Numerical integration based on trivariate $C^2$ quartic spline quasi-interpolants

In this paper we consider the space generated by the scaled translates of the trivariate C^2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals. We give weights and nodes of the above rules and we analyse their properties. Finally, some numerical tests and comparisons with other known integration formulas are presented.