I think I'd stay in cartesian coordinates for this one. What is your integral, including limits?
There you go. Nice work.
One way to avoid silly mistakes is to pretend that someone is holding a gun to your temple, and if you make a mistake, he's going to pull the trigger. Or, you could work at a high-energy physics lab, with all those 10,000V wires running around. If you make a mistake, you're dead.
So there are some nice, cheerful ways to avoid mistakes.