  In present work, a relation that connects the integrated correlation function
of a trapped two-particle system to infinite volume particles scattering phase
shift is derived. It has the potential to provide an alternative approach for
extracting two-particle scattering phase shift from integrated correlation
function in lattice simulation at small Euclidean time region. Both (i)
perturbation calculation of 1+1 dimensional lattice Euclidean field theory
model of fermions interacting with a contact interaction and (ii) Monte Carlo
simulation of a 1D exactly solvable quantum mechanics model are carried out to
test the proposed relation. In contrast to conventional two-step approach of
extracting energy levels from temporal correlation function in lattice
simulation at large Euclidean time first and then applying L\"uscher formula to
convert energy levels into scattering phase shifts, we show that the difference
of integrated correlation functions between interacting and noninteracting
trapped systems converges rapidly to infinite volume limit that is given in
terms of scattering phase shifts at small Euclidean time region.
