  Characterizing the atmospheres of planets orbiting M dwarfs requires
understanding the spectral energy distributions of M dwarfs over planetary
lifetimes. Surveys like MUSCLES, HAZMAT, and FUMES have collected
multiwavelength spectra across the spectral type's range of Teff and activity,
but the extreme ultraviolet flux (EUV, 100 to 912 Angstroms) of most of these
stars remains unobserved because of obscuration by the interstellar medium
compounded with limited detector sensitivity. While targets with observable EUV
flux exist, there is no currently operational facility observing between 150
and 912 Angstroms. Inferring the spectra of exoplanet hosts in this regime is
critical to studying the evolution of planetary atmospheres because the EUV
heats the top of the thermosphere and drives atmospheric escape. This paper
presents our implementation of the differential emission measure technique to
reconstruct the EUV spectra of cool dwarfs. We characterize our method's
accuracy and precision by applying it to the Sun and AU Mic. We then apply it
to three fainter M dwarfs: GJ 832, Barnard's Star, and TRAPPIST-1. We
demonstrate that with the strongest far ultraviolet (FUV, 912 to 1700
Angstroms) emission lines, observed with Hubble Space Telescope and/or Far
Ultraviolet Spectroscopic Explorer, and a coarse X-ray spectrum from either
Chandra X-ray Observatory or XMM-Newton, we can reconstruct the Sun's EUV
spectrum to within a factor of 1.8, with our model's formal uncertainties
encompassing the data. We report the integrated EUV flux of our M dwarf sample
with uncertainties between a factor of 2 to 7 depending on available data
quality.
