> f(xi,eta):=(729*sqrt(3)/2)*((eta-2*xi*eta-eta^2)^2+(1/3)*(2*xi-2*xi^2-
> eta+eta^2-2*xi*eta)^2):
> K[1010]:=Int(Int(f,eta=0..1-xi),xi=0..1)=int(int(f(xi,eta),eta=0..1-xi
> ),xi=0..1);
> 

                          1    1 - xi
                         /    /                        1/2
                        |    |                     81 3
            K[1010] :=  |    |        f deta dxi = -------
                        |    |                       20
                       /    /
                         0    0

> g(x,y):=(1/16)*(2-x-y)*subs({xi=(1+x)*(3-y)/8,eta=(1+y)*(3-x)/8},f(xi,
> eta)):
> g(x,y):=simplify(%):
> g(x,y):=collect(%,x):
> g(x,y):=collect(%,y):
> k[1010]:=Int(Int(g,y=-1..1),x=-1..1)=int(int(g(x,y),y=-1..1),x=-1..1);
> 

                              1    1
                             /    /                1/2
                            |    |             81 3
                k[1010] :=  |    |   g dy dx = -------
                            |    |               20
                           /    /
                             -1   -1

# Werte des Integranden in den 3 x 3 GAUSS-Punkten:
> a:=sqrt(3/5);
> 

                                     1/2
                                   15
                              a := -----
                                     5

> G[1]:=subs({x=-a,y=-a},g(x,y)):
> G[1]:=factor(%);
> 

                                   1/2           1/2
                           255879 5      427923 3
                 G[1] := - ----------- + -----------
                             3200000       640000

> G[2]:=subs({x=a,y=-a},g(x,y)):
> G[2]:=factor(%);
> 

                                         1/2
                                 177147 3
                         G[2] := -----------
                                   640000

> G[3]:=subs({x=a,y=a},g(x,y)):
> G[3]:=factor(%);
> 

                                  1/2           1/2
                          255879 5      427923 3
                  G[3] := ----------- + -----------
                            3200000       640000

> G[4]:=subs({x=-a,y=a},g(x,y)):
> G[4]:=factor(%);
> 

                                         1/2
                                 177147 3
                         G[4] := -----------
                                   640000

> G[5]:=subs({x=0,y=-a},g(x,y)):
> G[5]:=factor(%);
> 

                                  1/2           1/2
                          597051 5      174231 3
                  G[5] := ----------- + -----------
                            1024000       102400

> G[6]:=subs({x=a,y=0},g(x,y)):
> G[6]:=factor(%);
> 

                                   1/2           1/2
                           597051 5      174231 3
                 G[6] := - ----------- + -----------
                             1024000       102400

> G[7]:=subs({x=0,y=a},g(x,y)):
> G[7]:=factor(%);
> 

                                   1/2           1/2
                           597051 5      174231 3
                 G[7] := - ----------- + -----------
                             1024000       102400

> G[8]:=subs({x=-a,y=0},g(x,y)):
> G[8]:=factor(%);
> 

                                  1/2           1/2
                          597051 5      174231 3
                  G[8] := ----------- + -----------
                            1024000       102400

> G[9]:=subs({x=0,y=0},g(x,y)):
> G[9]:=factor(%);
> 

                                        1/2
                                  2187 3
                          G[9] := ---------
                                    16384

# Wichtungsfaktoren:
> w[1..4]:=25/81;  w[5..8]:=40/81;  w[9]:=64/81;
> 

                                        25
                           w[1 .. 4] := --
                                        81


                                        40
                           w[5 .. 8] := --
                                        81


                                      64
                              w[9] := --
                                      81

> GAUSS_Quadratur:=w[1..4]*sum(G[i],i=1..4)+w[5..8]*sum(G[j],j=5..8)+w[9
> ]*G[9];
> 

                                             1/2
                                         81 3
                      GAUSS_Quadratur := -------
                                           20

# Dieser Wert stimmt mit dem exatkten Integralwert berein.
> 
# 
