Amen to that.

I would:
1. Remove the power cord from the power supply.
2. Remove the cover from the computer chassis.
3. Remove the RAM modules from the Chassis.
4. Use a compressed air resource to blow out the entire chassis, especially the RAM slots and power supply.
5. Physically touch each of the wires that connect the power supply to anything. Hard Drive(s) CD drive(s) etc.
6. Remove the CPU heat sink.
7. Remove the CPU.
8. Clean the CPU and the CPU heat sink.
9. Replace the CPU.
10. Replace the heat sink compound.
11. Replace the heat sink.
12. Replace ONE of the RAM modules in the primary RAM slot.
13. Replace the CMOS battery.
14. Replace the power cord.
15. Boot the computer to the BIOS.
16. Make changes to the BIOS as necessary (date, time, etc.)
17. Save the changes you made., and insert a CD that can be booted from, such as UBCD or HIRENs CD in the CD drive before rebooting the computer.
18. Re-boot the computer and run a memory test on the RAM module you installed.
19. If this module passes an hour of tests with zero (o) errors. I would shut the computer down and swap a different RAM module with the one in the primary RAM slot.
20. After a successful hour I would swap another module until I had tested all four RAM modules.
21. If I had no RAM failures, I would expect that I had corrected whatever was causing the computer to shut down.
22. If while I was cleaning I noticed anything that looked abnormal, I would report back with what I had seen.
24. As far as replacing capacitors on anything is concerned, if you are not comfortable with the cold end of a soldering iron, I would replace ANY component including the motherboard before getting burned. AMAZON, EBay, and ALIEXPRESS are your friends.

One hour is meaningless. Some RAM modules take an hour or more to make one pass. Length of time required depends on RAM speed and RAM quantity. One pass is cursory. Full strength testing doesn't even happen until pass 2. If you're only testing one module at a time, you're only testing the module itself, not the memory controller, and not the other however many memory slots the board has. If you haven't made 4 passes, or populated all slots, you can't call the testing thorough.

We're all posting away but we haven't heard from the OP at all. Let's let him do something with his box and come back. I knew one guy who had too much boioze and a low temper threshold, and taught himself a lesson. He took a hammer(!) to his box, solved his problem that way, and barely stopped himself vandalising the monitor as well.

EDIT: He's better now.

If the computer is shutting down hard after 20-40 mins of use then it could be because of overheating. The BIOS will have a shutdown temperature defined and as soon as you hit it the system will shutdown hard.