This work presents a detailed modeling-based analysis of integrated micro-ring resonators used for absorption spectroscopy. Generally, sensors based on micro-ring resonators detect changes in the real part of the sensing medium refractive index, at critical coupling. In absorption spectroscopy, however, micro-ring resonators are used to measure changes in the imaginary part of the index and are most sensitive away from critical coupling, with separate maxima in the under- and over-coupled regimes. In this work, we present a detailed analysis of the under-coupled regime, explaining the relationships between sensitivity, mode confinement, and losses. The analysis is based on reverse-symmetry waveguides to increase the proportion of mode power in the sensing medium and incorporates a realistic model of propagation losses based on experimental measurements of sidewall roughness. The analysis demonstrates that the resonant nature of the sensor is most effective at small radii compared to a non-resonant structure of equal size and shows a behavior of diminishing returns at larger device sizes regarding sensitivity and elevated proportions of mode power in the evanescent field.