To calculate the electric field due to a non-uniformly charged spherical shell with a volume charge density of ρ = -kr, one should integrate over the volume of the sphere to determine the total charge enclosed. Gauss's law can then be applied to find the electric field, taking into account the azimuthal symmetry of the problem. The approach involves treating E.dA as E(dA) due to this symmetry. This method simplifies the calculations and leads to an accurate determination of the electric field. Ultimately, integrating the charge density is crucial for applying Gauss's law effectively.