(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). 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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[ 112125, 3473]*) (*NotebookOutlinePosition[ 147774, 4696]*) (* CellTagsIndexPosition[ 147699, 4690]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Lab 3 - Applications of Integration", FontSize->24, FontWeight->"Bold"], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: swenson@math.umn.edu, gantner@math.umn.edu" }], "Text", TextAlignment->Center, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell["Introduction", "Section", FontSize->14, Background->None], Cell[TextData[{ "Our goal in this part of the lab is to find the volume of liquid in a \ cylindrical tank which is tilted at some angle from the horizontal. To be \ specific, we start with the cylinder with radius 2 m and length 10 m, whose \ axis is the vertical line through (2,0,0): we will put the bottom of the \ cylinder at z=0 and the top at z=10. We can plot this cylinder using ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ". We've filled in most of the commands here, including the function. \ Fill in the appropriate bounds and evaluate the cell to see the cylinder. \ (Notice that we name the plot ", StyleBox["cyl", FontWeight->"Bold"], ".)" }], "Text"], Cell[BoxData[ \(\(cyl = ContourPlot3D[\((x - 2)\)^2 + y^2 - 4, {x, , }, {y, , }, {z, , }, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\", "\"}];\)\)], "Input"], Cell["\<\ The surface of the liquid in our tank should be given by a \ horizontal plane: here is an example.\ \>", "Text"], Cell[BoxData[{ \(plane1 = ParametricPlot3D[{x, y, 4}, {x, \(-1\), 5}, {y, \(-3\), 3}]\), "\[IndentingNewLine]", \(Show[cyl, plane1, AxesLabel \[Rule] {x, y, z}]\)}], "Input", CellTags->"zerodeg"], Cell[TextData[{ "In this case, the volume of the liquid is \[Pi]", Cell[BoxData[ \(TraditionalForm\`\(r\^2\) h\)]], " = 16\[Pi]. (See why?)" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Tilting Cylinders Is Too Hard", "Section", FontSize->14, Background->None], Cell["\<\ To find the volume of liquid when the cylinder is tilted, we will \ need to use integration. Evaluate the next cell to see an example of a \ tilted cylinder, with a horizontal plane representing the surface of a \ liquid.\ \>", "Text"], Cell[BoxData[{ \(\(u = Pi/4;\)\), "\[IndentingNewLine]", \(cyl2 = ParametricPlot3D[{Cos[u]*\((2*Cos[t] + 2)\) - Sin[u]*s, 2*Sin[t], Sin[u]*\((2*Cos[t] + 2)\) + Cos[u]*s}, {s, 0, 10}, {t, 0, 2 Pi}]\), "\[IndentingNewLine]", \(plane2 = ParametricPlot3D[{x, y, 4}, {x, \(-6\), 4}, {y, \(-2\), 2}]\), "\[IndentingNewLine]", \(Show[cyl2, plane2, \ AxesLabel \[Rule] {x, y, z}, ViewPoint -> {0, \ \(-3\), \ 0}, BoxRatios \[Rule] {1, 1, 1}]\)}], "Input"], Cell["\<\ This picture is not very useful for calculation: we don't know what \ to integrate, nor over what region! [It was also tricky to reparametrize the \ cylinder.]\ \>", "Text"], Cell[TextData[{ "We will take a different approach: again, we imagine the cylinder tilting, \ but to simplify the calculations, we tilt the plane instead, and reuse ", StyleBox["cyl", FontWeight->"Bold"], ", our original cylinder. To put this another way, we are tilting the \ coordinate axes along with the cylinder." }], "Text"], Cell[BoxData[{ \(plane3 = ParametricPlot3D[{x, y, 6 - x}, {x, \(-2\), 6}, {y, \(-2\), 2}]\), "\[IndentingNewLine]", \(Show[cyl, plane3, AxesLabel \[Rule] {x, y, z}, BoxRatios \[Rule] {1, 1, 1}, ViewPoint -> {0, \ \(-3\), \ 0}]\)}], "Input"], Cell["\<\ If you tilt your head,you can see that this graph shows the same \ part of the cylinder as in the tilted-cylinder graph above. If you're not \ sure about that, evaluate the next cell to see the cylinder in a \ LiveGraphics3D window. Tilt the picture until the plane is flat. (HINT: hold \ the shift button down, click on the picture, and move your mouse to the \ left.) It should look the same as the earlier picture.\ \>", "Text"], Cell[BoxData[ \(\(ShowLive[cyl, plane3, AxesLabel \[Rule] {x, y, z}, BoxRatios \[Rule] {1, 1, 1}, ViewPoint -> {0, \ \(-3\), \ 0}];\)\)], "Input"], Cell[TextData[{ "We like this new viewpoint because it makes calculation much easier: we \ can find the volume of liquid by integration. Specifically, we want the \ volume of the region below our plane, and above the bottom of the tank.\n\n\ Therefore, the region of integration should be the bottom of the tank, which \ is in the xy-plane, and given by the inequality ", Cell[BoxData[ \(TraditionalForm\`\((x - 2)\)\^2 + y\^2\)]], "\[LessEqual] 4, and the integrand should be the height of the plane." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Exercises", "Section", CellFrame->False, FontSize->14, Background->None], Cell[TextData[{ StyleBox["Exercise 1", FontWeight->"Bold"], "\na) Suppose that the fluid level in the tank is 7 m on the left edge of \ the tank (where x=0) and 5 m on the right edge (where x=4). Find the \ equation of the plane of the liquid, and use a double integral to find the \ volume of liquid in the tank. [Hint: you should use a \"dy dx\" iterated \ integral, where the bounds on y depend on x, and are given by the equation of \ the base of the cylinder.] Also find the angle at which the tank is tipped \ from its upright position. For consistency, assume that the ", ButtonBox["tank", ButtonData:>"zerodeg", ButtonStyle->"Hyperlink"], " in the ", StyleBox["Introduction", FontWeight->"Bold"], " above is tipped at a 0\[Degree] angle. [Hint: use a \"side elevation\" \ sketch, and use trigonometry to find the angle between your plane and the \ plane z=5.]\n\nb) Suppose now that the fluid level is 7 m at the left edge of \ the tank, but that there isn't enough fluid to reach the right edge; instead, \ the fluid covers the base of the tank only out to the plane x=3. [You can \ produce a graph with the commands below: you'll have to paste them into \ another cell.]\n", Cell[BoxData[{ \(plane4 = ParametricPlot3D[{x, y, 7 - 7 x/3}, {x, 0, 3}, {y, \(-2\), 2}]\), "\n", \(Show[cyl, plane4, AxesLabel \[Rule] {x, y, z}, ViewPoint -> {0, \ \(-3\), \ 0}]\)}], "Input", CellFrame->False], "\nFind the equation of the plane of the liquid,and use a double integral \ to find the volume of liquid in the tank. Also find the angle at which the \ tank is tipped from its upright position. [", StyleBox["Warning", FontWeight->"Bold"], ": the region of integration has changed -- if you reuse the bounds from \ problem 1, you'll include the \"negative volume\" from the region where z<0.]\ \n\nc) Suppose that the tank is tipped so far that the liquid touches both \ ends of the tank, touching the top (at z=10) out to the plane x=1, and \ covering the bottom out to the plane x=3. Find the volume of liquid in the \ tank using two double integrals. Explain why you can't just use one double \ integral. Lastly, find the angle at which the tank is tipped from its \ upright position.\n\nd) Suppose, finally, that the liquid touches the top of \ the tank out to the plane x=3, and reaches a level of 3 m on the right edge \ of the tank. Find the volume of liquid in the tank using two double \ integrals, and find the angle at which the tank is tipped from its upright \ position.\n\ne) Suppose that our tank is tilted at an angle of 80\[Degree] \ from its upright position. Where could you mark the surface of the tank to \ indicate when the tank is exactly one-third full? [Hint: use trigonometry to \ decide which of the first four exercises most resembles this problem. \ Calculate the volume in terms of some unknown variable, and use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["FindRoot", FontFamily->"Courier", FontWeight->"Bold"], " command, which you can look up in the help browser.]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell["\<\ In exercise 2, rather than using the tank that was described in the \ introduction, we will use an oblong tank. When intersected with the \ xy-plane, the tank will be an ellipse which goes through the points (0,0,0), \ (4,2,0), (4,-2,0), and (8,0,0). This ellipse is centered at (4,0,0), and \ every point on the ellipse satisfies the equation (1/4)(x-4)^2 + y^2 = 4. We \ will assume that the tank has height 10. The new tank looks like\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tank\ = \ ContourPlot3D[\((1/4)\)*\((x - 4)\)^2\ + \ y^2\ - \ 4, \ {x, 0, 8}, \ {y, \(-2\), 2}, \ {z, 0, 10}, \ AxesLabel \[Rule] {x, y, z}, \ Axes \[Rule] True]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.26566 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics3D %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.134744 1.26566 -1.30104e-18 1.26566 [ [.2521 1.27735 -3.1377 0 ] [.2521 1.27735 2.8623 9 ] [.42233 1.23522 -3.04275 0 ] [.42233 1.23522 2.95725 9 ] [.60027 1.19116 -2.94186 0 ] [.60027 1.19116 3.05814 9 ] [.78647 1.14504 -2.83448 0 ] [.78647 1.14504 3.16552 9 ] [.98152 1.0967 -2.71994 0 ] [.98152 1.0967 3.28006 9 ] [.6018 1.27025 -4.90311 0 ] [.6018 1.27025 5.09689 9.8125 ] [.00463 1.11217 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This is the same direction as in the examples above, but here it \ actually matters which direction we tip it, because the tank is no longer \ circular.\ \>", "Text"], Cell[TextData[{ StyleBox["Exercise 2\n", FontWeight->"Bold"], StyleBox["\na", FontVariations->{"CompatibilityType"->0}], ") Suppose that the fluid level in the tank is 7 m on the left edge of the \ tank (where x=0) and 5 m on the right edge (where x=8). Find the equation of \ the plane of the liquid, and use a double integral to find the volume of \ liquid in the tank. [Hint: you should use a \"dy dx\" iterated integral, \ where the bounds on y depend on x, and are given by the equation of the base \ of the cylinder.] Also find the angle at which the tank is tipped from its \ upright position. For consistency, assume that the tank in the paragraph \ before part a) above is tipped at a 0\[Degree] angle. [Hint: use a \"side \ elevation\" sketch, and use trigonometry to find the angle between your plane \ and the plane z=5.]\n\nb) Suppose now that the fluid level is 7 m at the left \ edge of the tank, but that there isn't enough fluid to reach the right edge; \ instead, the fluid covers the base of the tank only out to the plane x=4. \ [You can produce a graph with the commands below: you'll have to paste them \ into another cell.]\n", Cell[BoxData[{ \(plane5 = ParametricPlot3D[{x, y, 7 - 7 x/4}, {x, 0, 4}, {y, \(-3\), 3}]\), "\n", \(Show[tank, plane5, AxesLabel \[Rule] {x, y, z}, ViewPoint -> {0, \ \(-4\), \ 0}]\)}], "Input", CellFrame->False], "\nFind the equation of the plane of the liquid,and use a double integral \ to find the volume of liquid in the tank. Also find the angle at which the \ tank is tipped from its upright position. [", StyleBox["Warning", FontWeight->"Bold"], ": the region of integration has changed -- if you reuse the bounds from \ problem 1, you'll include the \"negative volume\" from the region where z<0.]\ \n\nc) Suppose that the tank is tipped so far that the liquid touches both \ ends of the tank, touching the top (at z=10) out to the plane x=2, and \ covering the bottom out to the plane x=4. Find the volume of liquid in the \ tank using two double integrals. Explain why you can't just use one double \ integral. Lastly, find the angle at which the tank is tipped from its \ upright position.\n\nd) Suppose, finally, that the liquid touches the top of \ the tank out to the plane x=6, and reaches a level of 4 m on the right edge \ of the tank. Find the volume of liquid in the tank using two double \ integrals, and find the angle at which the tank is tipped from its upright \ position.\n\ne) Suppose that our tank is tilted at an angle of 80\[Degree] \ from its upright position. Where could you mark the surface of the tank to \ indicate when the tank is exactly one-third full? [Hint: use trigonometry to \ decide which of the first four exercises most resembles this problem. \ Calculate the volume in terms of some unknown variable, and use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["FindRoot", FontFamily->"Courier", FontWeight->"Bold"], " command, which you can look up in the help browser.]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961]], Cell[CellGroupData[{ Cell["Credits", "Subsection", Background->None], Cell["\<\ This lab was inspired by a lab written at Lafayette College in \ Pennsylvania. Cindy Kaus (cindy.kaus@metrostate.edu) wrote the first version \ that we used at the University of Minnesota. James Swenson \ (swenson@math.umn.edu) completely rewrote the text in Spring 2002 but kept \ the same exercises. In Spring 2004 Jonathan Rogness (rogness@math.umn.edu) \ went through and added a few Live commands and polished things up a bit; Ryan \ Gantner (gantner@math.umn.edu) added new exercises, where the cylinder is no \ longer circular. This lab (or portions thereof) copyrighted by their respective authors. \ We've all agreed to use the Creative Commons \ Attribution-NonCommercial-ShareAlike License. You can find more information \ on this license at http://creativecommons.org/licenses/by-nc-sa/1.0/ Although it's not specifically required by the license, I'd appreciate it if \ you let me know at rogness@math.umn.edu if you use parts of our labs, just so \ I can keep track of it. Please send me any questions or comments!\ \>", \ "Text"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{906, 761}, WindowMargins->{{Automatic, -6}, {Automatic, 60}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{"user002", "rogness"}, \ "Newlab.nb.ps", CharacterEncoding -> "iso8859-1"], "Magnification"->1}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. 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