The curve on a crystal momentum  versus energy plot can be folded back to this region in a reduced scheme.
Numerical integrals over this region can be done by using the equal y spaced mesh scheme of  Monkhorst and Pack or the tetrahedron method with Blöchl's correction. The irreducible wedge of this  region is the minimal portion needed to describe the entire region. The curve on a crystal momentum  versus energy plot can be folded back to this region in a reduced scheme. One of order n can be obtained  by finding the volume enclosed while crossing n minus one Bragg planes and the first one is the  Wigner-Seitz cel of the reciprocal lattice. For 10 points, identify these "zones" that are perpendicular bisectors of the reciprocal lattice vectors.
Numerical integrals over this region can be done by using the equal y spaced mesh scheme of  Monkhorst and Pack or the tetrahedron method with Blöchl's correction.
first Brillouin zone