|We investigate the off-limb active region and quiet Sun corona using spectroscopic data. The active region is clearly seen in several spectral lines formed at range of temperatures which was found to be diffuse in nature. We derive electron density and temperature using line ratio methods, and non-thermal velocity in the off-limb active region and quiet Sun up to height of 130 Mm. Non-thermal velocities in active regions first show increase and later decrease with height as extracted from warm spectral lines, whereas hot spectral lines show consistent decrease with height. However, in the quiet Sun region, non-thermal velocities are found to be mostly decreasing with height in all the observed spectral lines. Using these physical parameters, we calculate Alfv´en wave energy flux in both the active region and quiet Sun. These energy fluxes are found to be significantly decreasing with height, and thus, indicate their damping. To our knowledge, these results provide the first spectroscopic evidence of damping of Alfv´en waves in the active region and quiet Sun corona with height. Furthermore, we derive damping length of Alfv´en waves in both the regions. Damping lengths obtained from various spectral lines are found to be in the range of 25-200 Mm. These damping lengths also show some temperature dependence. Obtained results suggest that thermal conduction might play an important role in the damping of Alfv´en waves in the lower corona.