{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 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 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 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 1 1 2 2 2 1 1 1 1 }1 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 1 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 {PARA 4 "" 0 "" {TEXT -1 30 "Module 9 : Intergral Calculus" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 35 "901 : Th e Definition of an\027Integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 18 "O B J E C T I V E " }}{PARA 0 "" 0 "" {TEXT -1 266 "In the project, we will learn about areas under curves, how Ri emann sums converge to the area, how the choice of left, right, or mid point of the subintervals has no ultimate effect, and then we will go \+ through a four step procedure to set up Riemann sums on our own." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "S E T U \+ P" }}{PARA 0 "" 0 "" {TEXT -1 252 "In this project we will use the fol lowing command packages. Type and execute this line before begining th e project below. If you re-enter the worksheet for this project, be su re to re-execute this statement before jumping to any point in the wor ksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart; with(plots): w ith(student):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name change coords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "____________________ ______________________________________________________________" }} {PARA 4 "" 0 "" {TEXT -1 27 "A. Positive & Negative Area" }}{PARA 0 " " 0 "" {TEXT -1 82 "__________________________________________________ ________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 364 "First we ll look at the concept of area under a curve and see how it can be bot h positive and negative. To investigate this idea we will use the spec ial plot procedure below. Type it in precisely as you see it a single \+ comand plot. To create additiional lines without executing hit return \+ (macintosh) or shift return (windows). When you're done, enter the com amand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 " area_plot := proc(f1,a,b)\n local box,i,n,x1,x2,y 1,y2,A,delta: \n n:= 200; delta := (b-a)/n; x2 := a;\n for i from 1 \+ to n do\n x1 := evalf(x2); y1 := evalf(f(x1)); \n x2 := evalf (a + i*delta); y2 := evalf(f(x2));\n if( y1 > 0 )\n then A||i :=polygonplot( [[x1,0],[x1,y1],[x2,y2],[x2,0]],color=coral,style=patc hnogrid);\n else A||i:=polygonplot( [[x1,0],[x1,y1],[x2,y2],[x2, 0]],color=red, style=patchnogrid); \n fi; od;\n display([plot(f1( x), x = a..b, color=blue,thickness=2,discont=true), seq(A||i,i=1..n)]) ;\n end:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "To use this new function, define a function, and the left and rig ht end points of an interval. In this case, we will view the area betw een x = 1 and x = 3. The area is measured between the x-axis and the g raph of the function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " f := x -> 1 + x/2 + (x/3)*sin(x^2) ; a := 1; b := 3;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6 \"6$%)operatorG%&arrowGF(,(\"\"\"F-*&#F-\"\"#F-9$F-F-*(#F-\"\"$F-F1F-- %$sinG6#*$)F1F0F-F-F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " area_plot( f(x), a,b );" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6gw-%'CURVESG6%7`o7$$\" \"\"\"\"!$\"+G.\\!y\"!\"*7$$\"+3VfV5F-$\"+Un2I=F-7$$\"+$[D:3\"F-$\"+t& eE(=F-7$$\"+cI=C6F-$\"+BwH>>F-7$$\"+\"RBr;\"F-$\"+fv;k>F-7$$\"+Q'f)47F -$\"+.%3f+#F-7$$\"+5;[\\7F-$\"+:b@T?F-7$$\"+my]!H\"F-$\"+_#*\\t?F-7$$ \"+!GPHL\"F-$\"+5PR,@F-7$$\"+@1Bv8F-$\"+8xoA@F-7$$\"+AXt=9F-$\"+E(Go8# F-7$$\"+\"y_qX\"F-$\"+&)H-U@F-7$$\"+l+>+:F-$\"+P<+R@F-7$$\"+vW]V:F-$\" +())**e7#F-7$$\"+NfC&e\"F-$\"+tGL.@F-7$$\"+!=^Ji\"F-$\"+\"[tU2#F-7$$\" 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82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93 " "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "C urve 100" "Curve 101" "Curve 102" "Curve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" "Curve 109" "Curve 110" "Curve 111 " "Curve 112" "Curve 113" "Curve 114" "Curve 115" "Curve 116" "Curve 1 17" "Curve 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Cur ve 129" "Curve 130" "Curve 131" "Curve 132" "Curve 133" "Curve 134" "C urve 135" "Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146 " "Curve 147" "Curve 148" "Curve 149" "Curve 150" "Curve 151" "Curve 1 52" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Cur ve 164" "Curve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "C urve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176" "Curve 177" "Curve 178" "Curve 179" "Curve 180" "Curve 181 " "Curve 182" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 1 87" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192" "Curve 193" "Curve 194" "Curve 195" "Curve 196" "Curve 197" "Curve 198" "Cur ve 199" "Curve 200" "Curve 201" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 168 "In many cases, there are regions where t he curve weaves above and below the x axis. 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"Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 1 63" "Curve 164" "Curve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "Curve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Cur ve 175" "Curve 176" "Curve 177" "Curve 178" "Curve 179" "Curve 180" "C urve 181" "Curve 182" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 187" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192 " "Curve 193" "Curve 194" "Curve 195" "Curve 196" "Curve 197" "Curve 1 98" "Curve 199" "Curve 200" "Curve 201" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "________ ______________________________________________________________________ ____" }}{PARA 4 "" 0 "" {TEXT -1 23 "B. Integrable Functions" }}{PARA 0 "" 0 "" {TEXT -1 82 "_______________________________________________ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 267 "Althoug h a function needs to be both continuous and smooth to be differentiab le, the area can still be clearly defined even for functions which are neither smooth nor continuous. Such functions are termed integrable f or reasons which will be clear in the near future." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "\nArea also applies to d iscontinuous, piecewise defined functions, and functions which are not differentiable everywhere.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 " f:= x -> piecewise( x < 2, \+ -1, x<6, 2-x/2, \011\011\011x < 8, .5*x^2 - 6*x + 18, 1); \n" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrow GF(-%*piecewiseG6)29$\"\"#!\"\"2F0\"\"',&F1\"\"\"*&#F6F1F6F0F6F22F0\" \"),(*$)F0F1F6$\"\"&F2*&F4F6F0F6F2\"#=F6F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " a:=0; b := 10; area_plot( f(x), a, b );\n \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"#5" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6gw-%'CURVESG6(7S7$$\"\"!F)$!\"\"F)7$$\"+>v#*eV!#6F*7$$\" 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138" "Curve 139" "Curve 140" "Curve 141 " "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 1 47" "Curve 148" "Curve 149" "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Cur ve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "C urve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "Curve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176 " "Curve 177" "Curve 178" "Curve 179" "Curve 180" "Curve 181" "Curve 1 82" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 187" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192" "Curve 193" "Cur ve 194" "Curve 195" "Curve 196" "Curve 197" "Curve 198" "Curve 199" "C urve 200" "Curve 201" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f \+ := x-> sqrt( abs(x^2 - 9 )) - 2; \n a := -5; b := 5; area_plot( f(x ), a, b );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)ope 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`nFg`n-Fi_n6%7&Fh[tFe[t7$$\"+++++UF-F_in7$F]\\tF`_nFc`nFg`n-Fi_n6%7&F_ \\tF\\\\t7$$\"++++]UF-Ffhn7$Fd\\tF`_nFc`nFg`n-Fi_n6%7&Ff\\tFc\\t7$$\"+ ++++VF-F]hn7$F[]tF`_nFc`nFg`n-Fi_n6%7&F]]tFj\\t7$$\"++++]VF-Fdgn7$Fb]t F`_nFc`nFg`n-Fi_n6%7&Fd]tFa]t7$$\"+++++WF-F[gn7$Fi]tF`_nFc`nFg`n-Fi_n6 %7&F[^tFh]t7$$\"++++]WF-Fbfn7$F`^tF`_nFc`nFg`n-Fi_n6%7&Fb^tF_^t7$$Fh[r F-Fien7$Fg^tF`_nFc`nFg`n-Fi_n6%7&Fh^tFf^t7$$\"++++]XF-F`en7$F]_tF`_nFc `nFg`n-Fi_n6%7&F__tF\\_t7$$\"+++++YF-Fgdn7$Fd_tF`_nFc`nFg`n-Fi_n6%7&Ff _tFc_t7$$\"++++]YF-F^dn7$F[`tF`_nFc`nFg`n-Fi_n6%7&F]`tFj_t7$$\"+++++ZF -Fecn7$Fb`tF`_nFc`nFg`n-Fi_n6%7&Fd`tFa`t7$$\"++++]ZF-F\\cn7$Fi`tF`_nFc `nFg`n-Fi_n6%7&F[atFh`t7$$\"+++++[F-Fcbn7$F`atF`_nFc`nFg`n-Fi_n6%7&Fba tF_at7$$\"++++][F-Fjan7$FgatF`_nFc`nFg`n-Fi_n6%7&FiatFfat7$$\"+++++\\F -Faan7$F^btF`_nFc`nFg`n-Fi_n6%7&F`btF]bt7$$\"++++]\\F-F``n7$FebtF`_nFc `nFg`n-Fi_n6%7&FgbtFdbtFi^n7$Fj^nF`_nFc`nFg`n-%+AXESLABELSG6$Q\"x6\"Q! 6\"-%%VIEWG6$;F(Fj^n%(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" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37 " "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56 " "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "C urve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75 " "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "C urve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94 " "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" " Curve 101" "Curve 102" "Curve 103" "Curve 104" "Curve 105" "Curve 106 " "Curve 107" "Curve 108" "Curve 109" "Curve 110" "Curve 111" "Curve 1 12" "Curve 113" "Curve 114" "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "Curve 123" "Cur ve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "C urve 130" "Curve 131" "Curve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141 " "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 1 47" "Curve 148" "Curve 149" "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Cur ve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "C urve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "Curve 170" "Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176 " "Curve 177" "Curve 178" "Curve 179" "Curve 180" "Curve 181" "Curve 1 82" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 187" "Curve 188" "Curve 189" "Curve 190" "Curve 191" "Curve 192" "Curve 193" "Cur ve 194" "Curve 195" "Curve 196" "Curve 197" "Curve 198" "Curve 199" "C urve 200" "Curve 201" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "_________________________________________________________ _________________________" }}{PARA 4 "" 0 "" {TEXT -1 29 "C. Riemann S ums & Convergence" }}{PARA 0 "" 0 "" {TEXT -1 82 "____________________ ______________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 384 "Now well examine a scheme to compute the area \+ under curves. The idea is to divide the interval on the x axis into n \+ small but equal pieces, and then draw a rectangle that approximates th e area in each subinterval. The sum of all of the areas of the boxes a pproximates the area under the curve for this interval. As n gets larg er, the approximation to the area gets better and better. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 14 "BOXES AND AREA " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " f:= x -> x^3 -19*x^2 +99*x - 20; a := 1; b := 13; \+ n := 10;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%) operatorG%&arrowGF(,**$)9$\"\"$\"\"\"F1*&\"#>F1)F/\"\"#F1!\"\"*&\"#**F 1F/F1F1\"#?F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rightsum( f(x), x = a..b, n):\011\011Area := evalf(%,20);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AreaG$\"5+++++++'*G9!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "rightbox( f(x), x = a..b, n, \011ti tle= convert( evalf(Area,10), string) );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "60-%)POLYGONSG6$7&7$$\"\"\"\"\"!$F* F*7$F($\"+++)[;\"!\"(7$$\"+++++A!\"*F-7$F1F+-%&COLORG6&%$RGBG$\"\"(!\" \"$\"\"*F;F9-F$6$7&F47$F1$\"+++ki8F/7$$\"+++++MF3FB7$FEF+F5-F$6$7&FG7$ FE$\"+++'pI\"F/7$$\"+++++YF3FL7$FOF+F5-F$6$7&FQ7$FO$\"+++_,6F/7$$\"+++ ++eF3FV7$FYF+F5-F$6$7&Fen7$FY$\"#&)F*7$$F:F*Fjn7$F]oF+F5-F$6$7&F^o7$F] o$\"+++!3c'!\")7$$\"+++++#)F3Fco7$FgoF+F5-F$6$7&Fio7$Fgo$\"+++SMiFeo7$ $\"+++++%*F3F^p7$FapF+F5-F$6$7&Fcp7$Fap$\"+++gd&)Feo7$$\"++++g5FeoFhp7 $F[qF+F5-F$6$7&F]q7$F[q$\"+++sc9F/7$$\"++++!=\"FeoFbq7$FeqF+F5-F$6$7&F gq7$Feq$\"$`#F*7$$\"#8F*F\\r7$F_rF+F5-%'CURVESG6&7Y7$F($\"#hF*7$$\"3++ ++]ech7!#<$\"3c(4^oFWjm(!#;7$$\"3!*****\\(*G:*[\"F\\s$\"33_&=7k\\%f))F _s7$$\"3++++N$)4XsW\"e 45\"Fjs7$$\"3;++]Fy:fAF\\s$\"3M%H$)e8[@=\"Fjs7$$\"3q****\\d'*)o\\#F\\s $\"3fWQC%GTIC\"Fjs7$$\"3:++](>ZIu#F\\s$\"3'e*>+PNR#H\"Fjs7$$\"3-++]xOi (*HF\\s$\"3j[\\ZwUrH8Fjs7$$\"3%*****\\FPQ^KF\\s$\"3)4kI$Gb+a8Fjs7$$\"3 ;+++IrS7NF\\s$\"3U(yR08ylO\"Fjs7$$\"3.+++&o;Bu$F\\s$\"3WI1*Q,l!o8Fjs7$ $\"3x*******QS6+%F\\s$\"3ew0[*)G%*f8Fjs7$$\"3O+++]o-hUF\\s$\"34Yw.dqNU 8Fjs7$$\"3v******4cZ6XF\\s$\"3y>y0mq'))oam%*zF\\s$\"3oz3pn\"\\p!oF_s7$$\"3k*****\\JigC)F\\s$\"34aX8Z@) =^'F_s7$$\"3%*****\\P6E\\9'F_s7$$\"3;******fG0-!*F\\s$\"3nnCDsL++hF_s7$$\"3A+++]/;h#* F\\s$\"3Elejj]MchF_s7$$\"3Y****\\P/&f\\*F\\s$\"3C[-=0A(*3jF_s7$$\"3q++ +5zj_(*F\\s$\"3U.:ot_!ef'F_s7$$\"3=****\\<3;%***F\\s$\"3Ch\")**QI%*))p F_s7$$\"3-++v%=iY-\"F_s$\"3*Q!)f64')p`(F_s7$$\"3'******\\'[M\\5F_s$\"3 l!*\\'3A3u@)F_s7$$\"3'****\\PM&=v5F_s$\"3#\\J?I_MG4*F_s7$$\"32+++'zs+5 \"F_s$\"3:jMfw.K55Fjs7$$\"3,++]5Q_D6F_s$\"3gM8]`:fJ6Fjs7$$\"33++vxSw]6 F_s$\"3er:P5![2F\"Fjs7$$\"31+]()p1Oi6F_s$\"3(eq%Q1gDT8Fjs7$$\"31+++is& R<\"F_s$\"3E-5H[E.;9Fjs7$$\"31++Dl#[s=\"F_s$\"3lD-wob52:Fjs7$$\"3.++]o #R0?\"F_s$\"3+2q4f%\\Sg\"Fjs7$$\"3-++D+pU77F_s$\"3i^o\"f@Yep\"Fjs7$$\" 3++++KXJC7F_s$\"3=WCewIb#z\"Fjs7$$\"3-+](oA*)pB\"F_s$\"3cX3iQZ=,>Fjs7$ $\"3/++v@Rm\\7F_s$\"3Xm0Sj\\j:?Fjs7$$\"3%******>A&zh7F_s$\"3\")epw[4uI @Fjs7$$\"3/++DAl#RF\"F_s$\"3;]cp>jR^AFjs7$$\"35+]7hK'pG\"F_s$\"3y4p,* \\ltQ#FjsF^r-%'COLOURG6&F8$\"*++++\"FeoF+F+-%&STYLEG6#%%LINEG-%*THICKN ESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!6\"-%&TITLEG6#Q,1428.960000Ffdl-%% VIEWG6$;F(F_r%(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" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "You can see 10 re ctangles along with the graph of f(x). The area computation is the tot al area of the 10 rectangles. This number is a rough approximation to \+ the area under the curve. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 11 "CONVERGENCE" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can c arry this idea a bit further and look at increasing values of n and se e how the area changes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 " for k from 1 to 25 do\n print (cat(`Area for n=`,200*k), evalf(rightsum( f(x), x = a..b,200*k), 20)) ; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Area~for~n=200G$\"5++++++W x88!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Area~for~n=400G$\"5++++++O )3J\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Area~for~n=600G$\"5+++++ +;#*48!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Area~for~n=800G$\"5++++ ++4W48!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=1000G$\"5+++ ++wD:48!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=1200G$\"5++ ++++/'*38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=1400G$\"5 K;3/-^J#)38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=1600G$ \"5+++++D-s38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=1800G $\"5yxxxxx,k38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=2000 G$\"5+++++Whd38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=220 0G$\"5k#3!>PbP_38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=2 400G$\"5++++++,[38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n= 2600G$\"5[^x,8iJW38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n =2800G$\"5lK;3/-:T38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~ n=3000G$\"5+++++kSQ38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for ~n=3200G$\"5++++Dc+O38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~fo r~n=3400G$\"5TQ,kNt)Q$38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~ for~n=3600G$\"5WWWWWW+K38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area ~for~n=3800G$\"5!=N$Ry(>.$38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0A rea~for~n=4000G$\"5+++++O!)G38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 0Area~for~n=4200G$\"5)QpMn$=VF38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%0Area~for~n=4400G$\"5vH%)Q$z%=E38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=4600G$\"5%=2y.?Y]#38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=4800G$\"5+++++D+C38!#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%0Area~for~n=5000G$\"5++++/B/B38!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "In this way, we can se e that the area seems to be converging as n increased from 200 to 5,00 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 315 "You can get an even better idea of how \+ areas of the boxes converge to the area under the curve if we choose a sequence of values for n which is increasing geometrically. However, \+ this process requires a fast computer. Depending on the speed of your computer, may wish to increase or decrease the value of 12 below." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 " for k from 1 to 12 do\n print(cat(`Area for n=`,100*2^k), eva lf(rightsum(f(x), x = a..b,100*2^k),20)) ;od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Area~for~n=200G$\"5++++++Wx88!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/Area~for~n=400G$\"5++++++O)3J\"!#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%/Area~for~n=800G$\"5++++++4W48!#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%0Area~for~n=1600G$\"5+++++D-s38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=3200G$\"5++++Dc+O38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0Area~for~n=6400G$\"5+++D19+=38!#;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%1Area~for~n=12800G$\"5++Dc^.+438!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%1Area~for~n=25600G$\"5+D1*y3+X!38!#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%1Area~for~n=51200G$\"5DcE(>-]A!38!#; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2Area~for~n=102400G$\"51kJ\\0]7,3 8!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2Area~for~n=204800G$\"5-\"Ht8 ]i0!38!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2Area~for~n=409600G$\"5v ALM]7G+38!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Note that the areas appe ar to converge much quicker to more or less a common value. You can al so see that the differences between the numbers are decreasing toward \+ zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 4 "" 0 "" {TEXT -1 70 "_____________________________________ _________________________________" }}{PARA 4 "" 0 "" {TEXT -1 24 "D. L eft, Right, & Middle" }}{PARA 0 "" 0 "" {TEXT -1 82 "_________________ _________________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 357 "When we \+ drew boxes above, we computed the height of the small rectangles by ev aluating f(x) at the right endpoint of each subinterval. This is not t he only way of doing it. Just as easily, we could have evaluated f(x) \+ at the left endpoint or the midpoint of each subinterval. Lets look at how these other methods appear different and give different results. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "Care fully scrutinize the three plots. Look at the graph and where each sma ll rectangle meet. 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