You sure it's the DVD drive my slim makes a loud noise when I'm turning it of but that's the fan cooling it down, also if cloud storage is enabled the Xbox wont shut down completely until it has synced you will get a blinking green light then it will turn of, also if you have your download que set to download then shut down it will also do this. Hope some of this helps.
How long does it spin for? Do you have any peripherals plugged in? Does it do this everytime? Does it do it with/without a disc in? What type of Xbox is it ?
Mine does it as well.. Slim 4GB with 320GB HDD added to it.. Seems to do it 1 in 5 shut downs with disc inside, I found if I hit the power button 2-3 times it'll finally permanently shut down.. Not sure what causes it :S
Its a cooling down procedure, if the console has been on for a considerable amount of time and its at a certain temp, usually when you don't install a game. Its designed to do it helps with cooling through the whole xbox360. On a different note I have a problem with certain disc's that make a lot of noise when spinning up and I found out its when disc's are warped slightly, My BF3 disc makes hugely excessive noise until its loaded up as I have it installed ****es me right off, but the issue your talking about don't worry I had this checked out several times and best thing to do it make sure its in a good cool place, install your games (or most popular), stand it up if its horizontally. I have mine in a ok spot can get warm and its stood up but if the console has been on for a long time especially if its a game I have not installed it does it most times. Hope this helps and re assures you
Mine does the same thing but sometimes the disc is out or in. I just keep clicking the power but off.
My fable 2 games is realllly loud lol.. gonna check it out and see If it's slightly warped at all lol..
That's pretty much normal as long as it powers off after 30 seconds up to 1 min, if it goes any longer, I would be concerned. If your background downloads are turned [ON] that is the reason why most likely it is doing that.