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The integrated 21 cm H I emission profile of a galaxy encodes valuable information on the kinematics, spatial distribution, and dynamical state of its cold interstellar medium. The line width, in particular, reflects the rotation velocity of the galaxy, which, in combination with a size scale, can be used to constrain the dynamical mass of the system. We introduce a new method based on the concept of the curve of growth to derive a set of robust parameters to characterize the line width, asymmetry, and concentration of the integrated H I spectra. We use mock spectra to evaluate the performance of our method, to estimate realistic systematic uncertainties for the proposed parameters, and to correct the line widths for the effects of instrumental resolution and turbulence broadening. Using a large sample of nearby galaxies with available spatially resolved kinematics, we demonstrate that the newly defined line widths can predict the rotational velocities of galaxies to within an accuracy of $\lesssim 30$ km s$^{-1}$. We use the calibrated line widths, in conjunction with the empirical relation between the size and mass of H I disks, to formulate a prescription for estimating the dynamical mass within the H I-emitting region of gas-rich galaxies. Our formalism yields dynamical masses accurate to $\sim 0.3$ dex based solely on quantities that can be derived efficiently and robustly from current and future extragalactic H I surveys. We further extend the dynamical mass calibration to the scale of the dark matter halo.