(************** 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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See ", ButtonBox["license.nb ", ButtonData:>{"license.nb", None}, ButtonStyle->"Hyperlink"], "for details." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Indefinite Integrals"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "While ", StyleBox["Mathematica", FontSlant->"Italic"], " can calculate the derivative of any expression, it cannot find the \ integral of an arbitrary expression. In a sense, most functions do not have \ an integral that can be expressed in terms of elementary functions. You can \ find indefinite integrals with the ", StyleBox["Integrate[]", "Input"], " function. This function takes two arguments, the function to integrate, \ and the integration variable." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(Integrate[x\^2 + Sin[x], x]\)], "Input"], Cell["\<\ To get the typeset version: \t\[EscapeKey]int\[EscapeKey] (x \ \[ControlKey]\[LeftModified]6\[RightModified] 2 \[ControlKey]\[LeftModified]\ \[SpaceKey]\[RightModified] +Sin[x]) \[EscapeKey]dd\[EscapeKey] x\ \>", "Text",\ TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(\[Integral]\((x\^2 + Sin[x])\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "The indefinite integral of a function is only determined up to a constant. \ This constant is not shown in the output. If ", StyleBox["Mathematica", FontSlant->"Italic"], " is unable to find an expression for an integral in terms of the functions \ that it knows, it returns the input unchanged." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]Sin[Sin[x]] \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " knows a lot of obscure functions. (Many functions are defined in terms \ of an integral or in terms of the solution of a differential equation.) The \ result of an integration may involve such functions." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\(Sin[x]\/x\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(?SinIntegral\)\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also integrate expressions involving formal functions." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ RowBox[{\(f[x]\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], \(\[DifferentialD]x\)}]}]], "Input", AspectRatioFixed->True], Cell[TextData[{ "You must use some caution when integrating expressions that involve \ parameters. When you ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to integrate ", Cell[BoxData[ \(TraditionalForm\`x\^n\)]], ", it gives the ``general'' result and assumes that ", Cell[BoxData[ \(TraditionalForm\`n\ \[NotEqual] \(-1\)\)]], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\(x\^n\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "If you later substitute the value ", Cell[BoxData[ \(TraditionalForm\`n\ = \ \(-1\)\)]], " into the answer you do not get the correct value of the integral." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(% /. n \[Rule] \(-1\)\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "You can get the correct value of the integral only by substituting ", Cell[BoxData[ \(TraditionalForm\`n\ = \ \(-1\)\)]], " before you integrate." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\((x\^n /. n \[Rule] \(-1\))\) \[DifferentialD]x\)], "Input",\ AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Definite Integrals"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "For definite integrals, you specify the integration variable and the \ limits of integration in the second argument of ", StyleBox["Integrate[]", "Input"], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(Integrate[Sin[x], {x, 0, \[Pi]}]\)], "Input"], Cell["\<\ For the typeset version, \t\[EscapeKey]int\[EscapeKey] \[ControlKey]\[LeftModified]-\[RightModified] 0 \ \[ControlKey]\[LeftModified]5\[RightModified] \[EscapeKey]p\[EscapeKey] \ \[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified] Sin[x] \[EscapeKey]dd\ \[EscapeKey] x or \t\[EscapeKey]int\[EscapeKey] \[ControlKey]\[LeftModified]-\[RightModified] 0 \ \[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified] \[ControlKey]\ \[LeftModified]6\[RightModified] \[EscapeKey]p\[EscapeKey] \[ControlKey]\ \[LeftModified]\[SpaceKey]\[RightModified] Sin[x] \[EscapeKey]dd\[EscapeKey] \ x\ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(\[Integral]\_0\%\[Pi] Sin[x] \[DifferentialD]x\)], "Input"], Cell[TextData[{ "Again, if ", StyleBox["Mathematica", FontSlant->"Italic"], " is unable to evaluate the integral exactly in terms of the functions it \ knows, it will return the input unchanged." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\_0\%\[Pi] Sin[Sin[x]] \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell["\<\ However, you can get a numerical approximation to such a definite \ integral with the numerical function.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(N[%]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "If you want a numerical approximation in the first place, use the ", StyleBox["NIntegrate[]", "Input"], " function." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(NIntegrate[Sin[Sin[x]], {x, 0, \[Pi]}]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will notify you if an integral does not converge." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\_\(-1\)\%1\( 1\/x\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "You must be careful when you do definite integrals where the limits of \ integration are parameters. Below ", StyleBox["Mathematica", FontSlant->"Italic"], " warns you that the integral may not converge for all values of the \ parameters." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\_a\%b\( 1\/x\^2\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "Indeed, if you substitute the limits ", Cell[BoxData[ \(TraditionalForm\`a\ = \ \(-1\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`b\ = \ 1\)]], ", you get an incorrect result." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(% /. {a \[Rule] \(-1\), b \[Rule] 1}\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "If ", StyleBox["Mathematica", FontSlant->"Italic"], " had known the limits of integration, it would have warned that the \ integral diverges." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\_\(-1\)\%1\( 1\/x\^2\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Exercise 2", "Subsubsection"], Cell[TextData[{ "Find the area of the region below the curve ", Cell[BoxData[ \(TraditionalForm\`y = \(-x\^2\) - 2 x + 10\)]], " and above the curve ", Cell[BoxData[ \(TraditionalForm\`y = 2 x - 1\)]], "." }], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Solutions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solution 2", "Subsubsection"], Cell[TextData["First we find where the curves intersect."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(endPts = x /. Solve[\(-x\^2\) - 2\ x + 10 \[Equal] 2\ x - 1]\)], "Input",\ AspectRatioFixed->True], Cell[TextData["The approximate value of the intersections are:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(TableForm[N[endPts]]\)], "Input", AspectRatioFixed->True], Cell[TextData["Here is a plot of the region."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(Plot[{\(-x\^2\) - 2\ x + 10, 2\ x - 1}, {x, \(-6\), 2}]; \)\)], "Input", AspectRatioFixed->True], Cell[TextData["Now we find the area of the region."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\_\(endPts[\([1]\)]\)\%\(endPts[\([2]\)]\)\((\(-x\^2\) - 2\ x + 10 - \((2\ x - 1)\))\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[TextData[ "Mathematica is able to simplify the above result considerably."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Simplify[%]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[endPts]\)], "Input", AspectRatioFixed->True] }, Open ]] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1680}, {0, 963}}, WindowToolbars->"EditBar", CellGrouping->Automatic, WindowSize->{772, 768}, WindowMargins->{{2, Automatic}, {Automatic, 0}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"XAutomaticEncoding", Magnification->1.5 ] (******************************************************************* Cached data follows. 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