How to measure the number of database writes on SQL Server 2008 R2
I would look at using extended events. Truthfully I have not worekd with these and it's one of those things I have not sat down and tried to learn...yet. So I cannot give you an exact example of how you would impliment with your example.

The advantage to these over trace files is no performance hit. They were specifically designed with performance in mind. So letting it collect information over a long period should not be a problem. You can check out using it for monitoring system activity here. Jonathan Keyhayias did a good month-long series on using extended events. The first day gives a good overview of them here.

He also created a SSMS add-in that makes it a little easier to work with the Extended Events sessions.