Lösungen

1

LaengeC[f_, x_] := Integrate[Evaluate[Sqrt[D[f,x]^2+1]],x];
Laengeab[f_, x_,a_, b_] := Integrate[Evaluate[Sqrt[D[f,x]^2+1]],{x,a,b}];

a

f[x_] := x^2;
a = -1;
b = 1;

LaengeC[f[x], x]

Laengeab[f[x], x,a,b]

Laengeab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

b

f[x_] := E^x;
a = 0;
b = 100;

LaengeC[f[x], x]

Laengeab[f[x], x,a,b]

Laengeab[f[x], x,a,b] // N // Re

Plot[f[x],{x,a,b}];

c

f[x_] := Sin[x];
a = 0;
b = 2 Pi;

LaengeC[f[x], x]

Laengeab[f[x], x,a,b]

Laengeab[f[x], x,a,b] // N

Plot[f[x],{x,a,b}];

2

FlaecheC[r_, φ_] := 1/2 Integrate[r^2,φ];
Flaecheab[r_, φ_,a_, b_] := 1/2 Integrate[r^2,{φ,a,b}];

a

r[θ_] := E^φ;
a = 0;
b = 2 Pi;

FlaecheC[r[φ], φ]

Flaecheab[r[φ], φ,a,b]

Flaecheab[r[φ], φ,a,b]//N

ParametricPlot[{r[φ] Cos[φ],r[φ] Sin[φ]},{φ,a,b}];

b

r[φ_] := Log[φ];
a = 0;
b = 2 Pi;

FlaecheC[r[φ], φ]

Flaecheab[r[φ], φ,a,b]

Flaecheab[r[φ], φ,a,b]//N

ParametricPlot[{r[φ] Cos[φ],r[φ] Sin[φ]},{φ,a,b}];

c

r[φ_] := 1/(1+φ^2);
a = 0;
b = Pi/2;

FlaecheC[r[φ], φ]

Flaecheab[r[φ], φ,a,b]

Flaecheab[r[φ], φ,a,b]//N

ParametricPlot[{r[φ] Cos[φ],r[φ] Sin[φ]},{φ,a,b}];

3

LaengePC[r_, φ_] := Integrate[Evaluate[Sqrt[r^2+D[r,φ]^2]],φ];
LaengePab[r_, φ_,a_, b_] := Integrate[Evaluate[Sqrt[r^2+D[r,φ]^2]],{φ,a,b}];

a

r[θ_] := E^φ;
a = 0;
b = 2 Pi;

LaengeC[r[φ], φ]

Laengeab[r[φ], φ,a,b]

Laengeab[r[φ], φ,a,b]// N //Chop

ParametricPlot[{r[φ] Cos[φ],r[φ] Sin[φ]},{φ,a,b}];

b

r[φ_] := Log[φ];
a = 0;
b = 2 Pi;

LaengeC[r[φ], φ]

Laengeab[r[φ], φ,a,b]

Laengeab[r[φ], φ,a,b]//N

ParametricPlot[{r[φ] Cos[φ],r[φ] Sin[φ]},{φ,a,b}];

c

r[φ_] := 1/(1+φ^2);
a = 0;
b = Pi/2;

LaengeC[r[φ], φ]

Laengeab[r[φ], φ,a,b]

Flaecheab[r[φ], φ,a,b]//N

ParametricPlot[{r[φ] Cos[φ],r[φ] Sin[φ]},{φ,a,b}];

4

VolumenC[f_, x_] := Pi Integrate[f^2,x];
Volumenab[f_, x_,a_, b_] := Pi Integrate[f^2,{x,a,b}];

a

f[x_] := x^2;
a = -1;
b = 1;

VolumenC[f[x], x]

Volumenab[f[x], x,a,b]

Volumenab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

b

f[x_] := E^x;
a = 0;
b = 100;

LaengeC[f[x], x]

Volumenab[f[x], x,a,b]

Volumenab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

c

f[x_] := Sin[x];
a = 0;
b = Pi;

LaengeC[f[x], x]

Volumenab[f[x], x,a,b]

Volumenab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

5

OberflC[f_, x_] := 2 Pi Integrate[Evaluate[f Sqrt[1+D[f,x]^2]],x];
Oberflab[f_, x_,a_, b_] := 2 Pi Integrate[Evaluate[f Sqrt[1+D[f,x]^2]],{x,a,b}];

a

f[x_] := x^2;
a = -1;
b = 1;

OberflC[f[x], x]

Oberflab[f[x], x,a,b]

Oberflab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

b

f[x_] := E^x;
a = 0;
b = 100;

OberflC[f[x], x]

Oberflab[f[x], x,a,b]

Oberflab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

c

f[x_] := Sin[x];
a = 0;
b = Pi;

OberflC[f[x], x]

Oberflab[f[x], x,a,b]

Oberflab[f[x], x,a,b]//N

Plot[f[x],{x,a,b}];

6

Remove["Global`*"]

a

vec[t_]:= {x[t],y[t]};
x[t_]:= t^2; y[t_]:= t;
f[x_,y_]:= x^2+x y-y^2;
a=-1;
b=1;
LinienintegralC= Integrate[Evaluate[f[x[t],y[t]]*Sqrt[D[x[t],t]^2+D[y[t],t]^2]],t]

Linienintegralab= Integrate[Evaluate[f[x[t],y[t]]*Sqrt[D[x[t],t]^2+D[y[t],t]^2]],{t,a,b}]

N[%]

ParametricPlot[{x[t],y[t]},{t,a,b}];

ParametricPlot3D[{x[t],y[t],f[x[t],y[t]]},{t,a,b}];

b

vec[t_]:= {x[t],y[t],z[t]};
x[t_]:= Cos[t]; y[t_]:= Sin[t]; z[t_]:= 1;
f[x_,y_,z_]:= Cos[x^2] + y z;
a= 0;
b= 2 Pi;
LinienintegralC= Integrate[Evaluate[f[x[t],y[t],z[t]]*Sqrt[D[x[t],t]^2+D[y[t],t]^2]+D[z[t],t]^2],t]

Linienintegralab= Integrate[Evaluate[f[x[t],y[t],z[t]]*Sqrt[D[x[t],t]^2+D[y[t],t]^2]+D[z[t],t]^2],{t,a,b}]

N[%]

NIntegrate[Evaluate[f[x[t],y[t],z[t]]*Sqrt[D[x[t],t]^2+D[y[t],t]^2]+D[z[t],t]^2],{t,a,b}]

ParametricPlot3D[{x[t],y[t],z[t]},{t,a,b}];

Plot[f[x[t],y[t],z[t]],{t,a,b}];

c

Remove["Global`*"]

vec[t_]:= {x[t],y[t],z[t]};
x[t_]:= t^2; y[t_]:= t; z[t_]:= 1;
f[x_,y_,z_]:= {x+y,y+z,z-x};
a= 0;
b= 4;

<<Graphics`PlotField3D`

PlotVectorField3D[f[x,y,z] ,
   {x, -2, 2}, {y, -2, 2}, {z, -2, 2}];

ParametricPlot3D[{x[t],y[t],z[t]},{t,a,b}];

LinienintegralC= Integrate[Evaluate[{f[x[t],y[t],z[t]]}.{D[x[t],t],D[y[t],t],D[z[t],t]}],t]

LinienintegralC= Integrate[Evaluate[{f[x[t],y[t],z[t]]}.{D[x[t],t],D[y[t],t],D[z[t],t]}],{t,a,b}]

N[%]

d

Remove["Global`*"]

vec[t_]:= {x[t],y[t],z[t]};
x[t_]:= Cos[t]; y[t_]:= Sin[t]; z[t_]:= 1;
f[x_,y_,z_]:= {Cos[x^2],y z,x};
a= 0;
b= 2Pi;

<<Graphics`PlotField3D`

PlotVectorField3D[f[x,y,z] ,
   {x, -2, 2}, {y, -2, 2}, {z, -2, 2}];

ParametricPlot3D[{x[t],y[t],z[t]},{t,a,b}];

LinienintegralC= Integrate[Evaluate[{f[x[t],y[t],z[t]]}.{D[x[t],t],D[y[t],t],D[z[t],t]}],t]

LinienintegralC= Integrate[Evaluate[{f[x[t],y[t],z[t]]}.{D[x[t],t],D[y[t],t],D[z[t],t]}],{t,a,b}]

N[%]