{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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 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 "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 P lot" -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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 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 "Heading 2" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 258 " " 0 "" {TEXT -1 44 "Lesson 23: The Meaning of Area Under a Curve" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 24 "Positive & Negative Area" }}{PARA 0 "" 0 "" {TEXT -1 232 "First we'll look at \+ the concept of area under a curve and see how it can be both positive \+ and negative. To investigate this idea we will use the special plot pr ocedure below. Type it in precisely as you see it a single comand plot . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 " area_plot := proc(f1 ,a,b)\n local box,i,n,x1,x2,y1,y2,A,delta: \n n:= 2; delta := (b-a) /n; x2 := a;\n for i from 1 to n do\n x1 := evalf(x2); y1 := eva lf(f1(x1)); \n x2 := evalf(a + i*delta); y2 := evalf(f1(x2));\n \+ if( y1 > 0 )\n then A||i :=polygonplot( [[x1,0],[x1,y1],[x2,y2] ,[x2,0]],color=coral,style=patchnogrid);\n else A||i:=polygonplo t( [[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,di scont=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 right end points of an interval. In this \+ case, we will view the area between x = 1 and x = 3. The area is measu red between the x-axis and the graph of the function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl ots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "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 21 "area_plot( f, a, b );" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 {PLOTDATA 2 "6gw-%'CURVE SG6%7`o7$$\"\"\"\"\"!$\"3w)Hp#G.\\!y\"!#<7$$\"3ALLL3VfV5F-$\"34y1]Un2I =F-7$$\"3smm\"H[D:3\"F-$\"3%38\"4s&eE(=F-7$$\"3XLL$e0$=C6F-$\"3l(\\+Gi (H>>F-7$$\"3QLL$3RBr;\"F-$\"3!=$Q'ybnT'>F-7$$\"3imm\"zjf)47F-$\"3-W'GG S3f+#F-7$$\"3WLLe4;[\\7F-$\"3G+$*f9b@T?F-7$$\"3-++Dmy]!H\"F-$\"3#HX?>D *\\t?F-7$$\"3>LLezs$HL\"F-$\"3g0!*Q4PR,@F-7$$\"31++D@1Bv8F-$\"3.$HIJr( 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