The word 'book' is used in different ways. It can refer both to a whole work and to subsections of the work. Think of the Bible, which is spoken of as 'the Good Book' and which consists of several very different books - the Book of Psalms, the Book of Job, the Book of Genesis and so on.
The six books of LOTR recount one single story, with a beginning, a middle, and an end. The end brings to resolution all the different lines of the story in what most readers regard as a satisfactory conclusion. One would not advise anyone to read, say, Book 4 and Book 4 alone. Nor would one be satisfied with reading only the first Book. LOTR is a book made up of several books.