> f(xi,eta,zeta):=xi*eta*zeta;
> 

                   f(xi, eta, zeta) := xi eta zeta

> Digits:=15:
> Int(Int(Int(f,zeta=0..1-xi-eta),eta=0..1-xi),xi=0..1)=int(int(int(f(xi
> ,eta,zeta),zeta=0..1-xi-eta),eta=0..1-xi),xi=0..1);
> 

          1    1 - xi    1 - xi - eta
         /    /         /
        |    |         |
        |    |         |              f dzeta deta dxi = 1/720
        |    |         |
       /    /         /
         0    0         0

> 

# Transformation:
> xi(x,y,z):=(1+x)*(7-2*y-2*z+y*z)/24;
> 

                            (1 + x) (7 - 2 y - 2 z + y z)
             xi(x, y, z) := -----------------------------
                                         24

> eta(x,y,z):=(1+y)*(7-2*z-2*x+z*x)/24;
> 

                            (1 + y) (7 - 2 z - 2 x + z x)
            eta(x, y, z) := -----------------------------
                                         24

> zeta(x,y,z):=(1+z)*(7-2*x-2*y+x*y)/24;
> 

                             (1 + z) (7 - 2 x - 2 y + x y)
            zeta(x, y, z) := -----------------------------
                                          24

> with(linalg):  
> JACOBI_Matrix:=jacobian([xi(x,y,z),eta(x,y,z),zeta(x,y,z)],[x,y,z]):
Warning, the protected names norm and trace have been redefined and
unprotected

> J(x,y,z):=det(JACOBI_Matrix):
> 
> g(x,y,z):=J(x,y,z)*subs({xi=(1+x)*(7-2*y-2*z+y*z)/24,eta=(1+y)*(7-2*z-
> 2*x+z*x)/24,zeta=(1+z)*(7-2*x-2*y+x*y)/24},f(xi,eta,zeta)):
> Digits:=15:
> Int(Int(Int(g,z=-1..1),y=-1..1),x=-1..1)=int(int(int(g(x,y,z),z=-1..1)
> ,y=-1..1),x=-1..1);
> 

                     1    1    1
                    /    /    /
                   |    |    |
                   |    |    |   g dz dy dx = 1/720
                   |    |    |
                  /    /    /
                    -1   -1   -1


# Funktionswerte des Integranden:
> a:=sqrt(3/5);
> 

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

> G[1]:=subs({x=-a,y=-a,z=-a},g(x,y,z)):
> G[1]:=expand(%):
> G[2]:=subs({x=a,y=-a,z=-a},g(x,y,z)):
> G[2]:=expand(%):
> G[3]:=subs({x=a,y=a,z=-a},g(x,y,z)):
> G[3]:=expand(%):
> G[4]:=subs({x=-a,y=a,z=-a},g(x,y,z)):
> G[4]:=expand(%):
> G[5]:=subs({x=-a,y=-a,z=a},g(x,y,z)):
> G[5]:=expand(%):
> G[6]:=subs({x=a,y=-a,z=a},g(x,y,z)):
> G[6]:=expand(%):
> G[7]:=subs({x=a,y=a,z=a},g(x,y,z)):
> G[7]:=expand(%):
> G[8]:=subs({x=-a,y=a,z=a},g(x,y,z)):
> G[8]:=expand(%):
> G[9]:=subs({x=0,y=-a,z=-a},g(x,y,z)):
> G[9]:=expand(%):
> G[10]:=subs({x=0,y=a,z=-a},g(x,y,z)):
> G[10]:=expand(%):
> G[11]:=subs({x=0,y=a,z=a},g(x,y,z)):
> G[11]:=expand(%):
> G[12]:=subs({x=0,y=-a,z=a},g(x,y,z)):
> G[12]:=expand(%):
> G[13]:=subs({x=a,y=0,z=-a},g(x,y,z)):
> G[13]:=expand(%):
> G[14]:=subs({x=-a,y=0,z=-a},g(x,y,z)):
> G[14]:=expand(%):
> G[15]:=subs({x=-a,y=0,z=a},g(x,y,z)):
> G[15]:=expand(%):
> G[16]:=subs({x=a,y=0,z=a},g(x,y,z)):
> G[16]:=expand(%):
> G[17]:=subs({x=-a,y=-a,z=0},g(x,y,z)):
> G[17]:=expand(%):
> G[18]:=subs({x=a,y=-a,z=0},g(x,y,z)):
> G[18]:=expand(%):
> G[19]:=subs({x=a,y=a,z=0},g(x,y,z)):
> G[19]:=expand(%):
> G[20]:=subs({x=-a,y=a,z=0},g(x,y,z)):
> G[20]:=expand(%):
> G[21]:=subs({x=a,y=0,z=0},g(x,y,z)):
> G[21]:=expand(%):
> G[22]:=subs({x=-a,y=0,z=0},g(x,y,z)):
> G[22]:=expand(%):
> G[23]:=subs({x=0,y=a,z=0},g(x,y,z)):
> G[23]:=expand(%):
> G[24]:=subs({x=0,y=-a,z=0},g(x,y,z)):
> G[24]:=expand(%):
> G[25]:=subs({x=0,y=0,z=a},g(x,y,z)):
> G[25]:=expand(%):
> G[26]:=subs({x=0,y=0,z=-a},g(x,y,z)):
> G[26]:=expand(%):
> G[27]:=subs({x=0,y=0,z=0},g(x,y,z)):
# Wichtungsfaktoren:
> w[1..8]:=125/729;   w[9..20]:=200/729;   w[21..26]:=320/729;  
> w[27]:=512/729;
> 

                                        125
                           w[1 .. 8] := ---
                                        729


                                        200
                          w[9 .. 20] := ---
                                        729


                                         320
                          w[21 .. 26] := ---
                                         729


                                      512
                             w[27] := ---
                                      729

> GAUSS_Quadratur:=w[1..8]*sum(G[i],i=1..8)+w[9..20]*sum(G[j],j=9..20)+w
> [21..26]*sum(G[k],k=21..26)+w[27]*G[27];
> 

                       GAUSS_Quadratur := 1/720


# Dieses Ergebnis stimmt mit dem exakten Integralwert berein. 
> 
