There's a MUCH MUCH MUCH easier way.
1. Find the Jamoiya Video Downloader site as linked above.
2. Save the target video URL as a .flv
3. Download
SUPER Video Converter
4. Import the target file into SUPER
5. Select "mp3 (Audio Only)" from the Output Container box
6. Make sure that Sampling Freq = 44100 and Channels = 2 and Bitrate = 128 (well, this is up to you, but 128 would be the best for YouTube vids IMO.
7. Click Encode and find the new file in the C drive (this should be the default folder)
^Alright, there's a lot more steps to this method, but trust me, it's easier than poking around with your soundcard (which might not have the ability to record outgoing audio internally).