Some part of me thinks that counting your hours shows that you're uninspired, that all you care about is snagging as many hours as possible.
Well, at my school, the students have to count their hours. We're supposed to fulfill a certain number of service hours, and if we don't keep track of our time sheets, we get put on social probation. I think it's important to keep track of how many hours you put into this internship - not because you want to snag hours, but just knowing how much time and effort you gave. If this is for school, it might be important to keep a record as well.