(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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[ 30566, 934]*) (*NotebookOutlinePosition[ 31335, 960]*) (* CellTagsIndexPosition[ 31291, 956]*) (*WindowFrame->Normal*) Notebook[{ Cell["Lab 7: Multiple Integrals", "Title", Background->RGBColor[0, 0, 1]], Cell[TextData[{ "Math 233\tFall 2001\t", StyleBox["Chapter 12, Sections 1 and 4", FontFamily->"Arial", FontSize->16, FontWeight->"Bold"], " " }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[StyleBox["Instructions!", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[TextData[{ "Work in ", StyleBox["groups of 2 or 3", FontWeight->"Bold"], " today, going through the entire notebook. If you have questions, flag me \ down. Write up your answers (complete sentences, please) for a ", StyleBox["group", FontWeight->"Bold"], " hand in on ", StyleBox["Monday", FontWeight->"Bold"], " of next week (", StyleBox["12 October)", FontWeight->"Bold"], ". \n\nHomework exercises that you should work on to build your \ multivariable integration skills are contained on the hand-out given today in \ Lab. For those wanting additional challenges, you should try the following \ problems: ", StyleBox["Section 12.1 (p. 985)", FontWeight->"Bold"], ": 24, 33, and 41. ", StyleBox["Section 12.4 (p. 1015)", FontWeight->"Bold"], ": 23 and 27. These problems won't be handed in, but you ", StyleBox["should", FontWeight->"Bold"], " try them all (and hopefully ", StyleBox["complete", FontWeight->"Bold"], " them all too if you wish to do well on the Exam we have on Tuesday, Nov \ 20). I'll post solutions soon." }], "Subsubtitle"] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "OBJECTIVE: Today, you will learn:\n\t1. how to evaluate double and triple \ integrals using ", StyleBox["Mathematica", FontSlant->"Italic"], ".\n\t2. how to use ", StyleBox["Mathematica", FontSlant->"Italic"], " to help you set up triple integrals \t" }], "Text"], Cell[CellGroupData[{ Cell["Technology Guidelines", "Subsection", CellDingbat->"\[LightBulb]"], Cell[TextData[{ StyleBox["NOTE: If you have just finished a module, restart ", CellFrame->True, Background->None], StyleBox["Mathematica", CellFrame->True, FontSlant->"Italic", 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], "\nTO STOP AN EXECUTION\n\tSelect the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu and click on ", StyleBox["Abort Evaluation.\n", FontSlant->"Italic"], "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \ 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", FontSlant->"Italic"], " selection under the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu.\nEXPERIENCING MAJOR PROBLEMS\n\tSave if appropriate, and \ then shut down ", StyleBox["Mathematica", FontSlant->"Italic"], " and start it up again." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part I: How to Integrate", "Section"], Cell["Single variable functions", "Subsubsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " does both definite and indefinite integrals. Furthermore, it does this \ very well. There's a story about a master's student that compared an earlier \ version of ", StyleBox["Mathematica", FontSlant->"Italic"], " to the table of integrals in the CRC handbook. The two differed on \ three antiderivates. Upon further investigation, it turned out that ", StyleBox["Mathematica", FontSlant->"Italic"], " was right and the CRC handbook was wrong!\nLet's see how to integrate. \ For indefinite integrals (antiderivatives), you can either use the \ 'Integrate' command or the \[Integral]\[DifferentialD]\[Placeholder] button \ from the ", StyleBox["BasicInput palette ", FontWeight->"Bold"], "[which is available from the ", StyleBox["File->Palettes", FontWeight->"Bold"], " pull-down menu]", ". Both have two arguments, the first being the expression to integrate \ and the second being the variable of integration. The integrand can be \ either a previously defined function, f[x], or an expression. Execute the \ input cell below to see how this works." }], "Text"], Cell[BoxData[{ \(\(Clear[h, x];\)\), "\n", \(\(h[x_] = Sin[3 x];\)\), "\n", \(Integrate[h[x], x]\), "\n", \(\[Integral]Sin[3 x] \[DifferentialD]x\)}], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(For\ definite\ integrals\), ",", " ", \(you\ can\ use\ ' Integrate'\), ",", RowBox[{"the", " ", RowBox[{\(\[Integral]\_\[Placeholder]\%\[Placeholder]\), RowBox[{ "\[Placeholder]", \(\[DifferentialD]\[Placeholder]\), " ", "button", " ", "on", " ", "the", " ", StyleBox["BasicInput", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox["palette", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], \(or\ '\), \(NIntegrate'\), " ", "to", " ", "force", " ", "Mathematica", " ", "to", " ", "do", " ", "a", " ", "numerical", " ", \(integration . \ \ The\), " ", "first", " ", "two", " ", "methods", " ", "try", " ", "to", " ", "first", " ", "find", " ", "an", " ", "antiderivative", " ", "and", " ", "if", " ", \(that'\), "s", " ", "not", " ", "possible", " ", "they", " ", "resort", " ", "to", " ", "a", " ", "numerical", " ", \(method . \ \ The\ '\), \(NIntegrate'\), " ", "command", " ", "goes", " ", "straight", " ", "to", " ", "calculating", " ", "the", " ", "definite", " ", "integral", " ", \(numerically . \ \ \ \[IndentingNewLine]\ \[IndentingNewLine]With\), " ", "both", " ", \(the\ '\), \(Integrate'\), " ", \(and\ '\), \(NIntegrate'\), " ", "commands"}]}]}], ",", \(the\ second\ argument\ \((variable\ of\ integration)\)\ is\ \ replaced\ by\ a\ list\ ' {x, a, b}'\ where\ the\ list\ contains\ the\ variable\ of\ integration\ \ followed\ by\ the\ lower\ and\ upper\ limits\ of\ integration . \ \ The\ \ palette\ button\ simply\ provides\ two\ additional\ boxes\ for\ the\ upper\ \ and\ lower\ limits\ of\ integration . \ \ Execute\ the\ input\ cells\ below\ \ to\ see\ how\ these\ \(\(work\)\(.\)\)\)}], TextForm]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[Sin[3 x], {x, 0, Pi/2}]\)], "Input"], Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]/2\)Sin[3 x] \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(NIntegrate[Sin[3 x], {x, 0, \[Pi]\/2}]\)], "Input"] }, Open ]], Cell[TextData[{ StyleBox["CAUTION", FontWeight->"Bold"], ": The '\[DifferentialD]x' in the palette form of integrals is different \ from 'dx' and using 'dx' will NOT work for the differential part of \ integrals. Normally this is not a problem since the template you get by \ clicking the palette button gives you the '\[DifferentialD]x', but if you \ accidently delete it, you can get it back by typing 'ddx'." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ NOTE: Sometimes things that should work just don't. For a first example, \ execute the two input cells below. Notice how the first works but the second \ does not. 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rather bitterly that \ there isn't anything to do! Click on the lower limit of integration for the \ outer integral - you know things are going well if the box \"fills in\" and \ has a black square inside of it after you've clicked. Once you enter a value, \ you can use the ", StyleBox["tab", FontWeight->"Bold"], " key to move from box to box: Fill in values for the previous cell to \ take the integral of (2x^2)*", Cell[BoxData[ \(TraditionalForm\`\@y\%3\)]], " for 0\[LessEqual]x\[LessEqual]y and 2\[LessEqual]y\[LessEqual]8. ", StyleBox["[reality check: you should end up with 1257.21 as the solution]", FontWeight->"Bold"], " [you'll get the exact mode calculation as a default, try tagging a ", StyleBox["//N", FontWeight->"Bold"], " onto the back of the command to get the numerical representation of the \ exact answer].\nUsing the ", StyleBox["Integrate", FontWeight->"Bold"], " command, we work from the \"outside to the inside\" - see the note \ below:" }], "Text"], Cell[TextData[{ StyleBox["CAUTION", FontWeight->"Bold"], ": A potentially confusing aspect of the ", StyleBox["integrate ", FontWeight->"Bold"], "command is that the limits are ordered from the ", StyleBox["outermost ", FontWeight->"Bold"], "to the ", StyleBox["innermost", FontWeight->"Bold"], ":\nIntegrate[ f[x, y], {x, x0, x1}, {y, y0, y1}] is used to evaluate the \ integral ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_x0\%x1\(\[Integral]\_y0\%y1\( f(x, y)\) \[DifferentialD]y \[DifferentialD]x\)\)]] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Go ahead and add the limits of integration in the proper order to solve the \ integral from above:\ \>", "Text"], Cell[BoxData[ \(Integrate[\(2 \( x\^2\) \@y\%3\)\(,\)]\)], "Input"], Cell[TextData[{ "The numerical integrate command ", StyleBox["NIntegrate", FontWeight->"Bold"], " works the same way:", " " }], "Text"], Cell[BoxData[ \(NIntegrate[\(2 \( x\^2\) \@y\%3\)\(,\)]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part I", "Section"], Cell[TextData[{ StyleBox["Question 1:", FontWeight->"Bold"], "\t", StyleBox["Use both of the exact integration techniques outlined above to \ evaluate the following integrals:\n\t1. the indefinite integral xy^2 dx dy\n\ \t2. the definite integral of xy^2 for 1-y \[LessEqual]x \[LessEqual]", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\@y\)]], StyleBox["and 1\[LessEqual]y\[LessEqual]2\n\t3. the definite integral of y \ sin(x) - x sin(y) for 0 \[LessEqual]y\[LessEqual]Pi/2 and 0\[LessEqual]x\ \[LessEqual]Pi/6", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]], Cell[TextData[{ StyleBox["Question 2:\tVerify that ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" has trouble finding the exact integral, then find an approximate \ value of each integral:\n\t1. cos(x^2-y^2) for 0\[LessEqual]y\[LessEqual]", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\@Pi\)]], " ", StyleBox["and 0\[LessEqual]x\[LessEqual]", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\@Pi\)]], "\n\t", StyleBox["2. sin(e^(xy)) for 0\[LessEqual]x\[LessEqual]1 and \ 0\[LessEqual]y\[LessEqual]1", FontWeight->"Bold"] }], "Text", FontWeight->"Plain"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Part II: A Real Problem!", "Section"], Cell[TextData[{ StyleBox["The Problem: ", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["Find the volume of the pyramid with base in the plane", FontFamily->"Courier"], " ", StyleBox["z = -6 ", FontFamily->"Terminal"], StyleBox["and sides formed by the three planes ", FontFamily->"Courier"], StyleBox["y = 0, y - x = 4, and \n2x + y + z =4.\nFirst, here is what \ things look like... well, most of it, anyway. I've plotted the plane z = -6 \ (which appears to be the base of this pyramid) and the plane 2x + y + z = 4. \ We'll have to think about the other pieces later (since there is no z in y - \ x = 4, it is tough to plot it using ", FontFamily->"Terminal"], StyleBox["Plot3D", FontFamily->"Terminal", FontWeight->"Bold"], StyleBox[")", FontFamily->"Terminal"], StyleBox[":", FontFamily->"Terminal"] }], "Text"], Cell[BoxData[{ \(\(plt1 = Plot3D[\(-6\), {x, \(-5\), \ 5}, {y, 0, 10}, AxesLabel \[Rule] {"\", "\", "\"}];\)\), "\ \[IndentingNewLine]", \(\(plt2 = Plot3D[\ 4 - 2 x - y, {x, \(-5\), \ 5}, {y, 0, 10}, AxesLabel \[Rule] {"\", "\", "\"}];\)\), "\ \[IndentingNewLine]", \(\(Show[plt1, plt2];\)\)}], "Input"], Cell["Let's set up the limits of integration!", "Subsubsection"], Cell[TextData[{ StyleBox["STEP 1: \n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["Find the limits on ", FontFamily->"Courier"], StyleBox["z", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[":\n ", FontFamily->"Courier"], StyleBox["What equations involve ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["z", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["?", FontFamily->"Courier", FontSlant->"Italic"], StyleBox[" z=-6 and 2x+y+z=4 are the only two.\n (so no other equations \ can possibly bound ", FontFamily->"Courier"], StyleBox["z", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["). \n ", FontFamily->"Courier"], StyleBox["Which one is a lower bound? ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["z=-6", FontFamily->"Courier"], StyleBox["\n Which one is an upper bound? ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["z=4-2x-y\n", FontFamily->"Courier"] }], "Text"], Cell[TextData[{ StyleBox["STEP 2:\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["Describe the region of integration with respect to ", FontFamily->"Courier"], StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", FontFamily->"Courier"], StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[":\n ", FontFamily->"Courier"], StyleBox["Set the limits for ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["z", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" equal to each other!", FontFamily->"Courier", FontSlant->"Italic"], StyleBox[" This lets us know\n what these limits look like when we have \ \"integrated out\" the\n effect of the variable ", FontFamily->"Courier"], StyleBox["z", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". We end up with:", FontFamily->"Courier"] }], "Text"], Cell[BoxData[ \(Simplify[\(-6\)\ == 4 - 2 x - y]\)], "Input"], Cell[TextData[{ StyleBox[" ", FontFamily->"Courier"], StyleBox["What other equations involve only ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["x", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" and ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["y", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["? ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["y=0 and y-x=4.\n ", FontFamily->"Courier"], StyleBox["Graph all these equations to visualize the region of \ integration:", FontFamily->"Courier", FontSlant->"Italic"] }], "Text"], Cell[BoxData[ \(\(Plot[{10 - 2 x, \ 0, \ x + 4}, {x, \(-5\), 5}, PlotRange -> {\(-1\), 10}, AxesLabel -> {x, y}];\)\)], "Input"], Cell[TextData[{ StyleBox["Now, the problem is identical to those in Section 12.1 - find \ which variable ", FontFamily->"Courier"], StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" or ", FontFamily->"Courier"], StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" to integrate first.\n\nIn this case, it is easier to integrate \ with respect to ", FontFamily->"Courier"], StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" first (", FontFamily->"Courier"], StyleBox["why?", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["):\n lower limit: x=y-4\n upper limit: x=5-y/2\n ", FontFamily->"Courier"] }], "Text"], Cell[TextData[{ StyleBox["Step 3: ", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["You can either visually inspect the graph and determine the \ limits for ", FontFamily->"Courier"], StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[", or you can follow the same procedure as above to get the limits \ for ", FontFamily->"Courier"], StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[":\n\n", FontFamily->"Courier"], StyleBox[" Set the limits on the previous integration equal:", FontFamily->"Courier", FontSlant->"Italic"], StyleBox[" Here we get:", FontFamily->"Courier"] }], "Text"], Cell[BoxData[ \(Simplify[y - 4 == 5 - y/2]\)], "Input"], Cell[TextData[{ StyleBox[" ", FontFamily->"Courier"], StyleBox["What other equations involve only ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["y", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["? ", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["y=0\n ", FontFamily->"Courier"], StyleBox["Since there are only 2, one is the lower bound and one is the\n \ upper bound (otherwise graph and check):", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["\n \n lower bound: y=0\n upper bound: y=6 ", FontFamily->"Courier"] }], "Text"], Cell[TextData[StyleBox["So the integral is:", FontFamily->"Courier"]], "Text"], Cell[BoxData[ \(\[Integral]\_0\%6\(\[Integral]\_\(y - 4\)\%\(5 - \ y/2\)\(\[Integral]\_\(-6\)\%\(4 - 2 x - y\)1 \[DifferentialD]z \ \[DifferentialD]x \[DifferentialD]y\)\)\)], "Input"], Cell[TextData[{ StyleBox["and the area is 162. Notice that if you want to check integrals \ in ", FontFamily->"Courier"], StyleBox["Mathematica", FontFamily->"Courier", FontSlant->"Italic"], StyleBox[", you can check each \"layer\" by cutting and pasting:", FontFamily->"Courier"] }], "Text"], Cell[BoxData[ \(\[Integral]\_\(-6\)\%\(4 - 2 x - y\)1 \[DifferentialD]z\)], "Input"], Cell[BoxData[ \(\[Integral]\_\(y - 4\)\%\(5 - y/2\)\((10 - 2\ x - y)\) \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(\[Integral]\_0\%6\((81 - 27\ y + \(9\ y\^2\)\/4)\) \[DifferentialD]y\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part II", "Section"], Cell[TextData[{ StyleBox["The Problem:\t", FontWeight->"Bold"], "Find the volume of the solid bounded by the graphs of f(x,y) = 1-x-y and \ g(x,y)=2-", Cell[BoxData[ \(TraditionalForm\`x\^2 - y\^2\)]], "." }], "Text", FontWeight->"Plain"], Cell["\<\ We don't know the region of integration! So let's graph the functions to see \ what things look like:\ \>", "Text"], Cell[BoxData[{ \(Clear[f, g]\), "\[IndentingNewLine]", \(\(f[x_, y_] = 1 - x - y;\)\), "\[IndentingNewLine]", \(\(g[x_, y_] = 2 - x^2 - y^2;\)\), "\[IndentingNewLine]", \(\(pf = Plot3D[f[x, y], {x, \(-3\)/2, 2}, {y, \(-3\)/2, 2}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(pg = Plot3D[g[x, y], {x, \(-3\)/2, 2}, {y, \(-3\)/2, 2}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[pf, pg, ViewPoint \[Rule] {0.010, \(-2.723\), 2.000}, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"], Cell[TextData[StyleBox["Question 3:\tWhich surface is the upper bound and \ which is the lower bound?", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]], Cell[TextData[StyleBox["Question 4:\tSet up and solve the integral by \ following the steps from Part II:", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]] }, Closed]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 680}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{825, 609}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, StyleDefinitions -> "DemoText.nb" ] (******************************************************************* Cached data follows. 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