A new Brillouin optical time-domain analysis (BOTDA) technique for acquiring the full Brillouin gain spectrum (BGS) at high speed is proposed and demonstrated. The method employs a frequency swept microwave source for the generation of the probe wave, so that the entire BOTDA measurement is taken within the duration of the frequency sweep itself. By properly setting the duration of the sweep, the repetition rate of the pump pulses and the number of averages, truly distributed and dynamic measurements of the BGS are possible using a set-up at a fraction of the cost and complexity of the previously reported fast-BOTDA methods.