MacBook Pro :: Early 2008 Boots To Black Screen?

I left my laptop on in my room while I went to the bathroom, and when I came out, I saw the message that told me I must restart the computer. The message froze the computer, so I shut it off by holding the power button and tried to turn it back on again, but it only boots to a black screen. I don't hear the loud hard drive noise, but I hear the fans. Without AC power but with the charged battery, the computer shuts down after a few seconds. The light on the button that opens the lid is dim-to-none when the lid is open and bright when the lid is closed.

I tried many troubleshooting methods discussed on this site, like removing and reseating the RAM and draining the battery, along with the things that the Apple website suggested, like resetting the SMC and PRAM, but nothing works.

I brought the computer into an Apple Store, and after the NVIDIA test was negative, the technician told me that the problem is most likely the logic board. I also took the computer into an Apple Authorized Service Provider, and the technician there said it could be the logic board, hard drive, or boot connector. He also said that there is a very slight chance that the graphics card could have caused the logic board to fail. He tried my computer with another battery and concluded that it's not the battery's fault.

The Apple Store is charging me about $330 for a flat-rate repair. Should I troubleshoot further using suggestions you may have or pay Apple for repairs? If I decide to have Apple repair my laptop, do I need to take it back into the store to pay the repair charges, or can I do it over the phone and mail the computer to Apple?