"\"\"&F+-%$SumG6$-%$sinG
6#,&*&F,!\"\"%\"iGF+F+#F+F'F+/F6;\"\"!\"\"*F+F+" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 308 74 "In general, if we construct n boxes and calculate t
heir total area, we get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
A[n]:=middlesum(sin(x),x=0..2,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%\"AG6#%\"nG,$*(\"\"#\"\"\"F'!\"\"-%$SumG6$-%$sinG6#,$*(F*F+,&%\"iGF+
#F+F*F+F+F'F,F+/F6;\"\"!,&F'F+F+F,F+F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 309 5 "Maple" }{TEXT 310 30 " knows a formula for this sum:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%\"nG!\"\",&*(-%$sinG6#*&F&F&F'F(F&,&*
$)-%$cosGF-F%F&F&F&F(F(-F36#F&F%F&*&F+F&F/F(F(F&F&" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 311 75 "The area can now be found by taking the limit as
 n goes to infinity, which " }{TEXT 313 5 "Maple" }{TEXT 314 22 " also
 knows how to do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(
%,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\")-%$c
osG6#F&F%F&!\"\"F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 34 "To 10 de
cimal places, the area is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Oo9;9!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 336 28 "Thus, to 10 decima
l places, " }{XPPEDIT 18 0 "Int(sin(x),x = 0 .. 2);" "6#-%$IntG6$-%$si
nG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1.416146836
;" "6#-%&FloatG6$\"+Oo9;9!\"*" }{TEXT -1 0 "" }{TEXT 337 1 "." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 87 "E
valuating such sums and limits directly, as in the above example, is s
ometimes beyond " }{TEXT 339 7 "Maple's" }{TEXT 340 114 " capability. \+
However, the required area can always be approximated to any number of
 decimal places by calculating " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"n
G" }{TEXT 341 332 " for a large enough choice of n. The question, howe
ver, is exactly how large must you take n in order to be guaranteed ac
curacy to the desired number of decimal places. This is a non-trivial \+
question which is addressed in advanced mathematics courses on Numeric
al Analysis. All we can do at this stage is investigate using examples
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{MARK "0 6 39" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }