(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 493521, 9550] NotebookOptionsPosition[ 465642, 8893] NotebookOutlinePosition[ 482377, 9193] CellTagsIndexPosition[ 482334, 9190] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ TAREA 6\.aa: C\[AAcute]lculo diferencial e integral\ \>", "Title", CellChangeTimes->{ 3.427778666641778*^9, {3.467914451796875*^9, 3.467914464328125*^9}}], Cell[TextData[{ "Ejercicio 1\.ba. 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Cell[TextData[{ "Ejercicio 2 \.ba.- Calcular los siguientes l\[IAcute]mites:\n\t1) ", Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "\[Infinity]"}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"sen", "(", FractionBox["1", "x"], ")"}], "+", RowBox[{"cos", "(", FractionBox["1", "x"], ")"}]}], ")"}], "x"], TraditionalForm]]], "\n\t2) ", Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "0"}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"cos", "(", RowBox[{"x", "\[CenterDot]", " ", SuperscriptBox["e", "x"]}], ")"}], "-", RowBox[{"cos", "(", RowBox[{"x", "\[CenterDot]", " ", SuperscriptBox["e", RowBox[{"-", "x"}]]}], ")"}]}], SuperscriptBox["x", "3"]], TraditionalForm]]], "\n\t3) ", Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "\[Infinity]"}]], TraditionalForm]]], " ", Cell[BoxData[ FractionBox[ RowBox[{ RadicalBox["x", 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Vamos a probarlo con el mismo comando Limit, \ pero a\[NTilde]adiendo la opci\[OAcute]n ", "Assumptions \[Rule] a > 0." }], "Text", CellChangeTimes->{{3.468674468234375*^9, 3.468674513328125*^9}, { 3.468674551921875*^9, 3.468674564640625*^9}, {3.468674634375*^9, 3.4686747419375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"\[Piecewise]", GridBox[{ { RowBox[{ SuperscriptBox["x", "a"], " ", RowBox[{"Sin", "[", FractionBox["1", "x"], "]"}]}], RowBox[{"x", "\[NotEqual]", "0"}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ",", RowBox[{"x", "\[Rule]", "0"}], ",", 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RowBox[{"-", "1"}], RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]], RowBox[{"|", "x", "|", RowBox[{"<", "1"}]}]}, {"9", RowBox[{"|", "x", "|", RowBox[{"\[GreaterEqual]", " ", "1"}]}]} }]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.46769098178125*^9, 3.46769124471875*^9}, { 3.467914831859375*^9, 3.467914836140625*^9}}], Cell["\<\ 1) Al aparecer el valor absoluto definimos la funci\[OAcute]n utilizando el \ comando Piecewise\ \>", "Text", CellChangeTimes->{{3.46867535059375*^9, 3.468675389328125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{"Piecewise", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Exp", "[", "x", "]"}], ",", RowBox[{"x", "<", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Exp", "[", RowBox[{"-", "x"}], "]"}], ",", RowBox[{"x", "\[GreaterEqual]", "0"}]}], "}"}]}], "}"}], "]"}]}], "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{ 3.46791489621875*^9, {3.468674875796875*^9, 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TraditionalForm]]], " y ", Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"-", "x"}]], TraditionalForm]]], " lo son. \nEn x=0 vamos a comprobar las derivadas laterales: " }], "Text", CellChangeTimes->{{3.46867500378125*^9, 3.468675081484375*^9}, { 3.468675217125*^9, 3.468675218125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}]}]], "Input", CellChangeTimes->{{3.46867505546875*^9, 3.468675086234375*^9}}], Cell[BoxData[ RowBox[{"\[Piecewise]", GridBox[{ { SuperscriptBox["\[ExponentialE]", "x"], RowBox[{"x", "<", "0"}]}, { RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "x"}]]}], RowBox[{"x", ">", "0"}]}, {"Indeterminate", TagBox["True", "PiecewiseDefault", AutoDelete->False, DeletionWarning->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}]], "Output", CellChangeTimes->{3.47230505146875*^9}] }, Open ]], Cell[TextData[{ Cell[BoxData[ FormBox[ UnderscriptBox[ RowBox[{"lim", " "}], RowBox[{"x", "\[Rule]", SuperscriptBox["0", "+"]}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"f", "[", "0", "]"}]}], "x"], TraditionalForm]]], " y ", Cell[BoxData[ FormBox[ UnderscriptBox[ RowBox[{"lim", " "}], RowBox[{"x", "\[Rule]", SuperscriptBox["0", "-"]}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"f", "[", "0", "]"}]}], "x"], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.468675140015625*^9, 3.468675228875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"f", "[", "0", "]"}]}], "x"], ",", RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"Direction", "\[Rule]", RowBox[{"-", "1"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.468675237421875*^9, 3.468675283109375*^9}}], Cell[BoxData[ RowBox[{"-", "1"}]], "Output", CellChangeTimes->{3.4723050515625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"f", "[", "0", "]"}]}], "x"], ",", RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"Direction", "\[Rule]", "1"}]}], "]"}]], "Input", CellChangeTimes->{{3.468675237421875*^9, 3.468675293546875*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{3.47230505159375*^9}] }, Open ]], Cell["\<\ Como estos l\[IAcute]mites son distintos, no existe el l\[IAcute]mite, o sea, \ no es derivable en x = 0.\ \>", "Text", CellChangeTimes->{{3.4686752970625*^9, 3.468675320015625*^9}}], Cell["\<\ 2) Igual que en 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GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}]], "Output", CellChangeTimes->{3.472305051625*^9}] }, Open ]], Cell["\<\ Otra vez, para todo x distinto de 0, la funci\[OAcute]n es derivable. en x=0 \ volvemos a calcular las derivadas por la derecha y por la izda. \ \>", "Text", CellChangeTimes->{{3.46867550034375*^9, 3.4686755598125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"f", "[", "0", "]"}]}], "x"], ",", RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"Direction", "\[Rule]", RowBox[{"-", "1"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.468675237421875*^9, 3.468675283109375*^9}}], Cell[BoxData["\[Infinity]"], "Output", CellChangeTimes->{3.47230505165625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"f", "[", "0", "]"}]}], "x"], ",", RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"Direction", "\[Rule]", "1"}]}], "]"}]], "Input", CellChangeTimes->{{3.468675237421875*^9, 3.468675293546875*^9}}], Cell[BoxData[ RowBox[{"-", "\[Infinity]"}]], "Output", CellChangeTimes->{3.4723050516875*^9}] }, Open ]], Cell["\<\ En este caso tampoco existe el l\[IAcute]mite luego la funci\[OAcute]n no es \ derivable en 0.\ \>", "Text", CellChangeTimes->{{3.468675577484375*^9, 3.468675595453125*^9}}], Cell["\<\ 3) En este caso los puntos en los que la derivada puede no existir son x=1 y \ x=-1.\ \>", "Text", CellChangeTimes->{{3.468675663390625*^9, 3.468675665046875*^9}, { 3.46867586784375*^9, 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Del mismo modo se \ comprueba en x = -1. 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Determinar las dimensiones de la secci\[OAcute]n que hacen m\[IAcute]nimo \ el per\[IAcute]metro que se encuentra en contacto con el fluido. 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CellChangeTimes->{{3.46791407478125*^9, 3.467914253578125*^9}, { 3.467914880296875*^9, 3.467914884890625*^9}, {3.4721480065*^9, 3.472148011390625*^9}, {3.472148058140625*^9, 3.4721480610625*^9}, { 3.47214817775*^9, 3.4721482203125*^9}, {3.47214830340625*^9, 3.472148317359375*^9}, {3.472148445015625*^9, 3.4721485074375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "1", "2"], RowBox[{ SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"x", "^", "2"}], "2"], ")"}], "2"], "*", "Pi", RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input", CellChangeTimes->{{3.472148543953125*^9, 3.472148584671875*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"31", " ", "\[Pi]"}], "20"]], "Output", CellChangeTimes->{3.472148588296875*^9, 3.47230505803125*^9}] }, Open ]], Cell[TextData[{ "Ejercicio 11 \.ba.- Calcular el \[AAcute]rea de la regi\[OAcute]n del plano \ comprendida entre la hip\[EAcute]rbola de ecuaci\[OAcute]n :\n\t", Cell[BoxData[ FormBox[ RowBox[{"\t", RowBox[{ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["a", "2"]], "-"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["y", "2"], SuperscriptBox["b", "2"]], TraditionalForm]]], "=1\ny la cuerda de ecuaci\[OAcute]n x=h, con h>a." }], "Text", CellChangeTimes->{{3.467914947609375*^9, 3.467915039109375*^9}, { 3.47214728365625*^9, 3.47214728365625*^9}}], Cell["\<\ En este caso no podemos hacer la representaci\[OAcute]n gr\[AAcute]fica ya \ que la curva viene dada en funci\[OAcute]n de un par\[AAcute]metro : a y la \ cuerda en funci\[OAcute]n de otro : h. Sin embargo podemos tomar dos valores \ concretos para tener una idea de la representaci\[OAcute]n gr\[AAcute]fica \ del \[AAcute]rea pedida, por ejemplo a=2, b=4 y h=4.\ \>", "Text", CellChangeTimes->{{3.472105347875*^9, 3.47210546378125*^9}, { 3.47214733296875*^9, 3.472147342296875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "/", "4"}], "-", RowBox[{ RowBox[{"y", "^", "2"}], "/", "16"}]}], "\[Equal]", "1"}], ",", RowBox[{"x", "\[Equal]", "4"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "7"}], ",", "7"}], "}"}], ",", RowBox[{"Axes", "\[Rule]", "True"}]}], "]"}], "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{ 3.467915042359375*^9, {3.47210547890625*^9, 3.47210552903125*^9}, { 3.47210557659375*^9, 3.47210560328125*^9}, {3.472147346609375*^9, 3.472147359734375*^9}}], Cell[BoxData[ GraphicsBox[GraphicsComplexBox[CompressedData[" 1:eJxllQtMU1cYx+8ttPZFKaUtpUUeyiaD+AqKUTLuZ5TF4oDBps4EmMOIiqLL MlxElrhscegyH5gNicvI1IkO3Rxz4jRyz3DMKIwNNOAGCWimwbfihI6Xo93p Nfx30ubmf/v7es53vvP9T0z+xuxVKkEQGsa+3mfM/vZaOcpMgm9E0HPu8li3 xUxdF4v3Te1x0dHsPnVXczBtkY8NNJW4qOtUUeepomD6rDH6WpvRRYMbPFm5 +cE83kWxuyPVE9TBJAYcaNxV7qQNnd4AE+1IvmLoTXLSR9FnXr23zcR5J4Wk b28UU0yUeUE6PeNoOH9vopxdDU/icsLpLUPf9Z53g2j4weYne+od1HFoffql diPnHJRGZy+drTVS4h//nChvC/tv/eVG6veUhf5yJIzuZNQkH79voOy8BZuD bthp7t7EsY+Bx9spoeD3a5cTDJTXUZzb8cRGad7wWAMVdt/scdy10eLI0zs2 HdbTrIFZDW9n2shYde94bqGelsZsNZ4ItPH8dCT/+tPMq1FW+i6tYEvp1zpq Hzn4Y1aIlVKq2j91fa7j81lJrE6Y676rpdHrDaWVrlDyTefWUm3coeWj8y0k JJ7+cnC6lhKzZ9bNm2yhpsyy7/tCtTzeQmJT66TW4gk0aq+e3TklhOq85ejU 0AZnfN2xDDOdm/TixPB6De1RLdSbXzBT76LqDk2VhsebacXT/OapOg3tbLnd X/uyiddTTdvqz/dVlhjJsHMo5L1lair9SlqdnGKkGbaTg9nz1Tzev++B/Kkl lj6z0NOpUt73jy4W36xXUfravOHtHwaQL71yFX1zoXf17J/H3jvPTS6OU9GU D+7vz4oTqWLOSzNau0UeL9CVJWcDRvaLtCzKeWfByhHJFx4rUuqFvUtf2+6R ni/q/djtEqn/t1fOjSR6pDce2cY+/niP5Fo/ry1SL9LK3JLm7nUD0qMDt+NS awQq27jwfXd1n1R88PBfE0sEzj+U4n9IOt+aKdCqzS1H2jJuSSse7HNeTRbo YmpY7jutN6RMU4VYFeHneyTnkpTlXRaBznzyekXk4w7JtCi+JlMl0MCcSte3 a85InStb1t0afCrxfAT4XYZ4Gf5fhvllWJ8M65chPxnyl2F/ZNg/GfZXhv2X oT4M6segvgzqz+B8sD+FaSfFnACaU3GjaFqTmsF5YnDeGJxHBueVwXlmcN4Z 9AODfmHQTwz6jUE/MuhXBv3MoN8Z+AHz+8UXaxZMqIq2sjVJl41bC/SctzHw Fwb+w8CfGPgXA39j4H8M/JGBfzLwVwb+y+ul+DMD/2bg7wz8n8H9wOD+YHC/ MLh/GNxPDO4vvr7/61mJ3uFSdLNvOIF3Ah8OvAN4B/BhwNuBtwNvA94KvBX4 UOAtwFuADwHeDPwzvbbAO0yKrqj0jiDgjcDrgdcBrwVeA7wa+EDgVcCLwAuK bvXlNyz79XRf/kPyeN4jj+f/Bv4x8A8VfdA3/y1F5/rWdxP4HkWn+v7vsqI3 +eIvAu8fyu8SxEvj+R5F8/kVzdcH/ENF8/wUzfMH3gP8EPDDwPvvIaU+iub1 o/F8IPBq4DXAa4HXAa8H3gh8EPAm4J89eX8omvcPcBbgLcCHAm8F3gq8DXg7 8Hbgw4B3AO8APhx4J/BO4F3AR9C/W9yPRQ== "], {{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109}]}, {Hue[0.9060679774997897, 0.6, 0.6], LineBox[{110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194}]}}], AspectRatio->1, Axes->True, Frame->True, PlotRange->{{0, 5}, {-7, 7}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{ 3.4721055301875*^9, {3.472105583578125*^9, 3.47210560484375*^9}, { 3.47214735153125*^9, 3.472147360546875*^9}, 3.472305058171875*^9}] }, Open ]], Cell[TextData[{ "Podemos calcular el \[AAcute]rea como el doble de la comprendida por encima \ del eje horizontal, teniendo en cuenta que el punto de corte de la hip\ \[EAcute]rbola con dicho eje es x=a , y que la ecuaci\[OAcute]n \ correspondiente a la rama de la hip\[EAcute]rbola de ordenadas positivas es : \ ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["b", "a"], SqrtBox[ RowBox[{ SuperscriptBox["x", "2"], "-", SuperscriptBox["a", "2"]}]]}], TraditionalForm]]], ".\nEn el c\[AAcute]lculo de la integral, utilizando el comando Simplify, \ hay que indicar que a es un valor real. Esto se hace utilizando la opci\ \[OAcute]n Element[a, Reals]. Tambi\[EAcute]n hay que indicar que a y h son \ mayores que 0, y que h-a es mayor que 0. Esto se hace con la opci\[OAcute]n ", "Assumptions \[Rule] a > 0 && h > 0 && h - a > 0." }], "Text", CellChangeTimes->{{3.472105655875*^9, 3.472105676625*^9}, { 3.47210572940625*^9, 3.4721058633125*^9}, {3.472147429703125*^9, 3.47214743415625*^9}, {3.472147699421875*^9, 3.472147879421875*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"a", ",", "h"}], "]"}]], "Input", CellChangeTimes->{{3.47210589646875*^9, 3.4721058998125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "=", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"2", RowBox[{"b", "/", "a"}], "*", RowBox[{ SubsuperscriptBox["\[Integral]", "a", "h"], RowBox[{ SqrtBox[ RowBox[{ RowBox[{"x", "^", "2"}], "-", RowBox[{"a", "^", "2"}]}]], RowBox[{"\[DifferentialD]", "x"}]}]}]}], ",", RowBox[{"Element", "[", RowBox[{"a", ",", "Reals"}], "]"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{ RowBox[{"a", ">", "0"}], "&&", RowBox[{"h", ">", "0"}], "&&", RowBox[{ RowBox[{"h", "-", "a"}], ">", "0"}]}]}]}], "]"}]}]], "Input", 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Para ello utilizamos el comando Solve\ \>", "Text", CellChangeTimes->{{3.472105063890625*^9, 3.472105116484375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "\[Equal]", RowBox[{"tg", "[", "x", "]"}]}], ",", "x"}], "]"}]}]], "Input", CellChangeTimes->{{3.472105123390625*^9, 3.472105153265625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", "1"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.472105154328125*^9, 3.47230510909375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"a", "=", RowBox[{"x", "/.", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.47210520275*^9, 3.472105209359375*^9}}], Cell[BoxData[ RowBox[{"-", "2"}]], "Output", CellChangeTimes->{3.472105210515625*^9, 3.472305109109375*^9}] }, Open ]], Cell["El valor del \[AAcute]rea ser\[AAcute] :", "Text", CellChangeTimes->{{3.47210521653125*^9, 3.47210522534375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "=", RowBox[{ SubsuperscriptBox["\[Integral]", "a", "1"], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"f", "[", "x", "]"}], "-", RowBox[{"tg", "[", "x", "]"}]}], ")"}], RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Input", CellChangeTimes->{{3.472105232703125*^9, 3.472105242046875*^9}, { 3.472105287109375*^9, 3.47210529315625*^9}}], Cell[BoxData[ FractionBox["27", "4"]], "Output", CellChangeTimes->{3.47210529503125*^9, 3.472305109171875*^9}] }, Open ]], Cell["\<\ Utilizamos el operador reemplazamiento para extraer la primera \ soluci\[OAcute]n del resultado :\ \>", "Text", CellChangeTimes->{{3.47210516709375*^9, 3.472105191515625*^9}}], Cell[TextData[{ "Ejercicio 13 \.ba.- Calcular el \[AAcute]rea de la regi\[OAcute]n del plano \ comprendida entre el eje OX y cada una de las siguientes curvas de \ ecuaciones, representando gr\[AAcute]ficamente las curvas previamente :\n\ta) \ ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["y", "2"], " ", "=", " ", RowBox[{ RowBox[{"4", SuperscriptBox["x", "2"]}], "-", SuperscriptBox["x", "4"], " "}]}], TraditionalForm]]], " , y \[GreaterEqual] 0\n\tb) y= sen(2x) , 0\[LessEqual] x \ \[LessEqual]4", "\[Pi] ." }], "Text", CellChangeTimes->{{3.4679151575625*^9, 3.46791532846875*^9}, { 3.47210424284375*^9, 3.47210425903125*^9}}], Cell["\<\ a) Realizamos la gr\[AAcute]fica en primer lugar utilizando el comando \ ContourPlot.\ \>", "Text", CellChangeTimes->{{3.472104262875*^9, 3.47210429221875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{"y", "^", "2"}], "\[Equal]", RowBox[{ RowBox[{"4", RowBox[{"x", "^", "2"}]}], "-", RowBox[{"x", "^", "4"}]}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "2"}], "}"}]}], "]"}], 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Podemos obtener \ tambi\[EAcute]n el valor aproximado\ \>", "Text", CellChangeTimes->{{3.47210288371875*^9, 3.4721029106875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "L1", "]"}]], "Input", CellChangeTimes->{{3.472102934109375*^9, 3.472102936453125*^9}}], Cell[BoxData["1.4789428575445975`"], "Output", CellChangeTimes->{3.472102939046875*^9, 3.472305110796875*^9}] }, Open ]], Cell[TextData[{ "2) La longitud de una curva plana dada en forma param\[EAcute]trica ", Cell[BoxData[ FormBox[ RowBox[{"\[Piecewise]", "\[NoBreak]", GridBox[{ { RowBox[{"x", "="}], RowBox[{"x", "(", "t", ")"}]}, { RowBox[{"y", "="}], RowBox[{"y", "(", "t", ")"}]} }]}], TraditionalForm]], FormatType->"TraditionalForm"], " es :\n\tL=", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x", "'"}], RowBox[{"(", "t", ")"}]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"y", "'"}], RowBox[{"(", "t", ")"}]}], ")"}], "2"]}]], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.472102942484375*^9, 3.47210304421875*^9}, 3.4721037891875*^9, {3.472103830796875*^9, 3.472103831796875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"x", "[", "t_", "]"}], "=", RowBox[{"r", "*", RowBox[{"(", RowBox[{"t", "-", RowBox[{"Sin", "[", "t", "]"}]}], ")"}]}]}], ";", RowBox[{ RowBox[{"y", "[", "t_", "]"}], "=", RowBox[{"r", "*", RowBox[{"(", RowBox[{"1", "-", RowBox[{"Cos", "[", "t", "]"}]}], ")"}]}]}], ";", RowBox[{"L2", "=", RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "Pi"}]], RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}], "2"], "+", SuperscriptBox[ RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}], "2"]}]], RowBox[{"\[DifferentialD]", "t"}]}]}]}]}]], "Input", CellChangeTimes->{{3.472103064671875*^9, 3.472103195125*^9}, 3.4721038705*^9} ], Cell[BoxData[ RowBox[{"8", " ", SqrtBox[ SuperscriptBox["r", "2"]]}]], "Output", CellChangeTimes->{3.472103201546875*^9, 3.472103881859375*^9, 3.47230511128125*^9}] }, Open ]], Cell["\<\ En este caso no podemos obtener el valor aproximado. Mathematica ha realizado \ el c\[AAcute]lculo simb\[OAcute]lico dando el resultado en funci\[OAcute]n \ del par\[AAcute]metro r.\ \>", "Text", CellChangeTimes->{{3.472103888421875*^9, 3.47210396534375*^9}}, TextJustification->1.], Cell[TextData[{ "Ejercicio 15 \.ba.- Representar las superficies : ", Cell[BoxData[ FormBox[ SubscriptBox["S", "1"], TraditionalForm]]], ": 4", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " +", Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]]], "- 4 = 0, ", Cell[BoxData[ FormBox[ SubscriptBox["S", "2"], TraditionalForm]]], ": y+z-4 = 0.\nCalcular el volumen limitado por las superficies anteriores y \ ", Cell[BoxData[ FormBox[ SubscriptBox["S", "3"], TraditionalForm]]], ": z = 0." }], "Text", CellChangeTimes->{{3.467962709265625*^9, 3.46796286734375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"4", 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