What does the region look like? When changing the order of integration, the limits will likely change as well. That is, in general you do NOT have this:
You have to sketch out the region. In the first integral, which is like yours, you view the inner integral as happening first. In addition, you think of the region first as being bounded below and above by two different functions of x. Then, the outer integral gives you a number by integrating from one number to another.
After you've switched the order of integration, you have something that looks more like this:
and now you must think of the inner integral's limits as enclosing the left function up to the right function. Then you get a y-interval in the outer integral to close it off.
I don't feel like I'm explaining this very clearly. Does this make sense?