> F[i]:=8*a^2*Int(Int(N[i],xi=0..1),eta=0..1)=8*a^2*int(int((81/4)*
> xi*(1-xi)*(1-3*xi)*eta*(1-eta)*(1-3*eta),xi=0..1),eta=0..1);
> 

                             1    1
                            /    /                     2
                        2  |    |                   9 a
             F[i] := 8 a   |    |   N[i] dxi deta = ----
                           |    |                    8
                          /    /
                            0    0

> F[_i]:=2*a^2*Int(Int(N[_i],x=-1..1),y=-1..1)=2*a^2*(int(int((81/4)*
> ((1+x)/2)*(1-(1+x)/2)*(1-3*(1+x)/2)*((1+y)/2)*(1-(1+y)/2)*(1-3*(1+y)/2
> ),x=-1..1),y=-1..1));
> 

                               1    1
                              /    /                   2
                          2  |    |                 9 a
              F[_i] := 2 a   |    |   N[_i] dx dy = ----
                             |    |                  8
                            /    /
                              -1   -1

# Lage der 2 x 2 GAUSS-Punkte im Master-Quadrat:
> p[1..4]:=Matrix(2,2,[[P[4]=(-b,b),P[3]=(b,b)],[P[1]=(-b,-b),P[2]=(b,-b
> )]]);
> 

                        [P[4] = (-b, b)     P[3] = (b, b) ]
           p[1 .. 4] := [                                 ]
                        [P[1] = (-b, -b)    P[2] = (b, -b)]

> b:=1/sqrt(3);
> 

                                    1/2
                                   3
                              b := ----
                                    3

# Wichtungsfaktoren:
> w[1..4]:=1;
> 

                            w[1 .. 4] := 1

# Werte des Integranden in den Gauss-Punkten:
> N(x,y):=(81/4/64)*(1+x)*(1-x)*(-1-3*x)*(1+y)*(1-y)*(-1-3*y);
> 

            81 (1 + x) (1 - x) (-1 - 3 x) (1 + y) (1 - y) (-1 - 3 y)
 N(x, y) := --------------------------------------------------------
                                      256

> n[1]:=subs({x=-1/sqrt(3),y=-1/sqrt(3)},N(x,y)):
> n[1]:=simplify(%):
> n[1]:=expand(%);
> 

                                          1/2
                                       9 3
                        n[1] := 9/16 - ------
                                         32

> n[2]:=subs({x=1/sqrt(3),y=-1/sqrt(3)},N(x,y)):
> n[2]:=simplify(%):
> n[2]:=expand(%);
> 

                                      -9
                              n[2] := --
                                      32

> n[3]:=subs({x=1/sqrt(3),y=1/sqrt(3)},N(x,y)):
> n[3]:=simplify(%):
> n[3]:=expand(%);
> 

                                          1/2
                                       9 3
                        n[3] := 9/16 + ------
                                         32

> n[4]:=subs({x=-1/sqrt(3),y=1/sqrt(3)},N(x,y)):
> n[4]:=simplify(%):
> n[4]:=expand(%);
> 

                                      -9
                              n[4] := --
                                      32

> GAUSS_Quadratur:=2*a^2*sum(n[k],k=1..4);
> 

                                             2
                                          9 a
                       GAUSS_Quadratur := ----
                                           8

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