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Cell[CellGroupData[{
Cell[" Project - Riemann Sum", "Title",
 CellChangeTimes->{{3.449681240091041*^9, 3.449681247628324*^9}}],

Cell["\<\
In this project you will use some of the graphics primitives to create a \
function that produces a nice animation illustrating the connection between \
Riemann sums, the area underneath the curve and the definite integral.\
\>", "Text",
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell["Overview", "Subsection"],

Cell[TextData[{
 "When introducing integration, one starts with the idea of the area \
underneath a curve. To approximate this area, one draws a sequence of \
approximating rectangles, whose height is related to the function. The \
general approach is to divide the interval [a, b], for which the area under \
the curve is to be determined, into ",
 StyleBox["n",
  FontSlant->"Italic"],
 " equally long intervals, each of length (",
 StyleBox["b-a",
  FontSlant->"Italic"],
 ")",
 StyleBox["/n",
  FontSlant->"Italic"],
 ". These subintervals form the base of their respective rectangle. Then one \
of three ways can be chosen to determine the height of the rectangle:\n\t1) \
function value at left endpoint of the subinterval\n\t2) function value at \
the midpoint of the subinterval\n\t3) function value at the right endpoint of \
the subinterval.\n\nIn each case, the area under the curve is approximated by \
the sum of the areas of the rectangles. As ",
 StyleBox["n\[Rule]\[Infinity]",
  FontSlant->"Italic"],
 ", the Riemann sum ",
 Cell[BoxData[
  FormBox[
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     RowBox[{"k", "=", "0"}], "n"], 
    RowBox[{
     RowBox[{"f", "(", 
      SubsuperscriptBox["x", "k", "*"], ")"}], 
     SubscriptBox["\[CapitalDelta]x", 
      RowBox[{"k", "  "}]]}]}], TraditionalForm]]],
 "approaches the area under the curve, and this limit is defined as the \
integral ",
 Cell[BoxData[
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    SubsuperscriptBox["\[Integral]", "a", "b"], 
    RowBox[{
     RowBox[{"f", "(", "x", ")"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]],
 ".\n\nYou are to write a function that has as inputs the interval limits ",
 StyleBox["a",
  FontSlant->"Italic"],
 " and ",
 StyleBox["b",
  FontSlant->"Italic"],
 ".  You can assume that the function ",
 StyleBox["f",
  FontSlant->"Italic"],
 " itself is defined before the function ",
 StyleBox["RiemannAnimation",
  FontWeight->"Bold"],
 " is called. Your function will use the left endpoint method. As output it \
will print a table which indicates the value of ",
 StyleBox["n ",
  FontSlant->"Italic"],
 "together with the approximate area and the difference \[CapitalDelta] \
between the approximation and the actual area, where the latter is computed \
from the integral (and is the same for all values of ",
 StyleBox["n",
  FontSlant->"Italic"],
 "). \n\n\t\t",
 Cell[BoxData[GridBox[{
    {"n", 
     RowBox[{"approx", ".", " ", "area"}], "\[CapitalDelta]"},
    {"10", "\[Placeholder]", "\[Placeholder]"},
    {"20", "\[Placeholder]", "\[Placeholder]"},
    {"\[VerticalEllipsis]", "\[Placeholder]", "\[Placeholder]"}
   },
   GridBoxDividers->{
    "Columns" -> {{False}}, "ColumnsIndexed" -> {}, "Rows" -> {False, 
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 "\n\t\t\nSecondly, your function should create 10 graphs (for ",
 StyleBox["n",
  FontSlant->"Italic"],
 " = 10, 20,..., 100) of the function together with the approximating \
rectangles  to be used for an animation. "
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Cell["\<\
Details (possible way to break down the task, but you may choose to do it \
differently)\
\>", "Subsection",
 CellChangeTimes->{{3.41876613122019*^9, 3.418766148593437*^9}}],

Cell["\<\
The project can be broken down into these smaller steps, which result from \
thinking about which pieces are needed to complete the whole.\
\>", "Text"],

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Cell["Step 1: Creating the subintervals", "Subsubsection"],

Cell[TextData[{
 "When drawing the rectangles under the function, we usually divide the given \
interval [a, b] into ",
 StyleBox["n",
  FontSlant->"Italic"],
 " subintervals\n\n\t[",
 Cell[BoxData[
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 ", ",
 Cell[BoxData[
  FormBox[
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 "], [",
 Cell[BoxData[
  FormBox[
   SubscriptBox["x", "1"], TraditionalForm]]],
 ", ",
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   SubscriptBox["x", "2"], TraditionalForm]]],
 "], [",
 Cell[BoxData[
  FormBox[
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 ", ",
 Cell[BoxData[
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 "], ....,  [",
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    RowBox[{"n", "-", "2"}]], TraditionalForm]]],
 ", ",
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 "], [",
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 ", ",
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 "], \n\nwhere ",
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 "= a, and ",
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 " Create a list of subintervals for given ",
 StyleBox["a, b, ",
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 "and ",
 StyleBox["n",
  FontSlant->"Italic"],
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 "       "
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Cell["Step 2: Creating a single rectangle", "Subsubsection"],

Cell[TextData[{
 "Now that the endpoints of the subintervals have been computed, we need to \
create a rectangle for each subinterval. The width of the rectangle is the \
length of each subinterval, i.e.  ",
 StyleBox["w",
  FontSlant->"Italic"],
 " = ",
 Cell[BoxData[
  FormBox[
   StyleBox[
    FractionBox[
     RowBox[{"b", "-", "a"}], "n"],
    FontSize->15], TraditionalForm]]],
 ",  and the height is determined by the method. Let's consider the \
subinterval with endpoints ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["x", 
    RowBox[{"i", " "}]], TraditionalForm]]],
 "and ",
 Cell[BoxData[
  FormBox[
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     RowBox[{"i", "+", "1"}]], "."}], TraditionalForm]]],
 " The height ",
 StyleBox["h",
  FontSlant->"Italic"],
 " of the rectangle is given by ",
 StyleBox["h = f(",
  FontSlant->"Italic"],
 Cell[BoxData[
  FormBox[
   SubscriptBox["x", "i"], TraditionalForm]]],
 ").\n\nWrite a function that draws a rectangle of width ",
 StyleBox["w",
  FontSlant->"Italic"],
 " and height ",
 StyleBox["f(",
  FontSlant->"Italic"],
 Cell[BoxData[
  FormBox[
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 ") above the interval [",
 Cell[BoxData[
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     RowBox[{
      SubscriptBox["x", "i"], ",", 
      SubscriptBox["x", 
       RowBox[{"i", "+", "1"}]]}], "]"}], " "}], TraditionalForm]]],
 " in a color of your choice. You should use the position of the lower left \
corner of the rectangle and the height of the rectangle as variables for your \
function, and assume that the other quantities (",
 StyleBox["f, a, b,",
  FontSlant->"Italic"],
 " and ",
 StyleBox["n",
  FontSlant->"Italic"],
 ") are known outside of your function."
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  StyleBox[
   RowBox[{
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Cell["Step 3: Creating all the rectangles", "Subsubsection"],

Cell[TextData[{
 "Once your function ",
 StyleBox["drawRectangle",
  FontWeight->"Bold"],
 " works, you can use the built-in function ",
 StyleBox["Map",
  FontWeight->"Bold"],
 " to apply the function easily to a list consisting of pairs {left \
endpoint,",
 StyleBox[" f",
  FontSlant->"Italic"],
 "(left endpoint)} (which you can derive from the list of all the endpoints \
created in Step 1). You have now created the rectangles that approximate the \
area under the curve. Give this graphics object a name."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Step 4: Combining the rectangle graph with a graph of the function and \
computing the approximate area\
\>", "Subsubsection"],

Cell[TextData[{
 "Combine the above graphics object with a graph of the function, which \
should be drawn slightly thicker in a color of your choice. Also compute the \
total area for all the rectangles, given by ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    RowBox[{"w", 
     RowBox[{
      UnderoverscriptBox["\[Sum]", 
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 " which is the approximate area under the curve. You have now created all \
the pieces for a function that uses the left endpoint method for \
approximation."
}], "Text",
 CellChangeTimes->{{3.4187663215306187`*^9, 3.418766337594346*^9}}]
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Cell["Step 5: Putting the pieces together", "Subsubsection"],

Cell[TextData[{
 "Now you can combine all the different pieces into a function called ",
 StyleBox["Riemann Animation[a_,b_]",
  FontWeight->"Bold"],
 "  which uses the left endpoint method to create a sequence of graphs of the \
function together with the approximating rectangles, and also prints out the \
table described above. You can assume that the function ",
 StyleBox["f",
  FontSlant->"Italic"],
 " is given outside the function ",
 StyleBox["Riemann Animation[a_,b_]",
  FontWeight->"Bold"],
 "."
}], "Text",
 CellChangeTimes->{{3.418766355410693*^9, 3.4187663767003098`*^9}, {
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Cell["Step 6: Usage statement and error checking", "Subsubsection",
 CellChangeTimes->{{3.4187663851506777`*^9, 3.4187663861411057`*^9}}],

Cell[TextData[{
 "Write a usage statement for the function ",
 StyleBox["RiemannAnimation ",
  FontWeight->"Bold"],
 "and make sure your function behaves properly when given bad input (i.e., \
the lower limit ",
 StyleBox["a ",
  FontSlant->"Italic"],
 "is bigger than upper limit ",
 StyleBox["b",
  FontSlant->"Italic"],
 ") or gives an error statement in that case. "
}], "Text",
 CellChangeTimes->{
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