Question : Problem: Computer randomly shutting off

My computer keeps shutting off randomly. It seems to have been working all week just fine. I didn't install any new software or hardware either so I'm thinking it is a hardware issue.

The computer shuts off about 15 mins after I'm using it, especially when I'm playing World of Warcraft. I check my video temperature and it's at 49 degrees Celsius consistently when playing the game. However, this morning my computer was shut off, and I didn't have any game running on my computer over night.

The event logs aren't indicating any problems what I can see at the time it shuts down. I'm not even sure what to look for in the Event Log that could be causing this. I ran WhoCrashed but it said there's been no crashes on my computer. My temperatures seem just fine consistently too. Could this be my power supply going bad or unable to support it? I mean it's been working fine for a week since I built this system.

Power Supply: Broadway Com Corp 600W ATX
Motherboard: ECS A780GM-A AM2+/AM2 AMD 780G HDMI ATX AMD
Memory: G.SKILL 4GB 240-Pin DDR2 SDRAM DDR2 800 (PC2 6400)
Video Card: SAPPHIRE Radeon HD 3870 512MB 256-bit GDDR4 PCI Express 2.0
Optical Drive: LG Super multi
Hard Drive: Seagate Barracuda 7200.10 250gb
AMD Athlon 64 X2 5000 Brisbane Dual Core
Windows XP x64 SP2

Temperatures:
Motherboard TMPIN0: 10 °C
Motherboard TMPIN1: 37°C

CPU, Core #0: 14°C
CPU, Core #1: 10°C

ATI Radeon HD 3800: 40°C (49°C when playing WoW)

Hard Drive: 37°C

Answer : Problem: Computer randomly shutting off

I had a similar problem. I built a pc with a cheap 600watt PSU. Anytime I would play a graphics intensive game it would just shut off. After ruling out everything else, I swapped the power supply for a nice Antec and haven't had the problem since.
I have a power supply tester and the old one passes every time - so as far as I know the only way to diagnose this is to change the power supply and see if it works.
Hope that helps!