"We have Rick's general Europe book, which covers every city we're visiting except Munich (which I wish it did!)."
It's not clear what book you do have? It sounds like you only have one book: which I'm guessing would be Best of Europe? If that's correct, then you only have the most general book and the coverage of individual cities in Best of Europe is quite limited.
Rick has 2 levels of more in depth coverage: country books and city books for certain cities, including the 3 you're going to.
In the Italy book, there may be 800 pages but Rome is only a chapter with maybe 120 pages. And the city book, Rome, has 400 pages just on Rome.
Whether the extra coverage is worth the cost, considering the days you're in each city, is a matter of personal preference, which you can decide for yourself after browsing inside the city books.
The way to browse inside each book is:Go to Google Book at http://books.google.com/advanced_book_searchonce on the site, search for a 2008 or 2009 edition of the city book (you probably can't browse inside a 2010 edition because of copyright restrictions that Rick has waived on many, if not all, of the slightly older editions)
The coverage difference is slight between, for example, a 2008 and 2010 Rome book; they're updated for prices but coverage will be the same for a 2008 and 2010 Rome book. Of course, if you decide to buy the city books, you'll get the 2010 edition, but for your purpose of deciding whether it's worth it in the first place, this should work for you.
If you currently only have Best of Europe, and you're going to London, Paris, and Rome, but you're not going to other parts of the UK, France, or Italy--then you may decide to buy the three city books but not the country books.