Mechanical stretching of covalent bonds, for example when a fibrous polymer is loaded in tension, results in their stretching vibrational bands in the infrared or Raman spectrum being shifted to lower frequency. Conversely stretching a hydrogen bond shifts the stretching vibrational mode of the donor covalent X-H bond to higher frequency. These band shifts are small and difficult to detect in complex regions of the spectrum where differently affected bands overlap. This paper describes a method of integrating the difference spectra (spectrum under tensile strain minus spectrum at zero tensile strain) to recover the shape of the bands that are shifted and the spectral variation in bandshift. The application of this method to two sets of vibrational spectra of cellulose under tension is described. In one example, C-O-C stretching bands of highly crystalline tunicate cellulose were observed to shift to lower frequency under axial strain. In the other example, a group of overlapping O-D stretching bands in partially deuterated cellulose showed varied bandshifts under axial strain, some bandshifts being positive as expected due to extension of axially oriented hydrogen bonds while others were negative. The possibility of constructing spectral plots of bandshift has the potential to clarify the interpretation of overlapped, shifting bands in the vibrational spectra of polymers under tension.