  The galaxy-wide stellar initial mass function (gwIMF) of a galaxy in
dependence of its metallicity and star formation rate (SFR) can be calculated
by the integrated galactic IMF (IGIMF) theory. Lacchin et al. (2019) apply the
IGIMF theory for the first time to study the chemical evolution of the
ultra-faint dwarf (UFD) satellite galaxies and failed to reproduce the data.
Here, we find that the IGIMF theory is naturally consistent with the data. We
apply the time-evolving gwIMF calculated at each timestep. The number of type
Ia supernova explosions per unit stellar mass formed is renormalised according
to the gwIMF. The chemical evolution of Bo\"otes I, one of the best observed
UFD, is calculated. Our calculation suggests a mildly bottom-light and
top-light gwIMF for Bo\"otes I, and that this UFD has the same gas-consumption
timescale as other dwarfs but was quenched about 0.1 Gyr after formation, being
consistent with independent estimations and similar to Dragonfly 44. The
recovered best fitting input parameters in this work are not covered in the
work of Lacchin et al. (2019), creating the discrepancy between our
conclusions. In addition, a detailed discussion of uncertainties is presented
addressing how the results of chemical evolution models depend on applied
assumptions. This study demonstrates the power of the IGIMF theory in
understanding the star-formation in extreme environments and shows that UDFs
are a promising pathway to constrain the variation of the low-mass stellar IMF.

Astrophysics of Galaxies