Note that the Windows "MemTest" referenced above is a completely different program from Memtest86 or Memtest86+, and the author acknowledges that it is not a complete test (there is a pay-for Pro version that does a bit more, but it's still not worth it, and you have to pay by snail-mail and wait for a floppy to arrive anyway!).
The bee's knees is Memtest86 or Memtest86+ (this is a modified version of Memtest86 that detects more cpus and chipsets correctly, but does the same tests). In either case you make a floppy or CD image (use a known-good machine, of course!) and boot it to run the test (note, this is not a DOS program - it is running on the bare metal of the machine).
Microsoft also make a memory diagnostic that runs from a bootable floppy or CD - you can get it from here. Not a bad idea, as more different diagnostics give a better chance of finding problems.
PS - good luck with the finals!