Integration

Integralfunktion und Stammfunktion

∫Sin[x]\[DifferentialD]x

(* Integrate[Sin[x],x] *)

(∫_0^π)Sin[x]\[DifferentialD]x

(* Integrate[Sin[x],{x,0,π] *)

Plot[Sin[x],{x,0,π},Filling->Axis,PlotRange->{{-1,6},{-1.5,1.5}}]

Dynamic[Manipulate[
Show[Plot[f[x],{x,0,2π},PlotStyle->{Black,Thin}],Plot[f[x],{x,-0.001,a},Filling->Axis],PlotLabel->Row[{Style["F(a)",Italic],"=",NumberForm[∫f[x]\[DifferentialD]x/.{x->"a"},4],"="NumberForm[(∫_0^a)f[x]\[DifferentialD]x,4]}] ,AxesOrigin->{0,0}],{a,0,2π},FrameLabel->Row[{Style["f(x)",Italic],"=",PopupMenu[Dynamic[f],{Sin,Sinh,
Exp,Sqrt,Tan,Log}]}]]]

Uneigentliche Integrale

(∫_1^∞)(1/x)\[DifferentialD]x

(∫_1^∞)(1/(x^2))\[DifferentialD]x

Integrate[(1/(x^p)),{x,1,∞},Assumptions->(p>2)]

Flachenmessung

Ellipse

Manipulate[Show[ParametricPlot[{a Cos[t],b Sin[t]},{t,0,2 π}],PlotRange->{{-5,5},{-5,5}},PlotLabel->Row[{Style["F",Italic],"= abπ =",NumberForm[Abs[(∫_0^(2π))b Sin[t]D[a Cos[t],t]\[DifferentialD]t],4]}]
],{a,1,4},{b,1,4}]

Zykloide

x[t_]=t- Sin[t]
y[t_]=1-Cos[t]
Integrate[y[t] * D[x[t],t],{t,0,2π}]

Manipulate[
Show[ParametricPlot[{x[t],y[t]},{t,0,2π},PlotStyle->{Black, Thin}],
ParametricPlot[{t- Sin[t],r(1-Cos[t])},{t,-0.001,s},{r,0,1},Mesh->20,MeshStyle->None],
ParametricPlot[{s- Sin[t],1-Cos[t]},{t,0,2π}],Graphics[{PointSize[Large],Point[{s- Sin[s],1-Cos[s]}]}],
PlotRange->{{-1,7.5},{-1,3}},PlotLabel->Row[{Style["F",Italic],"=",NumberForm[(∫_0^s)y[t] * D[x[t],t]\[DifferentialD]t,4]}]
],{s,0,2π}]

Astriode

Manipulate[
Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2π},PlotStyle->{Black, Thin}],
ParametricPlot[{((Cos[s])^3)+(1/4)Cos[t],((Sin[s])^3)+(1/4)Sin[t]},{t,0,2π}],
ParametricPlot[{((Cos[t])^3),((Sin[t])^3)},{t,0,s}],
Graphics[{PointSize[Large],Point[{((Cos[s])^3),((Sin[s])^3)}]}],
PlotRange->{{-1.5,1.5},{-1.5,1.5}}
],{s,-0.0001,2π}]

x[t_]=((Cos[t])^3)
y[t_]=((Sin[t])^3)
2*Integrate[y[t] * D[x[t],t],{t,0,π}] 
(1/2)Abs[Integrate[y[t] * D[x[t],t]-x[t] * D[y[t],t],{t,0,2π}]] (*als Sektorflache*)

N[%]

Manipulate[
Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2π},PlotStyle->{Black, Thin}],
ParametricPlot[{((Cos[s])^3)+(1/4)Cos[t],((Sin[s])^3)+(1/4)Sin[t]},{t,0,2π}],
ParametricPlot[{((Cos[t])^3),r ((Sin[t])^3)},{t,0,s},{r,0,1},Mesh->40,MeshStyle->None],
Graphics[{PointSize[Large],Point[{((Cos[s])^3),((Sin[s])^3)}]}],
PlotRange->{{-1.5,1.5},{-1.5,1.5}},
PlotLabel->Row[{Style["F",Italic],"=",NumberForm[Abs[(∫_0^s)y[t] * D[x[t],t]\[DifferentialD]t],4]}]
],{s,-0.0001,2π}]

Manipulate[
Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2π},PlotStyle->{Black, Thin}],
ParametricPlot[{((Cos[s])^3)+(1/4)Cos[t],((Sin[s])^3)+(1/4)Sin[t]},{t,0,2π}],
ParametricPlot[{r ((Cos[t])^3),r ((Sin[t])^3)},{t,0,s},{r,0,1},Mesh->40,MeshStyle->None],
Graphics[{PointSize[Large],Point[{((Cos[s])^3),((Sin[s])^3)}]}],
PlotRange->{{-1.5,1.5},{-1.5,1.5}},
PlotLabel->Row[{Style["F",Italic],"=",NumberForm[(1/2)Abs[Integrate[y[t] * D[x[t],t]-x[t] * D[y[t],t],{t,0,s}]],4]}]
],{s,-0.0001,2π}]

Rotationskörpers

Dynamic[Manipulate[

Show[{ParametricPlot3D[{x,0 ,f[x]},{x,-1.5,1.5},PlotStyle->Thick],
ParametricPlot3D[{x,-f[x] Sin[t],f[x] Cos[t]},{t,0,a},{x,-1.5,1.5},Mesh->40,MeshStyle->None]},

PlotRange->{{-1.5,1.5},{-1.5,1.5},{-1.5,1.5}},Axes->True,

PlotLabel->Row[{Style["V",Italic],"=",NumberForm[(∫_0^1.5)f[x]^2*Pi\[DifferentialD]x,4]}]
Row[{ Style["F",Italic],"=",NumberForm[NIntegrate[2 Pi f[x](Sqrt[1+D[f[x],x]^2]),{x,0,1.5}],4]}] 

],{a,0.001,2 Pi},FrameLabel->Row[{Style["f(x)",Italic],"=",PopupMenu[Dynamic[f],{Sin,Cosh,
Exp,Sqrt,Tan,Log}]}]]]