The averaged transmitted intensity of a cavity excited by a linearly frequency swept laser with finite line width is derived and presented as a sum over passes, analytical integrals (where the sum of passes is converted to a continuous time variable), and an approximate but computationally more stable stationary phase approximation expression. The transmitted waveform is used to derive the bias in extraction of the cavity decay rate from such a cavity transient for three different fitting models. Numerical simulation of cavity excitation gives statistical fluctuations in the transmitted intensity that leads to noise in the cavity decay rate. For a range of parameters spanning those likely to be encountered in real experiments, numerical results are presented. These demonstrate that the theoretical signal-to-noise ratio and thus sensitivity of swept cavity (or equivalently, frequency) CRDS is substantially below that for CRDS where one attenuates the laser either with current modulation or with an external modulator.