<p>I'm looking for a way to extract the audio part of a FLV file. </p>

<p>I'm recording from the user's microphone and the audio is encoded using the <a href="http://en.wikipedia.org/wiki/Nellymoser_Asao_Codec" rel="nofollow noreferrer">Nellymoser Asao Codec</a>. This is the default codec and there's no way to change this.</p>

This isn't an exact answer, but some relevant notes I've made from investigating FLV files for a business requirement.
Most FLV audio is encoded in the MP3 format, meaning you can extract it directly from the FLV container. If the FLV was created from someone recording from their microphone, the audio is encoded with the Nellymoser Asao codec, which is proprietary (IIRC).
I'd check out
libavcodec
, which handles FLV/MP3/Nellymoser natively, and should let you get to the audio.