These limits are so stupid! Buy a fishing/photographer's vest and watch them steam!
With these cabins designed to safely accommodate large quantities of carry-on, bags up to 20+ Kg should be allowed. On my last flight into the EU in NOV (not on UA), I had to deal with this situation. I was requested to weigh my carry-on and was, of course, over the limit at 14 kg. They requested that I check the bag at the gate, but I had wisely purchased a new fishing-vest from Bass Pro Shops which I was wearing in anticipation of trouble (thanks to warnings that I read here on FT), and removed my camera equipment, drugs and sundries, ipod, flashlights, etc., etc. and placed the items in my vest pockets. I then asked them to reweigh my bag and it came-in at exactly 8 kg. I asked them if it was OK and they said yes, but boy were they steamed at my little show!!!
Seriously people, this is one of those issues that are worth switching carriers over. It is a ridiculous restriction with most computers weighing 2-3kg and the bag itself 4-5Kg, they are essentially allowing you no useful carry-on allowance!!!!
Move to BA or another more reasonable carrier.