  The Cygnus region is a very bright and complex portion of the TeV sky, host
to unidentified sources and a diffuse excess with respect to conventional
cosmic-ray propagation models. Two of the brightest TeV sources, MGRO J2019+37
and MGRO J2031+41, are analyzed using Milagro data with a new technique, and
their emission is tested under two different spectral assumptions: a power law
and a power law with an exponential cutoff. The new analysis technique is based
on an energy estimator that uses the fraction of photomultiplier tubes in the
observatory that detect the extensive air shower. The photon spectrum is
measured in the range 1 to 200 TeV using the last 3 years of Milagro data
(2005-2008), with the detector in its final configuration. MGRO J2019+37 is
detected with a significance of 12.3 standard deviations ($\sigma$), and is
better fit by a power law with an exponential cutoff than by a simple power
law, with a probability $>98$% (F-test). The best-fitting parameters for the
power law with exponential cutoff model are a normalization at 10 TeV of
$7^{+5}_{-2}\times10^{-10}$ $\mathrm{s^{-1}\: m^{-2}\: TeV^{-1}}$, a spectral
index of $2.0^{+0.5}_{-1.0}$ and a cutoff energy of $29^{+50}_{-16}$ TeV. MGRO
J2031+41 is detected with a significance of 7.3$\sigma$, with no evidence of a
cutoff. The best-fitting parameters for a power law are a normalization of
$2.4^{+0.6}_{-0.5}\times10^{-10}$ $\mathrm{s^{-1}\: m^{-2}\: TeV^{-1}}$ and a
spectral index of $3.08^{+0.19}_{-0.17}$. The overall flux is subject to an
$\sim$30% systematic uncertainty. The systematic uncertainty on the power law
indices is $\sim$0.1. A comparison with previous results from TeV J2032+4130,
MGRO J2031+41 and MGRO J2019+37 is also presented.
