Lösungen/Solutions/E1S1AnUe/Ex15

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

f[x_] := Sinh[x];
l = Integrate[Sqrt[1 + Evaluate[D[f[x], x]]^2], x]

h[y_] := D[f[y], y];  h[y]

l = 
      NIntegrate[
        Sqrt[1 + (Evaluate[h[t]]^2 /. t -> x)], {x, -1, 1}, 
        Compiled -> False]

l = NIntegrate[Sqrt[1 + D[f[x], x]^2], {x, -1, 1}]

l = NIntegrate[Sqrt[1 + Cosh[x]^2], {x, -1, 1}]

Ueb/Ex 2      W

Integrate[
      Sqrt[E^(2  φ)  + E^(2  φ)], {φ, 
        0, 2  Pi}]

Ueb/Ex 3      W

Ob = 
      2 Pi Integrate[Sin[x] Sqrt[1 + Cos[x]^2], {x, 0,  Pi}]

Ob // N

Vol = Pi Integrate[Sin[x]^2, {x, 0,  Pi}]

Vol // N

Ueb/Ex 4      W

Ob = 
      2 Pi Integrate[E^x  Sqrt[1 + E^(2  x)], {x, -t, 0}] // 
        Expand

Ob = 
      2 Pi Integrate[
          E^x  Sqrt[1 + E^(2  x)], {x, -Infinity, 0}]

expr = 
      2 Pi Integrate[E^x  Sqrt[1 + E^(2  x)], {x, -t, 0}] // 
        Expand

expr /. E^(-t) -> 0

N[%]

Vol = Pi Integrate[(E^x)^2, {x, -t,  0}]

Vol /. E^(-2 t) -> 0

Ueb/Ex 5      W

Remove[Ob]

2 Pi Integrate[1/x  Sqrt[1 + (-1/x^2)^2], {x, 1, t}] // 
      Expand

Ob[t_] := 
      2 Pi Integrate[1/x  Sqrt[1 + (-1/x^2)^2], {x, 1, t}] // 
        Expand;  Ob[t]

Limit[Ob[t], t -> Infinity]

Vol = Pi Integrate[(1/x)^2, {x, 1, t}]

Vol /.  (1/t) -> 0

Das ist ein Beispiel eines Körpers mit endlichem Volumen,  aber unendlicher Oberfläche.