(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 38171, 1175]*) (*NotebookOutlinePosition[ 38879, 1200]*) (* CellTagsIndexPosition[ 38835, 1196]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Multiple (Iterated) Integrals", "Title", TextAlignment->Center, TextJustification->0, Background->RGBColor[0, 1, 0]], Cell["\<\ This notebook is by Steven Amgott. Please send any questions or comments to \ samgott1@swarthmore.edu. Feel free to use and distribute this notebook, but \ keep this author information in any copy you use or distribute.\ \>", "SmallText"], Cell[TextData[{ "In general, in this notebook, anything in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " is something you can, and possibly should, change. In order to easily \ set up double and triple integrals, it might be useful to open the ", StyleBox["MultivariableCalculus", FontColor->RGBColor[0, 0, 1]], " palette, if it is available. Look in ", StyleBox["File...Palettes", FontColor->RGBColor[0, 0, 1]], " to see if it is there. If not, you can copy or open it from the ", StyleBox["Natural Sciences + Engineering...Mathematics & \ Statistics...Mathematica Stuff...Extra Palettes", FontColor->RGBColor[0, 0, 1]], " folder in ", StyleBox["Class Folders", FontColor->RGBColor[0, 0, 1]], " (if connected to the Swarthmore network), or download it from ", ButtonBox["http://www.swarthmore.edu/NatSci/samgott1/Mathematica/Extra_\ Palettes.html", ButtonData:>{ URL[ "http://www.swarthmore.edu/NatSci/samgott1/Mathematica/Extra_Palettes.\ html"], None}, ButtonStyle->"Hyperlink"], ". " }], "Text"], Cell["\<\ The cell below is an initialization cell which turns off a few annoying \ warning messages.\ \>", "Text"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\n", \(\(Off[General::"\"];\)\)}], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["Double Integrals", "Section"], Cell[TextData[{ "There are several ways to get ", StyleBox["Mathematica", FontSlant->"Italic"], " to do a double integral. The easiest is to use the double integral \ button (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), ScriptLevel->0], RowBox[{"\[Placeholder]", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]\[Placeholder]\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]\[Placeholder]\), ScriptLevel->0]}]}]}], TraditionalForm]]], ") on the ", StyleBox["MultivariableCalculus", FontColor->RGBColor[0, 0, 1]], " palette. If you click this button it pastes a copy of the symbol at your \ cursor location in your notebook. You can then fill in the boxes with the \ appropriate information. (Use the mouse or the ", StyleBox["Tab", FontColor->RGBColor[1, 0, 0]], " key to cycle around the boxes in order to fill them in.) Here is an \ example." }], "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%1\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_0\%x\), ScriptLevel->0], RowBox[{\(x\^2\), " ", "y", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]}]], "Input"], Cell["\<\ Notice that the order of integration in the above example is y first, then x. \ The rightmost integral sign refers to the first of the integration \ variables, in this case y. This is the same as using the parenthesized \ version in the next cell.\ \>", "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%1\), ScriptLevel->0], RowBox[{ RowBox[{"(", RowBox[{ StyleBox[\(\[Integral]\_0\%x\), ScriptLevel->0], RowBox[{\(x\^2\), " ", "y", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0]}]}], StyleBox[")", ScriptLevel->0]}], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]], "Input"], Cell[TextData[{ "If you only have the ", StyleBox["Basic Input", FontColor->RGBColor[0, 0, 1]], " palette available, you can piece together the double integral symbol by \ clicking the ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), RowBox[{ StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], \(\[DifferentialD]\[Placeholder]\)}]}], TraditionalForm]]], " button, placing the cursor in the box for the function (marked in red in \ this sentence), and clicking the symbol ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\[Placeholder]\%\[Placeholder] \ \[Placeholder] \[DifferentialD]\[Placeholder]\)]], " once again." }], "Text"], Cell[TextData[{ "You can also get ", StyleBox["Mathematica", FontSlant->"Italic"], " to do the integral without palette symbols. Here is the syntax." }], "Text"], Cell[BoxData[ \(Integrate[x^2\ y, {x, 0, 1}, {y, 0, x}]\)], "Input"], Cell["\<\ Notice that in this case the second set of braces {y,0,x} corresponds to the \ first integration variable, and the first set {x,0,1} corresponds to the \ second integration variable. Be careful of the ordering of this integration \ variable information.\ \>", "Text"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find the area of the upper half of a unit circle (", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " = 1) by setting up a double integral. Do so in an input cell immediately \ following this one. Remember, to find the area of a region using double \ integrals, you integrate the function f(x,y) = 1 over the region." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Sometimes ", StyleBox["Mathematica", FontSlant->"Italic"], " might not be able to compute an exact answer to a double integral. In \ that case, it might return a single integral of a messy looking function or \ simply the original integral (unevaluated) . Here are two examples." }], "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%1\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_0\%x\), ScriptLevel->0], RowBox[{"x", " ", \(\[ExponentialE]\^\((y\^3)\)\), StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%1\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_0\%y\), ScriptLevel->0], RowBox[{\((\@Sin[\(x + x\^3\)\/2])\), StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0]}]}]}]], "Input"], Cell[TextData[{ "You could always try to use ", StyleBox["N", FontColor->RGBColor[1, 0, 0]], " to get a numerical answer. (Make sure the cell above this one is the \ last cell you evaluated before evaluating the cell below this one.)" }], "Text"], Cell[BoxData[ \(N[%]\)], "Input"], Cell[TextData[{ "You can also get a numerical answer more quickly using ", StyleBox["NIntegrate", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(NIntegrate[\@Sin[\(x + x\^3\)\/2], {y, 0, 1}, {x, 0, y}]\)], "Input"], Cell["Evaluate the next three cells to see the difference in time.", "Text"], Cell[BoxData[ RowBox[{"Timing", "[", RowBox[{"N", "[", RowBox[{ StyleBox[\(\[Integral]\_0\%1\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_0\%x\), ScriptLevel->0], RowBox[{"x", " ", \(\[ExponentialE]\^\((y\^3)\)\), StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]}], StyleBox["]", ScriptLevel->0]}], StyleBox["]", ScriptLevel->0]}]], "Input"], Cell[BoxData[ \(Timing[ N[Integrate[ x\ \[ExponentialE]\^\((y\^3)\), {x, 0, 1}, {y, 0, x}]]]\)], "Input"], Cell[BoxData[ \(Timing[ NIntegrate[ x\ \[ExponentialE]\^\((y\^3)\), {x, 0, 1}, {y, 0, x}]]\)], "Input"], Cell[TextData[{ "The reason ", StyleBox["NIntegrate", FontColor->RGBColor[1, 0, 0]], " is much faster is that ", StyleBox["Mathematica", FontSlant->"Italic"], " automatically goes to numerical methods to evaluate the integral, rather \ than trying symbolic methods first. The second cell is there to convince you \ that working with or without palette symbols doesn't make much difference in \ the timing. For some integrals, using ", StyleBox["NIntegrate", FontColor->RGBColor[1, 0, 0]], " to save time can be very important when only a numerical answer is \ needed." }], "Text"], Cell[CellGroupData[{ Cell["Center of Mass and Moments", "Subsection"], Cell[TextData[{ "For a region in the plane whose (area) density is given by the function \ \[Delta](x,y), the ", StyleBox["first moment with respect to the y-axis", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["y", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], ", is defined as the double integral over the region of the function x \ \[Delta](x,y). (The x is there as the distance from an arbitrary point in \ the region to the y-axis.) Similarly the ", StyleBox["first moment with respect to the x-axis", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["x", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], ", is the double integral over the region of the function y \[Delta](x,y). \ The ", StyleBox["x-coordinate of the center of mass", FontWeight->"Bold"], " of the region, ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["x", FontSlant->"Plain"], "_"], TraditionalForm]]], ", can be found by dividing ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["y", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], " by the mass M of the region (which is simply the double integral over the \ region of the density function \[Delta](x,y)). Similarly, ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["y", FontSlant->"Plain"], "_"], TraditionalForm]]], " =", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["x", FontSlant->"Plain"]], FontSlant->"Italic"], StyleBox["M", FontSlant->"Plain"]], TraditionalForm]]], " gives the ", StyleBox["y-coordinate of the center of mass", FontWeight->"Bold"], ". In the case that the density is constant, the center of mass of the \ region is also called the ", StyleBox["geometric center", FontWeight->"Bold"], " of the region. See an instructor for more on why these definitions make \ sense." }], "Text"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Find the geometric center of the upper half of the unit circle (", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " = 1) using the cell below. You will have to fill in the integration \ variables and bounds on the integrals. (You only need to figure out the \ bounds once; they are the same for each of the integrals. We can use \ \[Delta] as the density of the region, but you will notice that the final \ answer for the geometric center doesn't depend on \[Delta]. (Why doesn't \ it?)" }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[{\(Clear[M, My, Mx, xbar, ybar]\), "\n", RowBox[{"M", "=", RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{"\[Delta]", StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}], StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}]}]}]}]}], "\n", RowBox[{"My", "=", RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{"x", " ", "\[Delta]", StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}], StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}]}]}]}]}], "\n", RowBox[{"Mx", "=", RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{"y", " ", "\[Delta]", StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}], StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}]}]}]}]}], "\[IndentingNewLine]", \(xbar = My\/M;\), "\n", \(ybar = Mx\/M;\), "\n", \({xbar, ybar}\)}], "Input"], Cell[TextData[{ "We can also define something called a ", StyleBox["second moment", FontWeight->"Bold"], ", or ", StyleBox["moment of inertia", FontWeight->"Bold"], ". Although this can be done with respect to any point inside or outside \ the region, here we will only do it for the origin. For a region in the \ plane whose (area) density is given by the function \[Delta](x,y), the moment \ of inertia with respect to the origin, I, is defined as the double integral \ over the region of the function (", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], ")", " \[Delta](x,y). (The ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " is there as the square of the distance from an arbitrary point in the \ region to the origin.) This has physical significance; the moment of inertia \ is related to the kinetic energy involved in rotating the region around the \ point. " }], "Text"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Find the moment of inertia of the ", StyleBox["entire", FontVariations->{"Underline"->True}], " region (not just the upper half) bounded by the unit circle (", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " = 1), using the cell below. Assume the density \[Delta] of the region is \ constant. You will have to fill in the integration variables and bounds on \ the integrals." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{ StyleBox[ SubsuperscriptBox["\[Integral]", StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 1]]], ScriptLevel->0], RowBox[{ RowBox[{"(", RowBox[{ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], "TraditionalForm"], "+", FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], "TraditionalForm"]}], ")"}], "\[Delta]", StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}], StyleBox["\[ThinSpace]", ScriptLevel->0], RowBox[{ StyleBox["\[DifferentialD]", ScriptLevel->0], StyleBox["\[Placeholder]", ScriptLevel->0, FontColor->RGBColor[1, 0, 1]]}]}]}]}]], "Input"], Cell[TextData[{ "By the way, for those of you who do not have an interest in physics, these \ definitions have parallels in probability and statistics. If \[Delta](x,y) \ is a probability density function for jointly distributed random variables, \ then the first moments are related to ", StyleBox["expected values", FontWeight->"Bold"], " (including ", StyleBox["means", FontWeight->"Bold"], ") and second moments are related to ", StyleBox["variance", FontWeight->"Bold"], "." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Triple Integrals", "Section"], Cell[TextData[{ "As before, there are several ways to get ", StyleBox["Mathematica", FontSlant->"Italic"], " to do a triple integral. The easiest is to use the triple integral \ button (", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), ScriptLevel->0], RowBox[{"\[Placeholder]", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]\[Placeholder]\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]\[Placeholder]\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]\[Placeholder]\), ScriptLevel->0]}]}]}]}], TraditionalForm]]], ") on the ", StyleBox["MultivariableCalculus", FontColor->RGBColor[0, 0, 1]], " palette. If you click this button it pastes a copy of the symbol at your \ cursor location in your notebook. You can then fill in the boxes with the \ appropriate information. (Use the mouse or the ", StyleBox["Tab", FontColor->RGBColor[1, 0, 0]], " key to cycle around the boxes in order to fill them in.) Here is an \ example (which actually computes the volume of the tetrahedron bounded by the \ coordinate planes and the plane 2 x + y + z = 6)." }], "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%3\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_0\%\(6 - 2\ x\)\), ScriptLevel->0], RowBox[{ StyleBox[\(\[Integral]\_0\%\(6 - 2\ x - y\)\), ScriptLevel->0], RowBox[{"1", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]z\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]}]}]], "Input"], Cell["\<\ Notice that the order of integration in the above example is z first, then y, \ then x. The rightmost integral sign refers to the first of the integration \ variables, in this case z. This is the same as using the parenthesized \ version in the next cell.\ \>", "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%3\), ScriptLevel->0], RowBox[{ RowBox[{"(", RowBox[{ StyleBox[\(\[Integral]\_0\%\(6 - 2\ x\)\), ScriptLevel->0], RowBox[{ RowBox[{"(", RowBox[{ StyleBox[\(\[Integral]\_0\%\(6 - 2\ x - y\)\), ScriptLevel->0], RowBox[{"1", StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]z\), ScriptLevel->0]}]}], ")"}], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0]}]}], StyleBox["\[ThinSpace]", ScriptLevel->0], ")"}], StyleBox[\(\[DifferentialD]x\), ScriptLevel->0]}]}]], "Input"], Cell[TextData[{ "If you only have the ", StyleBox["Basic Input", FontColor->RGBColor[0, 0, 1]], " palette available, you can piece together the triple integral symbol by \ clicking the ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), RowBox[{ StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], \(\[DifferentialD]\[Placeholder]\)}]}], TraditionalForm]]], " button, placing the cursor in the box for the function (marked in red in \ this sentence), clicking the symbol ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\[Placeholder]\%\[Placeholder] \ \[Placeholder] \[DifferentialD]\[Placeholder]\)]], " once again, and then repeating this one more time." }], "Text"], Cell[TextData[{ "You can also get ", StyleBox["Mathematica", FontSlant->"Italic"], " to do the integral without palette symbols. Here is the syntax." }], "Text"], Cell[BoxData[ \(Integrate[ 1, {x, 0, 3}, {y, 0, 6 - 2\ x}, {z, 0, 6 - 2\ x - y}]\)], "Input"], Cell["\<\ Notice that in this case the third set of braces {z,0,6-2 x-y} corresponds to \ the first integration variable, the second {y,0,6-2 x} to the second \ integration variable, and the first set {x,0,3} corresponds to the third \ integration variable. Be careful of the ordering of this integration \ variable information.\ \>", "Text"], Cell[TextData[{ "As in the case of double integrals, if you only want numerical answers, \ you might wish to use ", StyleBox["NIntegrate", FontColor->RGBColor[1, 0, 0]], " instead of ", StyleBox["Integrate", FontColor->RGBColor[1, 0, 0]], ". The syntax for ", StyleBox["NIntegrate", FontColor->RGBColor[1, 0, 0]], " is the same as that for ", StyleBox["Integrate", FontColor->RGBColor[1, 0, 0]], ". (You might notice the decimal point after the answer. This is ", StyleBox["Mathematica", FontSlant->"Italic"], "'s way of letting you know that numerical work was done somewhere along \ the way.)" }], "Text"], Cell[BoxData[ \(NIntegrate[ 1, {x, 0, 3}, {y, 0, 6 - 2\ x}, {z, 0, 6 - 2\ x - y}]\)], "Input"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find the volume of the region bounded below by the paraboloid z = ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " and above by the paraboloid z = 8 - ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], ", using triple integrals. Again, remember that you integrate the function \ f(x,y) = 1 over the region to get the volume." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[CellGroupData[{ Cell["Center of Mass and Moments", "Subsection"], Cell[TextData[{ "This time we have first moments with respect to planes instead of lines. \ For a region in space whose (volume) density is given by the function \ \[Delta](x,y,z), the ", StyleBox["first moment with respect to the y-z plane", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["yz", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], ", is defined as the triple integral over the region of the function x \ \[Delta](x,y,z). (The x is there as the distance from an arbitrary point in \ the region to the y-z plane.) Similarly the ", StyleBox["first moment with respect to the x-z plane", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["xz", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], ", is the triple integral over the region of the function y \ \[Delta](x,y,z), and the ", StyleBox["first moment with respect to the x-y plane", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["xy", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], ", is the triple integral over the region of the function z \ \[Delta](x,y,z). The ", StyleBox["x-coordinate of the center of mass of the region", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["x", FontSlant->"Plain"], "_"], TraditionalForm]]], ", can be found by dividing ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["yz", FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], " by the ", StyleBox["mass", FontWeight->"Bold"], " M of the region (which is simply the triple integral over the region of \ the density function \[Delta](x,y,z)). Similarly, ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["y", FontSlant->"Plain"], "_"], TraditionalForm]]], " =", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["xz", FontSlant->"Plain"]], FontSlant->"Italic"], StyleBox["M", FontSlant->"Plain"]], TraditionalForm]]], " gives the ", StyleBox["y-coordinate of the center of mass", FontWeight->"Bold"], ", and ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["z", FontSlant->"Plain"], "_"], TraditionalForm]]], " =", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ SubscriptBox[ StyleBox["M", FontSlant->"Plain"], StyleBox["xy", FontSlant->"Plain"]], FontSlant->"Italic"], StyleBox["M", FontSlant->"Plain"]], TraditionalForm]]], " gives the ", StyleBox["z-coordinate of the center of mass", FontWeight->"Bold"], ". In the case that the density is constant, the center of mass of the \ region is also called the ", StyleBox["geometric center", FontWeight->"Bold"], " of the region. See an instructor for more on why these definitions make \ sense." }], "Text"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Find the center of mass of the region bounded below by the paraboloid z \ = ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " and above by the paraboloid z = 8 - ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], ", assuming the density \[Delta](x,y,z) = z. You will have to set up four \ integrals to do this. (You only need to figure out the bounds on the \ integrals once; they are the same for each of the integrals. If you did the \ previous exercise, you already have the bounds.)" }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "The notion of ", StyleBox["second moment", FontWeight->"Bold"], ", or ", StyleBox["moment of inertia", FontWeight->"Bold"], ", is now relative to lines rather than points. We will consider three of \ these. For a region in space whose (volume) density is given by the function \ \[Delta](x,y,z), the ", StyleBox["moment of inertia with respect to the x-axis", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["I", FontSlant->"Plain"], StyleBox["x", FontSlant->"Plain"]], TraditionalForm]]], ", is defined as the triple integral over the region of the function (", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["z", FontSlant->"Plain"], "2"], TraditionalForm]]], ")", " \[Delta](x,y,z). (The ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["z", FontSlant->"Plain"], "2"], TraditionalForm]]], " is there as the square of the distance from an arbitrary point in the \ region to the x-axis.) Similarly we define As before, this has physical \ significance; the moment of inertia is related to the kinetic energy involved \ in rotating the region around the axis. " }], "Text"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Find the moment of inertia of the region bounded below by the \ paraboloid z = ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], " and above by the paraboloid z = 8 - ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSlant->"Plain"], "2"], TraditionalForm]]], ", about each of the coordinate axes, assuming the density \[Delta](x,y,z) \ = z. You will have to set up three integrals to do this." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ As in the case for double integrals, there are corresponding uses of these \ ideas in statistics and probability, among other places.\ \>", "Text"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 720}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{818, 606}, WindowMargins->{{0, Automatic}, {Automatic, 0}} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 128, 3, 131, "Title"], Cell[1870, 56, 250, 4, 44, "SmallText"], Cell[2123, 62, 1080, 27, 90, "Text"], Cell[3206, 91, 115, 3, 33, "Text"], Cell[3324, 96, 141, 3, 50, "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[3490, 103, 35, 0, 59, "Section"], Cell[3528, 105, 1295, 33, 80, "Text"], Cell[4826, 140, 499, 15, 42, "Input"], Cell[5328, 157, 273, 5, 52, "Text"], Cell[5604, 164, 618, 19, 42, "Input"], Cell[6225, 185, 770, 20, 52, "Text"], Cell[6998, 207, 172, 5, 33, "Text"], Cell[7173, 214, 72, 1, 30, "Input"], Cell[7248, 217, 278, 5, 52, "Text"], Cell[7529, 224, 761, 23, 68, "Text"], Cell[8293, 249, 320, 7, 52, "Text"], Cell[8616, 258, 522, 15, 42, "Input"], Cell[9141, 275, 509, 15, 55, "Input"], Cell[9653, 292, 257, 6, 52, "Text"], Cell[9913, 300, 37, 1, 30, "Input"], Cell[9953, 303, 154, 5, 33, "Text"], Cell[10110, 310, 89, 1, 50, "Input"], Cell[10202, 313, 76, 0, 33, "Text"], Cell[10281, 315, 735, 21, 42, "Input"], Cell[11019, 338, 125, 3, 34, "Input"], Cell[11147, 343, 121, 3, 34, "Input"], Cell[11271, 348, 608, 15, 71, "Text"], Cell[CellGroupData[{ Cell[11904, 367, 48, 0, 47, "Subsection"], Cell[11955, 369, 2566, 79, 133, "Text"], Cell[14524, 450, 867, 23, 87, "Text"], Cell[15394, 475, 3839, 104, 217, "Input"], Cell[19236, 581, 1388, 40, 109, "Text"], Cell[20627, 623, 762, 23, 68, "Text"], Cell[21392, 648, 1540, 45, 41, "Input"], Cell[22935, 695, 518, 14, 71, "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[23502, 715, 35, 0, 39, "Section"], Cell[23540, 717, 1748, 41, 99, "Text"], Cell[25291, 760, 764, 22, 42, "Input"], Cell[26058, 784, 281, 5, 52, "Text"], Cell[26342, 791, 940, 26, 42, "Input"], Cell[27285, 819, 805, 20, 52, "Text"], Cell[28093, 841, 172, 5, 33, "Text"], Cell[28268, 848, 103, 2, 30, "Input"], Cell[28374, 852, 343, 6, 52, "Text"], Cell[28720, 860, 652, 19, 71, "Text"], Cell[29375, 881, 104, 2, 30, "Input"], Cell[29482, 885, 1010, 34, 68, "Text"], Cell[CellGroupData[{ Cell[30517, 923, 48, 0, 47, "Subsection"], Cell[30568, 925, 3731, 119, 172, "Text"], Cell[34302, 1046, 1107, 34, 87, "Text"], Cell[35412, 1082, 1588, 49, 90, "Text"], Cell[37003, 1133, 968, 32, 68, "Text"], Cell[37974, 1167, 157, 3, 33, "Text"] }, Closed]] }, Closed]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)