In the following sentence, is rivers capitalized or not?
The Delaware and Potomac rivers are beautiful.
This is a style question. Some style guides specifically recommend lowercasing the plural form rivers in exactly the situation that the OP raises. For example, Chicago Manual of Style, Fifteenth Edition (2003) has this:
8.57 Mountains, rivers, and the like. Names of mountains, rivers, oceans, islands, and so forth are capitalized. The generic term (mountain, etc. is also capitalized when used as a part of the name. In the plural, it is capitalized when it is part of a single name (Hawaiian Islands) and when it is used of two or more names both beginning with the generic term (Mounts Washington and Rainier). When the generic term comes second and applies to two or more names, it is usually lowercased (the Illinois and Chicago rivers).
And the Associated Press Stylebook (2002): has this:
river Capitalize as part of a proper name: the Mississippi River. Lowercase in other uses: the river, the Mississippi and Missouri rivers.
Other style guides offer no guidance on this point, and still others may recommend capitalizing the plural word. Your best bet is to find out which guide (if any) your publisher or institution generally follows, and to adjust your style accordingly.
It should be capitalized because "River" is part of the name of that particular body of water. There are two reasons why this is so- first, naming the type of a water feature can be arbitrary, e.g. creek or river, so calling that feature by a certain name is part of the naming process. Also, different bodies of water can share the same "first name", e.g. Delaware River and Delaware Bay.
When I was at school in the UK, some 40-odd years ago, I distinctly remember being taught that one way the word river was capitalised but the other, it could be used without the capital. But I cannot remember which. To my shame I have been a teacher of English for 42 years but haven't got round to looking it up. I know that American grammar is sometimes different from English grammar and note that here it recommends capitalising river both before and after the name except when generic and used with plurals after the names, I wonder if anyone else has any contribution? Sometimes there is also an idiomatic rule applying: for example, very few people would write: the Thames river. It is always referred to as the River Thames, for some unknown reason, at least when living in the Thames Valley region.