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Notebook[{
Cell["Math 225: Calculus III    Quiz 8    4/1&3/97", "Subsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData["<<Calculus`Math225`"], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "Set up a triple integral in cylindrical coordinates that gives the \
integral of the function ",
  StyleBox["f[x_,y_,z_] = x y", "Input"],
  StyleBox[" over the solid region between the paraboloid ", "Text"],
  StyleBox["2x^2 + y^2 + z == 8", "Input"],
  StyleBox[" and the parabolic sheet ", "Text"],
  StyleBox["z = y^2. ", "Input"],
  StyleBox["(Do not evaluate the integral).", "Text"]
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  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData["Solution"], "Subsubsection",
  Evaluatable->False,
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Cell[TextData[{
  "The top surface is ",
  StyleBox["z == 8 - 2x^2 - y^2", "Input"],
  " and the bottom surface is ",
  StyleBox["z == y^2", "Input"],
  ". The common region over which these surfaces are graphs is given by the \
intersection equation,\n",
  StyleBox["    8 - 2x^2 - y^2 == z == y^2\n", "Input"],
  "or ",
  StyleBox["x^2 + y^2 == 4", "Input"],
  ",  a circle of radius ",
  StyleBox["2", "Input"],
  ". Switching to cylindrical coordinates, we must replace ",
  StyleBox["x", "Input"],
  " by ",
  StyleBox["r Cos[t]", "Input"],
  " and ",
  StyleBox["y", "Input"],
  " by ",
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  " and add an extra factor of ",
  StyleBox["r", "Input"],
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  StyleBox["2", "Input"],
  " is described by ",
  StyleBox["0 <= r <= 2", "Input"],
  ", ",
  StyleBox["0 <= t <= 2Pi", "Input"],
  "."
}], "Text"],

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Cell["\<\
Integrate[ r Cos[t] r Sin[t] r, {t, 0, 2Pi}, {r, 0, 2},
{z, r^2Sin[t]^2, 8-2r^2Cos[t]^2-r^2Sin[t]^2}]\
\>", "Input",
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Cell[TextData[{
  "This value ",
  StyleBox["0", "Input"],
  " is easy to predict, since by symmetry, for every point, say ",
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  ", in the solid that gives a positive value of the funtion ",
  StyleBox["f[a,b] == a b", "Input"],
  " there is another point in the region, ",
  StyleBox["{-a,b}", "Input"],
  ",  that gives the  opposite negative value, ",
  StyleBox["f[-a,b] == -a b", "Input"],
  ". Although you did not have to do the integral, it was straightforward to \
carry out:"
}], "Text"],

Cell["\<\
Integrate[ r Cos[t] r Sin[t] r,
{z, r^2Sin[t]^2, 8-2r^2Cos[t]^2-r^2Sin[t]^2}]\
\>", "Input",
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["%//Simplify", "Input"],

Cell[BoxData[
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Cell["Integrate[%,{r, 0, 2}]", "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[TextData[{
  "Set up a triple integral in spherical coordinates that gives the integral \
of the function ",
  StyleBox["f[x_,y_,z_] = x y", "Input"],
  StyleBox[" over the solid region inside the  sphere ", "Text"],
  StyleBox["x^2 + y^2 + z^2 == 9", "Input"],
  StyleBox[" below the ", "Text"],
  StyleBox["xy", "Input"],
  StyleBox["-plane", "Text"],
  StyleBox[". ", "Input"],
  StyleBox["(Do not evaluate the integral).", "Text"]
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  CellMargins->{{Inherited, 112}, {Inherited, Inherited}},
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData["Solution"], "Subsubsection",
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Cell["sph = {x->r Cos[t]Sin[p], y-> r Sin[t]Sin[p], z-> r Cos[p]};", "Input"],

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