|I've always found unit testing at the core to be useless, and unless someone really screws up you won't find anything new.
Integration is much better as you described. Once you start assembling parts, then you will know what's broken.
"Computer games don't affect kids; I mean if Pac-Man affected us as kids, we'd all be running around in darkened rooms, munching magic pills and listening to repetitive electronic music."
-- Marcus Brigstocke, British Comedian