> R(xi):=(2-3*xi+2*xi^2+4*xi*(1-xi)*Theta[2])*(4-8*Theta[2])+(-3+4*xi+(4
> -8*xi)*Theta[2])^2;

  R(xi) :=

                        2
        (2 - 3 xi + 2 xi  + 4 xi (1 - xi) Theta[2]) (4 - 8 Theta[2])

                                            2
         + (-3 + 4 xi + (4 - 8 xi) Theta[2])

> N[1]:=1-3*xi+2*xi^2; N[2]:=4*xi*(1-xi); N[3]:=xi*(-1+2*xi);

                                              2
                       N[1] := 1 - 3 xi + 2 xi


                        N[2] := 4 xi (1 - xi)


                        N[3] := xi (-1 + 2 xi)

> Int(N[1]*R,xi=0..1) = int(N[1]*R(xi),xi=0..1); 

     1
    /
   |                   2
   |   (1 - 3 xi + 2 xi ) R dxi =
   |
  /
    0

        -1/60 (2 - 4 Theta[2]) (4 - 8 Theta[2])

                                2
         - 1/60 (4 - 8 Theta[2])  + 4/3 - 8/3 Theta[2]

                                2
         + 1/6 (-3 + 4 Theta[2])

> Int(N[2]*R,xi=0..1) = int(N[2]*R(xi),xi=0..1); 

     1
    /
   |
   |   4 xi (1 - xi) R dxi = 1/5 (2 - 4 Theta[2]) (4 - 8 Theta[2])
   |
  /
    0

                               2
         + 1/5 (4 - 8 Theta[2])  + (-3 + 4 Theta[2]) (4 - 8 Theta[2])

                                                       2
         + 16/3 - 32/3 Theta[2] + 2/3 (-3 + 4 Theta[2])

> Int(N[3]*R,xi=0..1) = int(N[3]*R(xi),xi=0..1); 

     1
    /
   |
   |   xi (-1 + 2 xi) R dxi = 3/20 (2 - 4 Theta[2]) (4 - 8 Theta[2])
   |
  /
    0

                                2
         + 3/20 (4 - 8 Theta[2])

         + 1/2 (-3 + 4 Theta[2]) (4 - 8 Theta[2]) + 4/3

                                               2
         - 8/3 Theta[2] + 1/6 (-3 + 4 Theta[2])

> evalf(solve(-(76/15)*Theta[2]+(16/15)*Theta[2]^2+73/30=0,Theta[2]));

                       4.207859788, 0.542140212

> evalf(solve(-(88/15)*Theta[2]-(32/15)*Theta[2]^2+62/15=0,Theta[2]));

                      -3.331559480, 0.581559480

> solve(-(16/15)*Theta[2]+(16/15)*Theta[2]^2+13/30=0,Theta[2]);

                               1/2                1/2
                 1/2 + 1/8 I 10   , 1/2 - 1/8 I 10

# Die exakte Lsung ist:
> Theta[e]:=sqrt(4-3*xi) - 1;

                                          1/2
                    Theta[e] := (4 - 3 xi)    - 1

# Am Knotenpunkt 1/2 ergibt sich daraus der Wert:
> Theta[e]:=evalf(subs(xi=1/2,%));

                       Theta[e] := 0.581138830

# Der Nherungswert  0.581559481 stimme bis zur vierten Stelle nach dem
# Komma mit der exakten Lsung berein!
