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Notebook[{

Cell[CellGroupData[{
Cell["\<\
Using Riemann Sums to Estimate Areas, Volumes, and Lengths of Arc\
\>", "Title",
  PageWidth->PaperWidth],

Cell[TextData[StyleBox["Chapter 4, Sections 3 through 5, and\nChapter 5, \
Sections 1 and 3",
  FontFamily->"Arial",
  FontSize->16,
  FontWeight->"Bold"]], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(\(Off[General::spell];\)\), "\[IndentingNewLine]", 
    \(\(Off[General::spell1];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(Clear[areas];\)\), "\[IndentingNewLine]", 
    \(\(areas := 
        Block[{}, \[IndentingNewLine]h = \((b - a)\)/n; \n\[CapitalDelta]A[
              i_] = f[a + i*h]*h; \n
          Print["\<The \[CapitalDelta]A[i] function is \>", \[CapitalDelta]A[
              i]]; \nrateList = 
            Table[{a + i\ h, \[CapitalDelta]A[i]\/h}, {i, 0, 
                n - 1}]; \[IndentingNewLine]Print[\*"\"\<The \!\(\
\[CapitalDelta]A[i]\/\[CapitalDelta]x\) function is \>\"", \
\[CapitalDelta]A[i]\/h]; \n
          p0 = ListPlot[rateList, 
              PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.015]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_i\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]A\_i\/\[CapitalDelta]x\)\>\""}]; \n
          areaSum[k_] := \((\[Sum]\+\(i = 0\)\%\(k - 1\)N[\[CapitalDelta]A[
                    i]])\); \n
          arealist = Table[{a + k*h, areaSum[k]}, {k, 0, n}]; \n
          p1 = ListPlot[arealist, PlotRange \[Rule] All, 
              PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.0175]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \
\*"\"\<\!\(A\_total\)(k)\>\""}]; \[IndentingNewLine]A\_approx[x_] = 
            Fit[arealist, fitlist, 
              x]; \[IndentingNewLine]Print["\<The function Aapprox[x] is \>", 
            A\_approx[x]]; \[IndentingNewLine]p2 = 
            Plot[A\_approx[x], {x, a, a + n\ h}, PlotRange \[Rule] All, 
              PlotStyle \[Rule] {RGBColor[0, 0, 1]}, 
              AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(A\_approx\)[x]\>\""}, 
              DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{p2, 
              p1}, AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(A\_approx\)[x] & \!\
\(A\_total\)(k)\>\""}, DisplayFunction \[Rule] $DisplayFunction]; \n
          msecant = 
            Table[{a + k\ h, \(areaSum[k + 1] - areaSum[k]\)\/h}, {k, 0, 
                n - 1}]; \n
          p4 = ListPlot[msecant, 
              PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.02]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(m\_secant\
\)\>\""}, DisplayFunction \[Rule] Identity]; \n
          Show[{p4, p0}, 
            AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]A\_i\/\[CapitalDelta]x\) & \!\(m\_secant\)\>\""}, 
            DisplayFunction \[Rule] $DisplayFunction];];\)\[IndentingNewLine]\
\), "\[IndentingNewLine]", 
    \(\(Clear[volumes];\)\), "\[IndentingNewLine]", 
    \(\(volumes := 
        Block[{}, \[IndentingNewLine]h = \((b - a)\)/n; \n\[CapitalDelta]V[
              i_] = \[Pi]\ f[a + i*h]\^2*h; \n
          Print["\<The \[CapitalDelta]V[i] function is \>", \[CapitalDelta]V[
              i]]; \nrateList = 
            Table[{a + i\ h, \[CapitalDelta]V[i]\/h}, {i, 0, 
                n - 1}]; \[IndentingNewLine]Print[\*"\"\<The \!\(\
\[CapitalDelta]V[i]\/\[CapitalDelta]x\) function is \>\"", \
\[CapitalDelta]V[i]\/h]; \n
          p0 = ListPlot[rateList, 
              PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.015]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_i\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]V\_i\/\[CapitalDelta]x\)\>\""}]; \n
          volumeSum[
              k_] := \((\[Sum]\+\(i = 0\)\%\(k - 1\)N[\[CapitalDelta]V[
                    i]])\); \n
          volumelist = Table[{a + k*h, volumeSum[k]}, {k, 0, n}]; \n
          p1 = ListPlot[volumelist, PlotRange \[Rule] All, 
              PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.0175]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \
\*"\"\<\!\(V\_total\)(k)\>\""}]; \[IndentingNewLine]V\_approx[x_] = 
            Fit[volumelist, fitlist, 
              x]; \[IndentingNewLine]Print["\<The function Vapprox[x] is \>", 
            V\_approx[x]]; \[IndentingNewLine]p2 = 
            Plot[V\_approx[x], {x, a, a + n\ h}, PlotRange \[Rule] All, 
              PlotStyle \[Rule] {RGBColor[0, 0, 1]}, 
              AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(V\_approx\)[x]\>\""}, 
              DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{p2, 
              p1}, AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(V\_approx\)[x] & \!\
\(V\_total\)(k)\>\""}, DisplayFunction \[Rule] $DisplayFunction]; \n
          msecant = 
            Table[{a + k\ h, \(volumeSum[k + 1] - volumeSum[k]\)\/h}, {k, 0, 
                n - 1}]; \n
          p4 = ListPlot[msecant, 
              PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.02]}, 
              
              AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(m\_secant\
\)\>\""}, DisplayFunction \[Rule] Identity]; \n
          Show[{p4, p0}, 
            AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]V\_i\/\[CapitalDelta]x\) & \!\(m\_secant\)\>\""}, 
            DisplayFunction \[Rule] $DisplayFunction];];\)\[IndentingNewLine]\
\), "\[IndentingNewLine]", 
    \(\(Clear[lengths];\)\), "\[IndentingNewLine]", 
    \(\(lengths := 
        Block[{}, \[IndentingNewLine]h = \((b - a)\)/n; \n\[CapitalDelta]L[
              i_] = \@\(1 + \((\(f[a + i\ h + h] - f[a + i\ h]\)\/h)\)\^2\)*
              h; \nPrint["\<The \[CapitalDelta]L[i] function is \>", \
\[CapitalDelta]L[i]]; \n
          rateList = 
            Table[{a + i\ h, \[CapitalDelta]L[i]\/h}, {i, 0, 
                n - 1}]; \[IndentingNewLine]Print[\*"\"\<The \!\(\
\[CapitalDelta]L[i]\/\[CapitalDelta]x\) function is \>\"", \
\[CapitalDelta]L[i]\/h]; \n
          p0 = ListPlot[rateList, 
              PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.015]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_i\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]L\_i\/\[CapitalDelta]x\)\>\""}]; \n
          lengthSum[
              k_] := \((\[Sum]\+\(i = 0\)\%\(k - 1\)N[\[CapitalDelta]L[
                    i]])\); \n
          lengthlist = Table[{a + k*h, lengthSum[k]}, {k, 0, n}]; \n
          p1 = ListPlot[lengthlist, PlotRange \[Rule] All, 
              PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.0175]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \
\*"\"\<\!\(L\_total\)(k)\>\""}]; \[IndentingNewLine]L\_exact[
              x_] = \[Integral]\_a\%x\(\@\( 
                        1 + \(f\^\[Prime]\)[u]\^2\)\) \[DifferentialD]u // 
              TrigToExp; \[IndentingNewLine]Print["\<The function Lexact[x] \
is \>", L\_exact[x]]; \[IndentingNewLine]p2 = 
            Plot[L\_exact[x], {x, a, a + n\ h}, PlotRange \[Rule] All, 
              PlotStyle \[Rule] {RGBColor[0, 0, 1]}, 
              AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(L\_exact\)[x]\>\""}, 
              DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[{p2, 
              p1}, AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(L\_exact\)[x] & \
\!\(L\_total\)(k)\>\""}, DisplayFunction \[Rule] $DisplayFunction]; \n
          msecant = 
            Table[{a + k\ h, \(lengthSum[k + 1] - lengthSum[k]\)\/h}, {k, 0, 
                n - 1}]; \n
          p4 = ListPlot[msecant, 
              PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.02]}, 
              AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(m\_secant\
\)\>\""}, DisplayFunction \[Rule] Identity]; \n
          Show[{p4, p0}, 
            AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]L\_i\/\[CapitalDelta]x\) & \!\(m\_secant\)\>\""}, 
            DisplayFunction \[Rule] $DisplayFunction];];\)\)}], "Input",
  Editable->False,
  PageWidth->PaperWidth,
  CellOpen->False,
  InitializationCell->True],

Cell[CellGroupData[{

Cell["Introduction", "Section",
  PageWidth->PaperWidth],

Cell[TextData[{
  "OBJECTIVE: Use Riemann sums to approximate areas, volumes, and lengths of \
arc. Construct accumulation functions, and see how the accumulated quantities \
converge on the antiderivative in the limit. \n\nIn this module, we explore \
the use of Riemann sums to estimate areas, volumes of revolution, and the \
length of a curve. We first write an expression for ",
  Cell[BoxData[
      FormBox[
        SubscriptBox[
          StyleBox[
            RowBox[{"\[CapitalDelta]", 
              StyleBox["Q",
                FontSlant->"Italic"]}]], "i"], TraditionalForm]]],
  ", where ",
  StyleBox["Q",
    FontSlant->"Italic"],
  " is the quantity of interest (area, volume, or arc length); then we look \
at ",
  Cell[BoxData[
      FormBox[
        FractionBox[
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            StyleBox[
              RowBox[{"\[CapitalDelta]", 
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                  FontSlant->"Italic"]}]], "i"], \(\[CapitalDelta]\ x\)], 
        TraditionalForm]]],
  "as the rate at which ",
  StyleBox["Q",
    FontSlant->"Italic"],
  " accumulates with respect to increments in ",
  StyleBox["x",
    FontSlant->"Italic"],
  ";",
  " we form a list of accumulated quantities, ",
  Cell[BoxData[
      \(TraditionalForm\`Q\_i\)]],
  "; and finally, we use the list of ",
  Cell[BoxData[
      \(TraditionalForm\`Q\_i'\)]],
  "s to reconstruct ",
  Cell[BoxData[
      FormBox[
        FractionBox[
          SubscriptBox[
            StyleBox[
              RowBox[{"\[CapitalDelta]", 
                StyleBox["Q",
                  FontSlant->"Italic"]}]], "i"], \(\[CapitalDelta]\ x\)], 
        TraditionalForm]]],
  "."
}], "Text",
  PageWidth->PaperWidth],

Cell[CellGroupData[{

Cell["Technology Guidelines", "Subsection",
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Cell[TextData[{
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    Background->None],
  StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \
on the right cell bracket and double click.",
    CellFrame->True,
    Background->None],
  "\nINITIALIZATION CELLS\n\tWhen asked if you want to \". . . automatically \
evaluate all the initialization cells in the \tnotebook . . . ,\" respond by \
pressing the \"Yes\" button.\nTO STOP AN EXECUTION\n\tSelect the ",
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Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \
anytime to any directory you choose, and it is wise to save often.\n\t \
However, before you do your final save, delete all your output by selecting \
the \n\t ",
  StyleBox["Delete All Output",
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  " selection under the ",
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then shut down ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
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}], "Text",
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}, Closed]]
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Cell[CellGroupData[{

Cell["Part I: Areas", "Section",
  PageWidth->PaperWidth],

Cell[TextData[StyleBox["Chapter 4, Sections 3 and 4",
  FontWeight->"Bold"]], "Text",
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  " In this exercise, \[CapitalDelta]",
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  " represents the area of the ",
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  " rectangle, as shown in the figure above.\n\nIn this Part, we will do the \
following:\n1. Form a function, ",
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  ", that gives the area of the ",
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  " and ",
  StyleBox["h",
    FontSlant->"Italic"],
  ".\n2. Form a rate-of-change function, ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[CapitalDelta]A(i)\)\/\(\[CapitalDelta]\ x\)\)]],
  ", that gives the rate of accumulation of area at ",
  Cell[BoxData[
      \(TraditionalForm\`x = a + i\ h\)]],
  ".\n3. Form a function, ",
  StyleBox["areaSum[k_]",
    FontWeight->"Bold"],
  ", that gives the sum of the areas of the rectangles, that is, the ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]A(i)\)]],
  "'s, as ",
  StyleBox["i",
    FontSlant->"Italic"],
  " ranges from 0 to ",
  StyleBox["k",
    FontSlant->"Italic"],
  ". This approximates the area under the graph of ",
  Cell[BoxData[
      \(TraditionalForm\`y = f(x)\)]],
  ", between ",
  Cell[BoxData[
      \(TraditionalForm\`x = a\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`x = a + k\ h\)]],
  ".\n4. Form a function that gives the slopes of the secant lines on the \
graph of the ",
  Cell[BoxData[
      \(TraditionalForm\`A(k)\)]],
  " function found in step 3, and show that this is the same as the \
rate-of-change function found in step 2. \n5. In the You Try It exercise, you \
explore what happens as the number of rectangles goes to infinity while ",
  StyleBox["h",
    FontSlant->"Italic"],
  " goes to 0.\n\n\nOur first objective is to write \[CapitalDelta]",
  Cell[BoxData[
      FormBox[
        SubscriptBox[
          StyleBox["A",
            FontSlant->"Italic"], "i"], TraditionalForm]]],
  ", the area of the ",
  Cell[BoxData[
      \(TraditionalForm\`i\^th\)]],
  " rectangle, in terms of",
  Cell[BoxData[
      FormBox[
        RowBox[{" ", 
          RowBox[{"f", "(", 
            FormBox[
              SubscriptBox[
                StyleBox["x",
                  FontSlant->"Italic"], "i"],
              "TraditionalForm"], ")"}]}], TraditionalForm]]],
  " and ",
  StyleBox["h ",
    FontSlant->"Italic"],
  "where ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FormBox[
            SubscriptBox[
              StyleBox["x",
                FontSlant->"Italic"], 
              StyleBox["i",
                FontSlant->"Italic"]],
            "TraditionalForm"], " ", "=", " ", \(a + i\ h\)}], 
        TraditionalForm]]],
  StyleBox[". ",
    FontSlant->"Italic"],
  "At first, we let ",
  Cell[BoxData[
      \(TraditionalForm\`a\  = \ \(-5\)\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\ \)\(b\  = \ 10\)\)\)]],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`n = 10\)]],
  "."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[{
    \(\(Clear[f, \[CapitalDelta]A, a, b, h, n];\)\), "\[IndentingNewLine]", 
    \(\(f[x_] = x;\)\), "\[IndentingNewLine]", 
    \(\(a = \(-5\);\)\), "\[IndentingNewLine]", 
    \(\(b = 10;\)\), "\[IndentingNewLine]", 
    \(\(n = 10;\)\), "\[IndentingNewLine]", 
    \(\(h = \((b - a)\)/n;\)\), "\[IndentingNewLine]", 
    \(\[CapitalDelta]A[i_] = f[a + i*h]*h\)}], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Note that if ",
  Cell[BoxData[
      \(TraditionalForm\`f(x\_i)\)]],
  " is negative and ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]\ x = h\)]],
  " is positive, then ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]A\_i\)]],
  " is negative. The areas can be positive, negative, or zero, and so we \
refer to them as \"signed areas.\" Areas above the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-axis are positive, and areas below the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-axis are negative, provided ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]\ x = h\)]],
  " is positive.\n\nNext, we form the rate-of-change function, ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[CapitalDelta]A(i)\)\/\[CapitalDelta]x\)]],
  ". The function ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[CapitalDelta]A(i)\)\/\[CapitalDelta]x\)]],
  " is the rate at which the total signed area of the rectangles accumulates \
with each increment of ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]\ x = h\)]],
  " added to ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". We generate a list of the rate-of-change values for each ",
  Cell[BoxData[
      \(TraditionalForm\`x\_i = a + i\ h\)]],
  ", plot the rate of change versus ",
  Cell[BoxData[
      \(TraditionalForm\`x\_i\)]],
  ", and then assign the graph to the symbol name ",
  StyleBox["p0",
    FontWeight->"Bold"],
  " for later reference."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(rateList = 
      Table[{a + i\ h, \[CapitalDelta]A[i]\/h}, {i, 0, n - 1}]\)], "Input",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p0 = 
        ListPlot[rateList, 
          PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.015]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_i\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]A\_i\/\[CapitalDelta]x\)\>\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "You have probably noticed that, for the area problem, the rate at which \
the total area of the rectangles accumulates with each added increment ",
  StyleBox["h ",
    FontSlant->"Italic"],
  "is simply ",
  Cell[BoxData[
      \(TraditionalForm\`f(x\_i)\)]],
  ", the height of the ",
  Cell[BoxData[
      \(TraditionalForm\`i\^th\)]],
  " rectangle. That is, ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[CapitalDelta]A(i)\)\/\[CapitalDelta]x = 
        f(x\_i)\)]],
  "."
}], "Text",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Now we form a function that calculates the left Riemann sum of the ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]A\_i'\)]],
  "s for the first ",
  StyleBox["k",
    FontSlant->"Italic"],
  " rectangles, where ",
  StyleBox["k",
    FontSlant->"Italic"],
  " can range between 0 and ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  "."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(areaSum[
          k_] := \[Sum]\+\(i = 0\)\%\(k - 1\)\[CapitalDelta]A[
            i];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "For example, we can use ",
  StyleBox["areaSum[ ] ",
    FontWeight->"Bold"],
  " to calculate the total signed area of the first four rectangles."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(areaSum[4]\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Or we can calculate the area of all ten rectangles. (Recall that we set ",
  Cell[BoxData[
      \(TraditionalForm\`n = 10\)]],
  ".)"
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(areaSum[10]\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Next, we form a list of the ordered pairs, in which the first element of \
each ordered pair is the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-coordinate of the left side of the last rectangle in the sum, and the \
second element in each ordered pair is the area of the first ",
  StyleBox["k",
    FontSlant->"Italic"],
  " rectangles."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(arealist = Table[{a + k*h, areaSum[k]}, {k, 0, n}]\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Now we plot the list of points and assign it to the symbol name ",
  Cell[BoxData[
      FormBox[
        StyleBox["p1",
          FontWeight->"Bold"], TraditionalForm]]],
  " for later use."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p1 = 
        ListPlot[arealist, 
          PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.015]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(A\_total\)(k)\
\>\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The function ",
  StyleBox["areaSum[ k ]",
    FontWeight->"Bold"],
  " looks as though it might be a quadratic polynomial, so now we use the ",
  StyleBox["Fit[ ] ",
    FontWeight->"Bold"],
  "command to see if there is a polynomial function that passes through the \
points. "
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(A\_approx[x_] = Fit[arealist, {1, x, x\^2}, x]\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Now we plot ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["A",
              FontWeight->"Bold",
              FontSlant->"Plain"], "approx"], "[", 
          StyleBox["x",
            FontWeight->"Bold",
            FontSlant->"Plain"], "]"}], TraditionalForm]]],
  " and the points in ",
  StyleBox["arealist",
    FontWeight->"Bold"],
  " to show that the points do in fact lie on the graph of the quadratic \
polynomial. First, we plot ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["A",
              FontWeight->"Bold",
              FontSlant->"Plain"], "approx"], "[", 
          StyleBox["x",
            FontWeight->"Bold",
            FontSlant->"Plain"], "]"}], TraditionalForm]]],
  " and then show the two graphs on the same plot."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p2 = 
        Plot[A\_approx[x], {x, a, a + n\ h}, 
          PlotStyle \[Rule] {RGBColor[0, 0, 1]}, 
          AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(A\_approx\)[x]\>\""}];\)\)], \
"Input",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(Show[{p2, p1}, 
        AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(A\_approx\)[x] & \
\!\(A\_total\)(k)\>\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell["The points appear to lie on the curve.", "Text",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The slopes of the secant lines taken from the ",
  StyleBox["areaSum[ k_ ] ",
    FontWeight->"Bold"],
  "graph above give the rate at which the total area of the rectangles \
accumulates with each ",
  StyleBox["h ",
    FontSlant->"Italic"],
  "increment of ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". Now let's calculate the slopes of the secant lines between consecutive \
pairs of points on the graph above and plot them."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(msecant = 
      Table[{a + k\ h, \(areaSum[k + 1] - areaSum[k]\)\/h}, {k, 0, 
          n - 1}]\)], "Input",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p4 = 
        ListPlot[msecant, 
          PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.02]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(m\_secant\)\>\
\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "But this is the same thing as ",
  Cell[BoxData[
      StyleBox[\(\(\[CapitalDelta]A \((i)\)\)\/\[CapitalDelta]x\),
        FontWeight->"Bold"]]],
  "that we calculated at the beginning of this exercise. To confirm this, we \
show ",
  StyleBox["msecants ",
    FontWeight->"Bold"],
  "and ",
  Cell[BoxData[
      StyleBox[\(\(\[CapitalDelta]A \((i)\)\)\/\[CapitalDelta]x\),
        FontWeight->"Bold"]]],
  " together on the same graph."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(Show[{p4, p0}, 
        AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \
\*"\"\<\!\(\[CapitalDelta]A\_i\/\[CapitalDelta]x\) & \
\!\(m\_secant\)\>\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "It looks as though we've come full circle. In the \"",
  "You Try It\" ",
  "exercise that follows, you explore what happens as ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", the number of rectangles, goes to infinity and as ",
  Cell[BoxData[
      \(TraditionalForm\`h \[Rule] 0\)]],
  "."
}], "Text",
  PageWidth->PaperWidth]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[{
  "You Try It: Part I ",
  Cell[BoxData[
      \(TraditionalForm\`-\)]],
  " Taking it to the Limit and Other Functions"
}], "Section",
  PageWidth->PaperWidth],

Cell[TextData[StyleBox["Chapter 4, Sections 3 and 4",
  FontWeight->"Bold"]], "Text",
  PageWidth->PaperWidth],

Cell[TextData[{
  "To help you with these exercises, we copy all of the commands from Part I \
into a single cell, the one that follows. In each exercise, you are asked to \
change some of the red entries and answer the questions. (Note that we have \
hidden all but the input commands and put them in a command called ",
  StyleBox["areas",
    FontWeight->"Bold"],
  ", the last command in the following cell. In addition, we suppress \
displaying the lists of values that are generated because they become too \
large when ",
  StyleBox["n",
    FontSlant->"Italic"],
  " is a big number.)\n\n1. For the function ",
  Cell[BoxData[
      \(TraditionalForm\`f(x) = x\)]],
  ", considered in Part I, increase the number of rectangles, ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", so that the total signed area of the rectangles approaches the total \
signed area under the graph of ",
  Cell[BoxData[
      \(TraditionalForm\`f(x)\)]],
  ". Try ",
  Cell[BoxData[
      \(TraditionalForm\`n = 25, \ 50, \ 100, \ 250, \ 500\)]],
  ". (For larger values of",
  StyleBox[" n",
    FontSlant->"Italic"],
  ", the evaluation of the commands will take a while because of the large \
number of ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]A\_i'\)]],
  "s that must be added to form each value of ",
  StyleBox["areaSum[ k ]",
    FontWeight->"Bold"],
  ". There are more efficient ways to calculate these sums, and these are \
investigated in the module \"",
  "Riemann, Trapezoids, and Simpson,\" included in this supplement.",
  " As ",
  StyleBox["n",
    FontSlant->"Italic"],
  " gets larger, what function is ",
  StyleBox["A[ x ]",
    FontWeight->"Bold"],
  " approaching? Also, as ",
  StyleBox["n ",
    FontSlant->"Italic"],
  "gets larger, ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]\ x = h\)]],
  StyleBox[" ",
    FontSlant->"Italic"],
  "gets closer to 0. What is the ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(lim\)\(\ \)\)\+\(\[CapitalDelta]\ x \[Rule] 
              0\)\) \[CapitalLambda]A\_i\/\(\[CapitalLambda]\ x\)\)]],
  "? In the limit as ",
  Cell[BoxData[
      \(TraditionalForm\`n \[Rule] \[Infinity]\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`h \[Rule] 0\)]],
  ", what is the relationship between ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalLambda]A\_i\/\(\[CapitalLambda]\ x\)\)]],
  "and ",
  StyleBox["A[ x ]",
    FontWeight->"Bold"],
  "?\n\n2. Repeat Exercise 1 for the following: ",
  Cell[BoxData[
      \(TraditionalForm\`f(x) = x\^2, \ a = \(-5\), \ b = 5, \)]],
  " and ",
  StyleBox["fitlist ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`\(\(=\)\({1, \ x, \ x\^2, \ x\^3}\)\)\)]],
  ".\n\n3. Repeat Exercise 1 for the following: ",
  Cell[BoxData[
      \(TraditionalForm\`f(x) = cos\ x, \ a = 0, \ b = 2  \[Pi]\)]],
  ", and ",
  StyleBox["fitlist ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`\(\(=\)\(\ \)\({1, \ Cos[x], \ Sin[x]\ }\)\)\)]],
  ".\n\n4. Repeat Exercise 1 for the following: ",
  Cell[BoxData[
      \(TraditionalForm\`f(x) = e\^x, \ a = \(-1\), \ b = 1, \)]],
  " and ",
  StyleBox["fitlist ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`\(\(=\)\({1, \ E\^x}\)\)\)]],
  ". (Note that in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", Euler's number ",
  StyleBox["e",
    FontSlant->"Italic"],
  " is represented by upper case ",
  StyleBox["E",
    FontSlant->"Italic"],
  ".)"
}], "Text",
  PageWidth->PaperWidth],

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      RowBox[{\(f[x_]\), "=", 
        StyleBox["x",
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    RowBox[{
      RowBox[{"a", "=", 
        StyleBox[\(-5\),
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    RowBox[{
      RowBox[{"b", "=", 
        StyleBox["10",
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      RowBox[{"n", "=", 
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Cell["Part II: Volumes", "Section",
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Cell[TextData[StyleBox["Chapter 4, Sections 3 and 4\nChapter 5, Section 1",
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Cell[TextData[{
  StyleBox["Note:",
    FontWeight->"Bold"],
  " In this exercise, \[CapitalDelta]",
  Cell[BoxData[
      FormBox[
        SubscriptBox[
          StyleBox["V",
            FontSlant->"Italic"], "i"], TraditionalForm]]],
  " represents the volume of the ",
  Cell[BoxData[
      FormBox[
        SuperscriptBox[
          StyleBox["i",
            FontSlant->"Italic"], "th"], TraditionalForm]]],
  " disk, as shown in the figure above.\n\nOur first objective is to form a \
function, ",
  StyleBox["volumeSum[ k ]",
    FontWeight->"Bold"],
  ", that estimates the volume of the solid that is generated when the region \
bounded by the graph of a function ",
  Cell[BoxData[
      \(TraditionalForm\`f(x)\)]],
  ", the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-axis, and the vertical lines ",
  Cell[BoxData[
      \(TraditionalForm\`x\  = \ a\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`x\  = \ \(x\_k = a + k\ h\)\)]],
  " is rotated about the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-axis. We do this by adding the ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]V\_i'\)]],
  "s, the volumes of the first ",
  StyleBox["k",
    FontSlant->"Italic"],
  " disks, where ",
  StyleBox["k",
    FontSlant->"Italic"],
  " can range from 0 up to ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", the number of disks of thickness ",
  StyleBox["h",
    FontSlant->"Italic"],
  " between ",
  Cell[BoxData[
      \(TraditionalForm\`x = a\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`x = b\)]],
  ".\n\nFor the function ",
  StyleBox["f",
    FontSlant->"Italic"],
  "(",
  StyleBox["x",
    FontSlant->"Italic"],
  ") = ",
  StyleBox["x",
    FontSlant->"Italic"],
  ", we write \[CapitalDelta]",
  Cell[BoxData[
      \(TraditionalForm\`V\_i\)]],
  ", the volume of the ",
  Cell[BoxData[
      \(TraditionalForm\`i\^th\)]],
  " disk, in terms of",
  Cell[BoxData[
      FormBox[
        RowBox[{" ", 
          RowBox[{"f", "(", 
            FormBox[
              SubscriptBox[
                StyleBox["x",
                  FontSlant->"Italic"], "i"],
              "TraditionalForm"], ")"}]}], TraditionalForm]]],
  " and ",
  StyleBox["h",
    FontSlant->"Italic"],
  ",",
  StyleBox[" ",
    FontSlant->"Italic"],
  "where ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          FormBox[
            SubscriptBox[
              StyleBox["x",
                FontSlant->"Italic"], 
              StyleBox["i",
                FontSlant->"Italic"]],
            "TraditionalForm"], " ", "=", " ", \(a + i\ h\)}], 
        TraditionalForm]]],
  StyleBox[". ",
    FontSlant->"Italic"],
  "At first, we let ",
  Cell[BoxData[
      \(TraditionalForm\`a\  = \ \(-5\)\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`b = 5\)]],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`n = 10\)]],
  "."
}], "Text",
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Cell[BoxData[{
    \(\(Clear[f, \[CapitalDelta]V, a, b, h, n];\)\), "\[IndentingNewLine]", 
    \(\(f[x_] = x;\)\), "\[IndentingNewLine]", 
    \(\(a = \(-5\);\)\), "\[IndentingNewLine]", 
    \(\(b = 5;\)\), "\[IndentingNewLine]", 
    \(\(n = 10;\)\), "\[IndentingNewLine]", 
    \(\(h = \((b - a)\)/n;\)\), "\[IndentingNewLine]", 
    \(\[CapitalDelta]V[i_] = \[Pi]\ \((f[a + i*h])\)\^2*h\)}], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The function ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[CapitalDelta]V(i)\)\/\[CapitalDelta]x\)]],
  " is the rate at which the total volume of the solid accumulates with each \
increment of ",
  StyleBox["h",
    FontSlant->"Italic"],
  " added to ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". We generate a list of the rate-of-change values for each ",
  Cell[BoxData[
      \(TraditionalForm\`x\_i = a + i\ h\)]],
  ", plot the rate of change versus ",
  Cell[BoxData[
      \(TraditionalForm\`x\_i\)]],
  ", and then assign the graph to the symbol name ",
  StyleBox["p0",
    FontWeight->"Bold"],
  " for later reference."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(rateList = 
      Table[{a + i\ h, \[CapitalDelta]V[i]\/h}, {i, 0, n - 1}]\)], "Input",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p0 = 
        ListPlot[rateList, 
          PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.0175]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_i\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]V\_i\/\[CapitalDelta]x\)\>\""}];\)\)], "Input",
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Cell[TextData[{
  "You have probably noticed that, for the volume problem, the rate at which \
the total volume of the disks accumulates with each added increment ",
  StyleBox["h ",
    FontSlant->"Italic"],
  "is ",
  Cell[BoxData[
      \(TraditionalForm\`\[Pi]\ \(f(x\_i)\)\^2\)]],
  ", the area of the circular end of the ",
  Cell[BoxData[
      \(TraditionalForm\`i\^th\)]],
  " disk. That is, ",
  Cell[BoxData[
      FormBox[
        RowBox[{\(\(\[CapitalDelta]V(i)\)\/\[CapitalDelta]x\), "=", " ", 
          FormBox[\(\[Pi]\ \(f(x\_i)\)\^2\),
            "TraditionalForm"]}], TraditionalForm]]],
  "."
}], "Text",
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Cell[TextData[{
  "Now we form a function that calculates the left Riemann sum of the ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]V\_i'\)]],
  "s for the first ",
  StyleBox["k",
    FontSlant->"Italic"],
  " disks where ",
  StyleBox["k",
    FontSlant->"Italic"],
  " can range between 0 and ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  "."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(volumeSum[
          k_] := \[Sum]\+\(i = 0\)\%\(k - 1\)\[CapitalDelta]V[
            i];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "We form a list of the ordered pairs, in which the first element of each \
ordered pair is the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-coordinate of the left side of the last disk in the sum, and the second \
element in each ordered pair is the volume of the first ",
  StyleBox["k",
    FontSlant->"Italic"],
  " disks."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(volumelist = Table[{a + k*h, volumeSum[k]}, {k, 0, n}]\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Let's plot the list of points and assign it to the symbol name ",
  Cell[BoxData[
      FormBox[
        StyleBox["p1",
          FontWeight->"Bold"], TraditionalForm]]],
  " for later use."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p1 = 
        ListPlot[volumelist, 
          PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.0175]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(V\_total\)(k)\
\>\""}, PlotRange \[Rule] All];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The function ",
  StyleBox["volumeSum[ k ]",
    FontWeight->"Bold"],
  " appears to be a cubic polynomial, so now we use the ",
  StyleBox["Fit[ ] ",
    FontWeight->"Bold"],
  "command to see if there is a cubic function that passes through the \
points. "
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(V\_approx[x_] = Fit[volumelist, {1, x, x\^2, x\^3}, x]\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "Now we plot ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["V",
              FontWeight->"Bold",
              FontSlant->"Plain"], "approx"], "[", 
          StyleBox["x",
            FontWeight->"Bold",
            FontSlant->"Plain"], "]"}], TraditionalForm]]],
  " and the points in ",
  StyleBox["volumelist",
    FontWeight->"Bold"],
  " to show that the points do in fact lie on the graph of the cubic \
polynomial. First, we plot ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          SubscriptBox[
            StyleBox["V",
              FontWeight->"Bold",
              FontSlant->"Plain"], "approx"], "[", 
          StyleBox["x",
            FontWeight->"Bold",
            FontSlant->"Plain"], "]"}], TraditionalForm]]],
  " and then show the two graphs on the same plot."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p2 = 
        Plot[V\_approx[x], {x, a, a + n\ h}, PlotRange \[Rule] All, 
          PlotStyle \[Rule] {RGBColor[0, 0, 1]}, 
          AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(V\_approx\)[x]\>\""}];\)\)], \
"Input",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(Show[{p2, p1}, 
        AxesLabel \[Rule] {"\<x\>", \*"\"\<\!\(V\_approx\)[x] & \
\!\(V\_total\)(k)\>\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell["The points appear to lie on the curve.", "Text",
  PageWidth->PaperWidth],

Cell[TextData[{
  "The slopes of the secant lines taken from the ",
  StyleBox["volumeSum[ k_ ] ",
    FontWeight->"Bold"],
  "graph give the rate at which the total area of the rectangles accumulates \
with each ",
  StyleBox["h ",
    FontSlant->"Italic"],
  "increment of ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". Now let's calculate the slopes of the secant lines between consecutive \
pairs of points on the graph above and plot them."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(msecants = 
      Table[{a + k\ h, \(volumeSum[k + 1] - volumeSum[k]\)\/h}, {k, 0, 
          n - 1}]\)], "Input",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(p4 = 
        ListPlot[msecants, 
          PlotStyle \[Rule] {RGBColor[1, 0, 0], PointSize[0.0225]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \*"\"\<\!\(m\_secant\)\>\
\""}];\)\)], "Input",
  PageWidth->PaperWidth],

Cell[TextData[{
  "But this is the same thing as ",
  Cell[BoxData[
      StyleBox[\(\(\[CapitalDelta]V \((i)\)\)\/\[CapitalDelta]x\),
        FontWeight->"Bold"]]],
  "that we calculated at the beginning of this exercise. To confirm this, we \
show ",
  StyleBox["msecants ",
    FontWeight->"Bold"],
  "and ",
  Cell[BoxData[
      StyleBox[\(\(\[CapitalDelta]V \((i)\)\)\/\[CapitalDelta]x\),
        FontWeight->"Bold"]]],
  " together on the same graph."
}], "Text",
  PageWidth->PaperWidth],

Cell[BoxData[
    \(\(Show[{p4, p0}, 
        AxesLabel \[Rule] {\*"\"\<\!\(x\_k\)\>\"", \
\*"\"\<\!\(\[CapitalDelta]V\_i\/\[CapitalDelta]x\) & \!\(m\_secants\)\>\""}];\
\)\)], "Input",
  PageWidth->PaperWidth],

Cell["It looks as though we've come full circle again.", "Text",
  PageWidth->PaperWidth]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[{
  "You Try It: Part II ",
  Cell[BoxData[
      \(TraditionalForm\`-\)]],
  " Taking it to the Limit and Another Function"
}], "Section",
  PageWidth->PaperWidth],

Cell["\<\
Chapter 4, Sections 3 and 4
Chapter 5, Section 1\
\>", "Text",
  PageWidth->PaperWidth,
  FontWeight->"Bold"],

Cell[TextData[{
  "To help you with these exercises, we again copy all of the commands from \
Part II into a single cell, the one that follows. In each exercise, you are \
asked to change some of the red entries and answer the questions. (Note that \
we have hidden all but the input commands and put them in a command called ",
  StyleBox["volumes",
    FontWeight->"Bold"],
  ", the last command in the following cell. In addition, we suppress \
displaying the lists of values that are generated because they become too \
large when ",
  StyleBox["n",
    FontSlant->"Italic"],
  " is a big number.)\n\n1. For the function ",
  Cell[BoxData[
      \(TraditionalForm\`f(x) = x\)]],
  ", considered in Part II, increase ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", the number of disks between ",
  Cell[BoxData[
      \(TraditionalForm\`x = a\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`x = b\)]],
  ", so that the total volume of the disks approaches the volume of \
revolution. Try ",
  Cell[BoxData[
      \(TraditionalForm\`n = 25, \ 50, \ 100, \ 250, \ 500\)]],
  ". (For larger values of",
  StyleBox[" n",
    FontSlant->"Italic"],
  ", the evaluation of the commands will take a while because of the large \
number of ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]V\_i'\)]],
  "s that must be added to form each value of ",
  StyleBox["volumeSum[ k ]",
    FontWeight->"Bold"],
  ".) As ",
  StyleBox["n",
    FontSlant->"Italic"],
  " gets larger, what function does ",
  StyleBox["V[ x ]",
    FontWeight->"Bold"],
  " appear to be approaching? Also, as ",
  StyleBox["n ",
    FontSlant->"Italic"],
  "gets larger, ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalDelta]\ x = h\)]],
  StyleBox[" ",
    FontSlant->"Italic"],
  "gets closer to 0. What is the ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(lim\)\(\ \)\)\+\(\[CapitalDelta]\ x \[Rule] 
              0\)\) \[CapitalLambda]V\_i\/\(\[CapitalLambda]\ x\)\)]],
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  Cell[BoxData[
      \(TraditionalForm\`n \[Rule] \[Infinity]\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`h \[Rule] 0\)]],
  ", what is the relationship between ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalLambda]V\_i\/\(\[CapitalLambda]\ x\)\)]],
  "and ",
  StyleBox["V[ x ]",
    FontWeight->"Bold"],
  "?\n\n2. Repeat Exercise 1 for the following: ",
  Cell[BoxData[
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  " and ",
  StyleBox["fitlist ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`\(\(=\)\({1, x, sin\ 2  x, cos\ 2  x}\)\)\)]],
  ". Explain the stair-step pattern in the graph of ",
  StyleBox["V[x]",
    FontWeight->"Bold"],
  ",",
  " and explain why we chose the functions in ",
  StyleBox["fitlist",
    FontWeight->"Bold"],
  " for the ",
  StyleBox["Fit[ ]",
    FontWeight->"Bold"],
  " command."
}], "Text",
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Cell[BoxData[{\(Clear[f, \[CapitalDelta]V, a, b, h, n];\), "\n", 
    RowBox[{
      RowBox[{\(f[x_]\), "=", 
        StyleBox["x",
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    RowBox[{
      RowBox[{"a", "=", 
        StyleBox[\(-5\),
          FontColor->RGBColor[1, 0, 0]]}], ";"}], "\n", 
    RowBox[{
      RowBox[{"b", "=", 
        StyleBox["5",
          FontColor->RGBColor[1, 0, 0]]}], ";"}], "\n", 
    RowBox[{
      RowBox[{"n", "=", 
        StyleBox["10",
          FontColor->RGBColor[1, 0, 0]]}], ";"}], "\[IndentingNewLine]", 
    RowBox[{
      RowBox[{"fitlist", "=", 
        RowBox[{"{", 
          StyleBox[\(1, x, x\^2, x\^3\),
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Cell[CellGroupData[{

Cell["Part III: Lengths of Arc", "Section",
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Cell[TextData[{
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  " In this exercise, \[CapitalDelta]",
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  " represents the length of the ",
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Cell[TextData[{
  "Our first objective is to form a function, ",
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  ", that estimates the length of the graph of a function, ",
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  Cell[BoxData[
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  "s, the lengths of the first ",
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  StyleBox["k",
    FontSlant->"Italic"],
  " can range from 0 up to ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", the number of line segments between ",
  Cell[BoxData[
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  Cell[BoxData[
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  "(",
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  StyleBox["a",
    FontSlant->"Italic"],
  " + ",
  StyleBox["ih",
    FontSlant->"Italic"],
  ". To do this, we use the Pythagorean theorem to calculate the length of \
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      \(TraditionalForm\`\((x\_i, \ f(x\_i))\)\)]],
  " and ends at the point ",
  Cell[BoxData[
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  Cell[BoxData[
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                  SubscriptBox[
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                                SubscriptBox[
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                            RowBox[{
                              StyleBox["f",
                                FontSlant->"Italic"], 
                              StyleBox["(",
                                FontSlant->"Italic"], \(x\_i\), ")"}]}], "h"],
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  StyleBox[".",
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  "\n\nNow let's put it into ",
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    FontSlant->"Italic"],
  "code. We let ",
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  ", ",
  Cell[BoxData[
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  "."
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Cell[BoxData[{
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    \(\(f[x_] = x\^2;\)\), "\[IndentingNewLine]", 
    \(\(a = \(-5\);\)\), "\[IndentingNewLine]", 
    \(\(b = 5;\)\), "\[IndentingNewLine]", 
    \(\(n = 10;\)\), "\[IndentingNewLine]", 
    \(\(h = \((b - a)\)/n;\)\), "\[IndentingNewLine]", 
    \(\[CapitalDelta]L[i_] = 
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Cell[TextData[{
  "The function ",
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with each increment of ",
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    FontSlant->"Italic"],
  " added to ",
  StyleBox["x",
    FontSlant->"Italic"],
  ". We generate a list of the rate-of-change values for each ",
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      \(TraditionalForm\`x\_i = a + i\ h\)]],
  ", plot the rate of change versus ",
  Cell[BoxData[
      \(TraditionalForm\`x\_i\)]],
  ", and then assign the graph to the symbol name ",
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Cell[BoxData[
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Cell[BoxData[
    \(\(p0 = 
        ListPlot[rateList, PlotRange \[Rule] {0, Max[rateList]}, 
          PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.02]}, 
          AxesLabel \[Rule] {\*"\"\<\!\(x\_i\)\>\"", \*"\"\<\!\(\
\[CapitalDelta]L\_i\/\[CapitalDelta]x\)\>\""}];\)\)], "Input",
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Cell[TextData[{
  "The rate at which the total length of the line segments accumulates with \
each added increment ",
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  "is \n\n",
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                      StyleBox["(",
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                      RowBox[{
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                      RowBox[{
                        StyleBox["f",
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Cell[TextData[{
  "Now we form a function that calculates the left Riemann sum of the ",
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            StyleBox[
              RowBox[{"\[CapitalDelta]", 
                StyleBox["L",
                  FontSlant->"Italic"]}]], "i"], "'"}], TraditionalForm]]],
  "s for the first ",
  StyleBox["k",
    FontSlant->"Italic"],
  " line segments where ",
  StyleBox["k",
    FontSlant->"Italic"],
  " can range between 0 and ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  "."
}], "Text",
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Cell[BoxData[
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Cell[TextData[{
  "We form a list of the ordered pairs, in which the first element of each \
ordered pair is the ",
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    FontSlant->"Italic"],
  "-coordinate of the left side of the last line segment in the sum, and the \
second element in each ordered pair is the length of the first ",
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    FontSlant->"Italic"],
  " line segments."
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Cell[BoxData[
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Cell[TextData[{
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