Two slight corrections:
Air (composition written by Bach)
An "air" is "a song-like vocal or instrumental composition. The term can also be applied to the interchangeable melodies of folk songs and ballads. It is a variant of the musical song form often referred to (in opera, cantata and oratorio) as aria." So while
many, many musicians with surname of Bach wrote airs,
this being the most famous one (written by
the most famous member of the Bach family), writing airs is not exclusive to them.
9th movement ( Beethoven )
Were you talking about
Beethoven's
Symphony No. 9 in D minor, Op. 125? Because as far as I can tell none of his compositions have a "ninth movement". The most famous part of his 9th Symphony, commonly referred to as
"Ode to Joy", is from the fourth movement of the piece.