(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 6901710, 189344]*) (*NotebookOutlinePosition[ 6902389, 189368]*) (* CellTagsIndexPosition[ 6902345, 189364]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Riemann Sums: A Symbolic and Graphical Approach", FontFamily->"Arial", FontSize->24, FontColor->RGBColor[0.996109, 0, 0]], StyleBox["\n", FontSize->12] }], "Title", TextAlignment->Center, FontFamily->"Arial"], Cell[TextData[{ "This Mathematica notebook allows the investigation of left, right, and \ midpoint Riemann sums to approximate the definite integral of y = f(x) on \ [a,b] as well as sums obtained from the trapezoid and Simpson's rules. \n\n\ The sums are expressed in terms of \[CapitalDelta]x and the limits are \ investigated as \[CapitalDelta]x approaches 0 (from the left and right). In \ addition, rates of convergence to the limit are examined as well as \ improvements to the approximations that give faster convergence.\n\n", StyleBox["IMPORTANT: ", FontWeight->"Bold"], "If you use different functions, you will need to make modifications in the \ PlotRange options in the functions that generate the plots.\n\nThis ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook generates the graphs, animations, and data in the example in the \ demo entitled Riemann Sums: A Symbolic and Graphical Approach." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(Clear[x, f, \[CapitalDelta]x, b, a, n, \[CapitalDelta]x]\)], "Input", FontFamily->"Arial"], Cell[TextData[StyleBox["Define the limits of integration.", FontSize->14]], "Text", FontFamily->"Arial"], Cell[BoxData[ \(a = 0; b = 2;\)], "Input", FontFamily->"Arial"], Cell["Define f.", "Text", FontFamily->"Arial", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(\(f[x_] = \(-3\) x^4 - 4 x^3 + x^2 - x + 10;\)\)], "Input", FontFamily->"Arial"], Cell["\<\ Save the plot of f. For a nice display, use a plotting interval slightly \ larger than [a,b].\ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(plotf := Plot[f[x], {x, \(-1\), 3}, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] Thickness[ .01]]\)], "Input", FontFamily->"Arial"], Cell["\<\ Maximum number of (equal width) rectangles is maxnumpts. \[CapitalDelta]x \ (depends on n) and is equal to (b-a)/n.\ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(\(maxnumpts = 30;\)\)], "Input", FontFamily->"Arial"], Cell[BoxData[ \(\(\[CapitalDelta]x[n_] = \((b - a)\)/n;\)\)], "Input", FontFamily->"Arial"], Cell["\<\ Define the function to generate rectangles for Right Hand Endpoint \ approximation.\ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(rherect[n_] := Graphics[{RGBColor[1, 0, 0], Thickness[ .01], Table[Rectangle[{a + \((k - 1)\)*\[CapitalDelta]x[n], 0}, {a + k*\[CapitalDelta]x[n], f[a + k*\[CapitalDelta]x[n]]}], {k, 1, n}]}, PlotLabel \[Rule] StyleForm[StringJoin["\", ToString[n]], FontSize \[Rule] 14\ , FontColor -> RGBColor[1, 0, 0], FontWeight \[Rule] "\"\ \ ], DisplayFunction \[Rule] Identity, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\< \>"}, PlotRange \[Rule] {{\(-1\), 2}, {\(-136\), 20}}]\)], "Input", FontFamily->"Arial"], Cell["\<\ Define the Right Hand Endpoint (RHE) Riemann sum as a function of n and \ \[CapitalDelta]x.\ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(RHE[\[CapitalDelta]x_, n_] = Sum[f[a + k*\[CapitalDelta]x]*\[CapitalDelta]x, {k, 1, n}]\)], "Input", FontFamily->"Arial"], Cell[BoxData[ \(\[CapitalDelta]x\ \((10\ n - 1\/2\ n\ \((1 + n)\)\ \[CapitalDelta]x + 1\/6\ n\ \((1 + n)\)\ \((1 + 2\ n)\)\ \[CapitalDelta]x\^2 - n\^2\ \((1 + n)\)\^2\ \[CapitalDelta]x\^3 - 1\/10\ n\ \((1 + n)\)\ \((1 + 2\ n)\)\ \((\(-1\) + 3\ n + 3\ n\^2)\)\ \[CapitalDelta]x\^4)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Define the function to show the approximations to the definite integral \ using approximation scheme ", StyleBox["approx.", FontWeight->"Bold"] }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(showapprox[approx_, n_, color_] := Graphics[{color, PointSize[ .03], Table[Point[{i, approx[\[CapitalDelta]x[i], i]}], {i, 1, n}]}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\< \>"}, PlotLabel \[Rule] StyleForm["\", FontSize \[Rule] 14, FontColor \[Rule] color, FontWeight \[Rule] "\"], DisplayFunction \[Rule] Identity, PlotRange \[Rule] {{0, maxnumpts}, {\(-145\), 20}}]\)], "Input", FontFamily->"Arial"], Cell["\<\ Show the animation of graphics array of rectangles and associated \ approximation to definite integral.\ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Show[ GraphicsArray[{Show[{rherect[n], plotf}, PlotRange \[Rule] {{\(-1\), 2}, {\(-75\), 20}}], showapprox[RHE, n, RGBColor[1, 0, 0]]}], DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 600], {n, 1, maxnumpts}]\)], "Input", 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