  The correct evaluation of gradients is at the cornerstone of the smoothed
particle hydrodynamics (SPH) technique. Using an integral approach to estimate
gradients has proven to enhance accuracy substantially. Such approach retains
the Lagrangian structure of SPH equations and is fully conservative. In this
paper we study, among other things, the connection between the choice of the
volume elements (VEs), which enters in the SPH summations, and the accuracy in
the gradient estimation within the integral approach scheme (ISPH). A new kind
of VEs are proposed which improve the partition of unit and are fully
compatible with the Lagrangian formulation of SPH, including the grad-h
corrections. Using analytic considerations, simple static toy models in 1D, and
a few full 3D test cases, we show that any improvement in the partition of unit
also leads to a better calculation of gradients when the integral approach is
used jointly. Additionally, we propose a simple-to-implement variant of the
ISPH scheme which is more adequate to handle sharp density contrasts.
