Mechanisms invoked to heat the solar corona to millions of degrees kelvin involve either magnetic waves or magnetic reconnections. Turbulence in the convection zone produces MHD waves, which travel upward and dissipate. Photospheric motions continuously build up magnetic energy, which is released through magnetic reconnection. In this paper, we concentrate on hot non-flaring plasma with temperatures of 5 MK < T < 10 MK because it is one of the few observables for which wave and reconnection models make different predictions. Wave models predict no (or little) hot plasma, whereas reconnection models predict it, although in amounts that are challenging to detect with current instrumentation. We used data from the X-ray Telescope (XRT) and the Atmospheric Imaging Assembly (AIA). We requested a special XRT observing sequence, which cycled through the thickest XRT filter several times per hour so we could average these images and improve the signal-to-noise. We did differential emission measure (DEM) analysis using the time-averaged thick-filter data as well as all available channels from both the XRT and AIA for regions observed on 2014 December 11. Whereas our earlier work was only able to determine that plasma with a temperature greater than 5 MK was present, we are now able to find a well-constrained DEM distribution. We have therefore added a strong observational constraint that must be explained by any viable coronal heating model. Comparing state-of-the-art wave and reconnection model predictions, we can conclude that reconnection is heating the hot plasma in these active regions.