My computer will suddenly turn off when under high stress, how do I tell which part is causing the problem?

Sounds like you got the necessary hardware. I'd question as to the physical configuration. By the looks of it this computer is either custom built or had some upgrades done. So in other words it wasn't intended to run this extreme. Sounds as though it might be just over heating. Check the inside and make sure it's all cleaned out, especially the cpu cooling section. Leave it exposed and put a high powered fan on it. See if it does the trick. If so, you gotta find a way to cool the system down.