The image above is a screen shot of a post that ran across my Facebook newsfeed on 7 August 2013 from Ancestry.com. I don't know who wrote it, but I do know one thing: I didn't write it.
The first step in the naturalization process is filing a declaration of intention. The naturalization petition (to which the post refers) is filed at the end of the process usually right before the individual is naturalized. The declaration of intent is sometimes referred to as the "first papers." The naturalization petition (particularly before 1906) usually provides sketchy information, although sometimes researchers get lucky and more detail is included. The declaration of intention (if still extant) is usually the document that is more likely to provide the detailed information referred to in the Ancestry.com post.
Genealogy is about details, accurate details. Errors in Ancestry.com user-submitted trees are one things--user-submitted material is going to contain errors. This is different.
And the word "naturalization" in the last sentence does not need to be in capitalized. But that's a separate issue.