Check with f(x,y,z) = 1

> F := 1;

> Int(Int(Int(F, z= 0 .. 3*y/4), y = x^2+2 .. 4 ) , x = 0 .. sqrt(2))
=
int(int(int(F, z= 0 .. 3*y/4), y = x^2+2 .. 4 ) , x = 0 .. sqrt(2));

Int(Int(Int(F, z= 0 .. 3*y/4), x = 0 .. sqrt(y-2) ) , y = 2 .. 4)
=
int(int(int(F, z= 0 .. 3*y/4), x = 0 .. sqrt(y-2) ) , y = 2 .. 4);

Int(Int(Int(F, y= x^2 + 2 .. 4 ), z = 0 .. 3*x^2/4 + 3/2 ), x = 0 .. sqrt(2)) +
Int(Int(Int(F, y= 4*z/3 .. 4 ), z = 3*x^2/4 + 3/2 .. 3 ), x = 0 .. sqrt(2))
=
int(int(int(F, y= x^2 + 2 .. 4 ), z = 0 .. 3*x^2/4 + 3/2 ), x = 0 .. sqrt(2)) +
int(int(int(F, y= 4*z/3 .. 4 ), z = 3*x^2/4 + 3/2 .. 3 ), x = 0 .. sqrt(2)) ;

Int(Int(Int(F, y= x^2 + 2 .. 4 ), x = 0 .. sqrt(2) ), z = 0 .. 3/2) +
Int(Int(Int(F, y= x^2 + 2 .. 4 ), x = sqrt(4*z/3 - 2) .. sqrt(2) ), z = 3/2 .. 3) +
Int(Int(Int(F, y= 4*z/3 .. 4 ), x = 0 .. sqrt(4*z/3 - 2) ), z = 3/2 .. 3)
=
int(int(int(F, y= x^2 + 2 .. 4 ), x = 0 .. sqrt(2) ), z = 0 .. 3/2) +
int(int(int(F, y= x^2 + 2 .. 4 ), x = sqrt(4*z/3 - 2) .. sqrt(2) ), z = 3/2 .. 3) +
int(int(int(F, y= 4*z/3 .. 4 ), x = 0 .. sqrt(4*z/3 - 2) ), z = 3/2 .. 3) ;

Int(Int(Int(F, x=0 .. sqrt(y-2)), z=0 .. 3*y/4),y= 2..4)
=
int(int(int(F, x= 0 .. sqrt(y-2)), z=0 .. 3*y/4),y= 2..4);

Int(Int(Int(F, x= 0 .. sqrt(y-2)), y= 2 .. 4), z=0..3/2) +
Int(Int(Int(F, x= 0 .. sqrt(y-2)), y= 4*z/3 .. 4), z= 3/2 .. 3)
=
int(int(int(F, x= 0 .. sqrt(y-2)), y= 2 .. 4), z=0..3/2) +
int(int(int(F, x= 0 .. sqrt(y-2)), y= 4*z/3 .. 4), z= 3/2 .. 3) ;

>