I understand that "mining" for bitcoins requires your computer to do work that you don't monitor, so my question is, how do you know what it is that your computer is doing? Has anyone verified what it is your computer is actually doing while it mines for bitcoins?
Yes, it is verified. If you wouldn't do the work requested by the proof-of-work then you wouldn't be able to obtain the block reward. Therefore the fact that you're rewarded by the system means you're doing the work. If you're a programmer you can also read the code of the program you're using to mine, and verify for yourself that it's actually doing the proof-of-work.