{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 1 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 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 "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 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 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 256 0 "" } {TEXT 257 18 "The Trapezoid Rule" }}{PARA 258 "" 0 "" {TEXT -1 17 " By Sylvain Muise" }}{PARA 259 "" 0 "" {TEXT -1 34 "" }{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "The trapezoid rule is one way of many to approximate definite integrals. It is foun d by finding the average between the left-point rules and right-point \+ rules defined as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 17 "Left-point rule: " }{XPPEDIT 18 0 "int(f(x),x = a \+ .. b) = L[n];" "6#/-%$intG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG&%\"LG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sum(f(x[i-1]),i = 1 .. n);" "6#-%$su mG6$-%\"fG6#&%\"xG6#,&%\"iG\"\"\"F.!\"\"/F-;F.%\"nG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*x" "6#*&%&DeltaG\"\"\"%\"xGF%" }}{PARA 0 "" 0 " " {TEXT -1 17 "Right-point rule:" }{XPPEDIT 18 0 "int(f(x),x = a .. b) = R[n];" "6#/-%$intG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG&%\"RG6#%\"nG" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "sum(f(x[i]),i = 1 .. n);" "6#-%$sumG6 $-%\"fG6#&%\"xG6#%\"iG/F,;\"\"\"%\"nG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*x" "6#*&%&DeltaG\"\"\"%\"xGF%" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "x[i] = a+i*Del ta*x;" "6#/&%\"xG6#%\"iG,&%\"aG\"\"\"*(F'F*%&DeltaGF*F%F*F*" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Delta*x = (b-a)/n;" "6#/*&%&DeltaG\"\"\" %\"xGF&*&,&%\"bGF&%\"aG!\"\"F&%\"nGF," }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "It is called the tr apezoid rule because of the shape of polygons which results. This Mapl e application demonstrates the trapezoid rule using graphs and animati ons." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "leftGraph := proc(f, a, b, n)\n local dX, boxes, i, x0, x1, gr aph:\n dX := (b-a)/n:\n boxes := []:\n for i from 1 to n do\n x0 := a+(i-1) * dX:\n x1 := a + i * dX:\n boxes := [op(boxes), plo ttools[polygon]([[x0, 0], [x0,f(x0)],[x1,f(x0)],[x1,0]], color=cyan)]: \n end do:\n graph := plot(f(x),x=a..b):\n plots[display]([graph,se q(boxes[i],i=1..n)], axes=normal)\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 381 "rightGraph := proc(f, a, b, n)\n local dX, box es, i, x0, x1, graph:\n dX := (b-a)/n:\n boxes := []:\n for i from \+ 1 to n do\n x0 := a+(i-1) * dX:\n x1 := a + i * dX:\n boxes : = [op(boxes), plottools[polygon]([[x0,0],[x0,f(x1)], [x1, f(x1)], [x1, 0]], color=cyan)]:\n end do:\n graph := plot(f(x), x=a..b):\n plot s[display]([graph, seq(boxes[i],i=1..n)], axes=normal)\nend proc:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 384 "trapGraph := proc(f, a, b, \+ n)\n local dX, traps, i, x0, x1, graph:\n dX := (b - a) /n:\n traps := []:\n for i from 1 to n do\n x0 := a + (i-1) * dX:\n x1 := \+ a + i * dX:\n traps := [op(traps), plottools[polygon]([[x0,0],[x0,f (x0)],[x1, f(x1)], [x1, 0]], color=cyan)]:\n end do:\n graph := plot (f(x),x=a..b):\n plots[display]([graph, seq(traps[i], i=1..n)], axes= normal)\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "anim ateTraps := proc(f, a, b, initN, finalN)\n local i, animations, temp: \n if initN <= finalN then\n for i from initN to finalN do\n \+ animations[i] := trapGraph(f,a, b, i):\n end do:\n temp := initN :\n temp2 := finalN:\n elif initN > finalN then\n temp := initN :\n for i from finalN to initN do\n animations[temp] := trapGr aph(f,a,b,i):\n temp := temp - 1:\n end do:\n temp := final N:\n temp2 := initN:\n end if:\n plots[display](seq(animations[i] ,i=temp..temp2),insequence=true, axes=normal)\nend proc:" }}{PARA 7 " " 1 "" {TEXT -1 74 "Warning, `temp2` is implicitly declared local to p rocedure `animateTraps`\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "First, define a function. Ex: f := x -> sin(x);" }}{PARA 0 "" 0 "" {TEXT -1 135 "The first three functions, leftGraph, rightGra ph, and trapGraph, display an image of the function and the correspond ing approximation. " }}{PARA 0 "" 0 "" {TEXT -1 33 "Parameters: ????Gr aph(f, a, b, n)" }}{PARA 0 "" 0 "" {TEXT -1 20 " f : the function" }}{PARA 0 "" 0 "" {TEXT -1 51 " a: the lower limit of integration/a pproximation" }}{PARA 0 "" 0 "" {TEXT -1 51 " b: the upper limit of integration/approximation" }}{PARA 0 "" 0 "" {TEXT -1 66 " n: the \+ number of trapezoids to use to approximate the integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The second function , animateTraps, displays an animation of the function and the trapezoi dal approximation with changing values of n." }}{PARA 0 "" 0 "" {TEXT -1 48 "Parameters: animateTraps(f, a, b, initN, finalN)" }}{PARA 0 "" 0 "" {TEXT -1 19 " f: the function" }}{PARA 0 "" 0 "" {TEXT -1 51 " a: the lower limit of integration/approximation" }}{PARA 0 "" 0 " " {TEXT -1 51 " b: the upper limit of integration/approximation" }} {PARA 0 "" 0 "" {TEXT -1 78 " initN: the initial number of trapezoi ds to use to approximate the integral" }}{PARA 0 "" 0 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