JESÚS AVALOS RODRÍGUEZ
Docente del Departamento Académico de Matemáticas
Facultad de Ciencias Físicas y Matemáticas
Universidad Nacional de Trujillo
E-mail: javalos@unitru.edu.pe
Embedded Files
TABLE OF INTEGRALS
TABLE OF INTEGRALS
ARC LENGHT 2
ARC LENGHT 2
ARC LENGTH CODE MATHEMATICA
ARC LENGTH CODE MATHEMATICA
==============================
Manipulate[
Module[{func, curve, arcLength, arcLengthValue, arcLengthComputation,
coords, points, polygonal, para2D, para3D, pol, polyLength},
Switch[mode, 1,
max = 2; func = curves;
curve =
Plot[func, {x, 0, max}, PlotRange -> All, AxesOrigin -> {0, 0}];
arcLength = NIntegrate[Sqrt[D[func, x]^2 + 1], {x, 0, max}];
arcLengthValue =
If[MemberQ[{x^3, Sin[Pi/2 x]}, curves],
NIntegrate[Sqrt[D[func, x]^2 + 1], {x, 0, max}],
Integrate[Sqrt[D[func, x]^2 + 1], {x, 0, max}]];
arcLengthComputation =
Row[{"\!\(\*SubsuperscriptBox[\(\[Integral]\), \(\(\\\ \\\ \
\)\(0\)\), \(2\)]\)", TraditionalForm[Sqrt[D[func, x]^2 + 1]],
it["d\[InvisibleSpace]x"],
If[MemberQ[{x^3, Sin[Pi/2 x]}, curves],
Row[{" \[TildeTilde] ", arcLength}],
Row[{" = ", arcLengthValue, " \[TildeTilde] ", arcLength}]]}];
coords = Table[{x, func}, {x, 0, max, max/n}];
points = Graphics[{PointSize[0.02], Point[coords]}];
polygonal = Graphics[Line[coords]],
2,
max = 2 Pi; para2D = curves;
curve = ParametricPlot[para2D, {t, 0, max}, PlotRange -> All];
arcLength =
NIntegrate[
Sqrt[D[para2D[[1]], t]^2 + D[para2D[[2]], t]^2], {t, 0, max}];
arcLengthValue =
If[MemberQ[{{2 Cos[t], Sin[t]}, {Cos[t], Sin[3 t]}}, curves],
NIntegrate[
Sqrt[D[para2D[[1]], t]^2 + D[para2D[[2]], t]^2], {t, 0, max}],
Integrate[
Sqrt[D[para2D[[1]], t]^2 + D[para2D[[2]], t]^2], {t, 0, max}]];
arcLengthComputation =
Row[{"\!\(\*SubsuperscriptBox[\(\[Integral]\), \(\(\\\ \\\ \
\)\(0\)\), \(2 \[Pi]\)]\)",
TraditionalForm[
Sqrt[D[para2D[[1]], t]^2 + D[para2D[[2]], t]^2]],
it["d\[InvisibleSpace]t"],
If[MemberQ[{{2 Cos[t], Sin[t]}, {Cos[t], Sin[3 t]}}, curves],
Row[{" \[TildeTilde] ", arcLength}],
If[MemberQ[{{Cos[t], Sin[t]}}, curves],
Row[{" = ", arcLengthValue, " \[TildeTilde] ", arcLength}],
Row[{" = ", arcLengthValue}]]]}];
coords = Table[para2D, {t, 0, max, max/n}];
points = Graphics[{PointSize[0.02], Point[coords]}];
polygonal = Graphics[Line[coords]],
3,
max = 4 Pi; para3D = curves;
curve =
ParametricPlot3D[para3D, {t, 0, max}, PlotRange -> All,
AspectRatio -> 1];
arcLength =
NIntegrate[
Sqrt[D[para3D[[1]], t]^2 + D[para3D[[2]], t]^2 +
D[para3D[[3]], t]^2], {t, 0, max}];
arcLengthValue =
If[MemberQ[{{Cos[t], Sin[t], Cos[t/2]}}, curves],
NIntegrate[
Sqrt[D[para3D[[1]], t]^2 + D[para3D[[2]], t]^2 +
D[para3D[[3]], t]^2], {t, 0, max}],
Integrate[
Sqrt[D[para3D[[1]], t]^2 + D[para3D[[2]], t]^2 +
D[para3D[[3]], t]^2], {t, 0, max}]];
arcLengthComputation =
Row[{"\!\(\*SubsuperscriptBox[\(\[Integral]\), \(\(\\\ \\\ \
\)\(0\)\), \(4 \[Pi]\)]\)",
TraditionalForm[
Sqrt[D[para3D[[1]], t]^2 + D[para3D[[2]], t]^2 +
D[para3D[[3]], t]^2]], it["dt"],
If[MemberQ[{{Cos[t], Sin[t], Cos[t/2]}}, curves],
Row[{" \[TildeTilde] ", arcLength}],
Row[{" = ", arcLengthValue, " \[TildeTilde] ", arcLength}]]}];
coords = Table[para3D, {t, 0, max, max/n}];
points = Graphics3D[{PointSize[0.02], Point[coords]}];
polygonal = Graphics3D[Line[coords]],
4,
max = 2 Pi; pol = curves;
curve = PolarPlot[pol, {\[Theta], 0, max}, PlotRange -> All];
arcLength =
NIntegrate[Sqrt[D[pol, \[Theta]]^2 + pol^2], {\[Theta], 0, max}];
arcLengthValue =
If[MemberQ[{}, curves],
NIntegrate[Sqrt[D[pol, \[Theta]]^2 + pol^2], {\[Theta], 0, max}],
Integrate[Sqrt[D[pol, \[Theta]]^2 + pol^2], {\[Theta], 0, max}]];
arcLengthComputation =
Row[{"\!\(\*SubsuperscriptBox[\(\[Integral]\), \(\(\\\ \\\ \
\)\(0\)\), \(2 \[Pi]\)]\)",
TraditionalForm[Sqrt[D[pol, \[Theta]]^2 + pol^2]], it["d"],
"\[Theta]",
If[MemberQ[{3 Sin[2 \[Theta]], 1 + 2 Cos[\[Theta]]}, curves],
Row[{" \[TildeTilde] ", arcLength}],
If[MemberQ[{\[Theta], \[Theta]^2}, curves],
Row[{" = ", arcLengthValue, " \[TildeTilde] ", arcLength}],
Row[{" = ", arcLengthValue}]]]}];
coords =
Table[{pol Cos[\[Theta]], pol Sin[\[Theta]]}, {\[Theta], 0, max,
max/n}];
points = Graphics[{PointSize[0.02], Point[coords]}];
polygonal = Graphics[Line[coords]]
];
polyLength =
Sum[Norm[N[coords[[i + 1]]] - N[coords[[i]]]], {i, 1, n}];
Text@Pane[
Column[{If[arc,
Show[curve, points,
PlotLabel -> Style[Row[{curves, " on [0, ", max, "]"}], 18],
ImageSize -> {300, 300}],
Show[curve, points, polygonal,
PlotLabel -> Style[Row[{curves, " on [0, ", max, "]"}], 18],
ImageSize -> {300, 300}]],
If[arc,
Style[Row[{"The length of the curve above is ",
TraditionalForm@arcLengthComputation, "."}], 18],
Style[Row[{"The length of the polygonal curve when subdividing\
\n[0, ", max, "] into ", it["n"], " = ", n ,
" equal subintervals is ", polyLength, "."}], 18]]},
Alignment -> Center], ImageSize -> {400, 400}]],
Row[{"choose a mode:"}],
ButtonBar[{Row[{it["y"], "=", it["f"], "(", it["x"], ")"}] :> {mode =
1; curves = curveLists[[mode, 1]]},
Row[{it["r"], "=", it["f"], "(\[Theta])"}] :> {mode = 4;
curves = curveLists[[mode, 1]]}, "parametric 2D" :> {mode = 2;
curves = curveLists[[mode, 1]]},
"parametric 3D" :> {mode = 3; curves = curveLists[[mode, 1]]}},
Appearance -> "Vertical"],
Delimiter,
Row[{"choose a curve:"}],
{{mode, 1}, ControlType -> None},
{{max, 2}, ControlType -> None},
{{curveList, curveLists[[1]]}, ControlType -> None},
{{curves, curveLists[[1, 1]], Null}, curveLists[[mode]],
ControlType -> PopupMenu, ImageSize -> 150},
Delimiter,
Row[{"choose a display:"}],
ButtonBar[{"arc length formula" :> {arc = True; n = 1},
"polygonal approximation" :> {arc = False}},
Appearance -> "Vertical"],
{{arc, True}, {False, True}, ControlType -> None},
{{n, 1}, 1, 50, 1, Enabled -> (arc == False),
Appearance -> "Labeled", ImageSize -> Small},
ControlPlacement -> Left,
TrackedSymbols -> True,
Initialization :> (it = Style[#, Italic] &;
curveLists = {{x^2, x^3, x^(3/2), Sqrt[x], Cosh[x],
Sin[Pi/2 x]}, {{Cos[t], Sin[t]}, {2 Cos[t], Sin[t]}, {Cos[t],
Sin[3 t]}, {t - Sin[t], 1 - Cos[t]}, {Cos[t]^3,
Sin[t]^3}}, {{Cos[t], Sin[t], t}, {Cos[t], Sin[t],
Cos[t/2]}}, {\[Theta], \[Theta]^2, 1 + Sin[\[Theta]],
3 Sin[2 \[Theta]], 1 + 2 Cos[\[Theta]]}}),
AutorunSequencing -> {5, 6}]
ARC LENGHT
ARC LENGHT
==============================
a = 0;
(*beginning point*)
b = 3; (*ending point*)
f[x_] := (1 - x)^2;(*function*)
points[x_] :=
Table[{i, f[i]}, {i, a,
b, (b - a)/
x}]; (*generates a list of the points for the line segments*)
distance[x_] :=
Sum[Sqrt[((b - a)/x)^2 + (f[i] - f[i - (b - a)/x])^2], {i,
a + (b - a)/x,
b, (b - a)/
x}];(*evaluates the approximate length of the line segment*)
MEAN VALUE THEOREM
MEAN VALUE THEOREM
================================
Manipulate[
Solutionab = x /. NSolve[f[x] == \[Alpha], x];
a1 = If[\[Alpha] < 50, Solutionab[[2]], Solutionab[[1]]];
b1 = If[\[Alpha] < 50, Solutionab[[3]], Solutionab[[2]]];
(*{a1,b1}=Select[Solutionab,-7\[LessEqual]#\[LessEqual]4&];*)
Solutionc1 =
y /. N[Solve[f[y] == Integrate[f[x], {x, a1, y}]/(y - a1), y]];
c1 = Select[Solutionc1, a1 + .1 < Re[#] < b1 &][[1]];
Solutionc2 =
y /. N[Solve[f[y] == Integrate[f[x], {x, y, b1}]/(b1 - y), y]];
c2 = Select[Solutionc2, a1 < Re[#] < b1 - .1 &][[1]];
(*{c}=Select[Solutionc,-7\[LessEqual]#\[LessEqual]4&& (#-a1)f[a1]\
\[Equal]Integrate[f[x],{x,a1,#}]&];*)
Grid[{
{Plot[{f[x], \[Alpha]} , {x, -7.5, 4.5}, Filling -> {1 -> Axis},
FillingStyle -> Directive[Opacity[0], Red],
AxesOrigin -> {0, 0}, PlotRange -> {{-7.5, 4.2}, {-15, 110}},
PlotStyle -> {{Thickness[.004], Blue}, {Thickness[.003], Dashed,
Red}},
Prolog -> {Text[Style["a", Medium, Italic, Blue], {a1, -10}],
Text[Style["b", Medium, Italic, Blue], {b1, -10}],
Text[Style["c", Medium, Italic, Blue], {c1, -10}],
{Gray, Dashed, Thickness[0.001], Line[{{a1, 0}, {a1, f[a1]}}]},
Text[
Style[Row[{Style["y", Italic], " = \[Alpha]"}], Medium,
Black], {3, \[Alpha] + 5}],
{Gray, Dashed, Thickness[0.001], Line[{{b1, 0}, {b1, f[b1]}}]},
{Blue, Opacity[.4], Rectangle[{a1, 0}, {c1, f[c1]}]},
{Yellow, Opacity[.5], Thickness[0.01],
Table[Line[{{a1 + h, 0}, {a1 + h, f[a1 + h]}}], {h, .1,
c1 - a1, .1}]},
{Gray, Dashed, Thickness[0.001], Line[{{c1, 0}, {c1, f[c1]}}]}},
Epilog -> {}, ImageSize -> {295, 200}],
Plot[{f[x], \[Alpha]} , {x, -7.5, 4.5}, Filling -> {1 -> Axis},
FillingStyle -> Directive[Opacity[0], Red],
AxesOrigin -> {0, 0}, PlotRange -> {{-7.5, 4.2}, {-15, 110}},
PlotStyle -> {{Thickness[.004], Blue}, {Thickness[.003], Dashed,
Red}},
Prolog -> {Text[Style["a", Medium, Italic, Blue], {a1, -10}],
Text[Style["b", Medium, Italic, Blue], {b1, -10}],
Text[Style["d", Medium, Italic, Blue], {c2, -10}],
{Gray, Dashed, Thickness[0.001], Line[{{a1, 0}, {a1, f[a1]}}]},
Text[
Style[Row[{Style["y", Italic], " = \[Alpha]"}], Medium,
Black], {3, \[Alpha] + 5}],
{Gray, Dashed, Thickness[0.001], Line[{{b1, 0}, {b1, f[b1]}}]},
{Blue, Opacity[.4], Rectangle[{c2, 0}, {b1, f[c2]}]}, {Yellow,
Opacity[.5], Thickness[0.01],
Table[Line[{{c2 + h, 0}, {c2 + h, f[c2 + h]}}], {h, 0,
b1 - c2, .1}]},
{Gray, Dashed, Thickness[0.001], Line[{{c1, 0}, {c1, f[c1]}}]}},
Epilog -> {}, ImageSize -> {295, 200}]},
{Text[Style["blue area = yellow area", 15, Blue]],
Text[Style["blue area = yellow area", 15, Blue]]
}}],
{{\[Alpha], 70, "\[Alpha]"}, 10, 100, Appearance -> "Labeled"},
SaveDefinitions -> True,
TrackedSymbols -> {\[Alpha]}
]
======================================
INTEGRATION AS A SUM
INTEGRATION AS A SUM
===================================================
Manipulate[
With[{ans = Integrate[func, {x, x1, x2}]},
Show[Graphics[
Plot[func, {x, -2, 2},
PlotLabel ->
Style[Framed[Row[{"Area = ", NumberForm[ans, {2, 4}]}]],
If[ans > 0, RGBColor[1, .47, 0], RGBColor[.12, .61, .78]], 12,
"Label"] ]],
Graphics[
If[x1 == x2, {},
Plot[func, {x, x1, x2}, Filling -> 0 ,
FillingStyle ->
If[ans > 0, RGBColor[1, .47, 0], RGBColor[.12, .61, .78]]]]],
ImageSize -> {350, 350}]], {{x1 , 0, "x1"}, -2, 2,
Appearance -> "Labeled"}, {{x2, 1, "x2"}, -2, 2,
Appearance -> "Labeled"} , {{func , f1,
"curve"} , {f1 -> "constant line", f2 -> "sine wave",
f3 -> "parabola"}, ControlType -> SetterBar},
SaveDefinitions -> True ]
===================================================
RIEMANN SUMS: A SIMPLE ILLUSTRATION
RIEMANN SUMS: A SIMPLE ILLUSTRATION
A Riemann sum is an approximation to the area between a curve and the x axis, made by adding together the areas of a set of rectangles. A summation calculation is involved, of the form Sum[f(x)./\x], where /\x is the width of each rectangle. The limit of the Riemann sum as /\x approaches zero is the (Riemann) integral of the function. This manipulation uses a left Riemann sum, in which the value of f(x) that is used is the one at the left edge of each rectangle. It is easy to adapt the code for right Riemann sums and for the trapezoidal rule; try it!
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