Is there a way to generate a volume report for a bounded volume surface?
I think only if you have the Volumes Dashboard - subscription in 2012, standard in 2013
If you don't have the Volumes Dashboard (i.e. 2011 or earlier) you can use the Bounded Volumes command. In order to use the command you'll need a volume surface and a polyline representing the area of interest.
I assumed the OP wanted a real report rather than a command line cut and paste which is the only output available from this method