The three Banu Musa brothers are almost indistinguishable and most of the information is at this link. However, there is some information specific to the youngest brother: al-Hasan ibn Musa ibn Shakir.
He wrote The elongated circular figure which is a work on the ellipse. This book is lost except for a fragment in Hebrew of a compilation by Ibn al-Samh. From this fragment Rashed in  deduces that al-Hasan had two objectives. One objective was to measure a curved area while the other was to study the geometric properties of curves. Rashed claims, as we have suggested above, that while Archimedes' texts were being translated into Arabic for the first time, the Banu Musa (perhaps al-Hasan in particular) was trying to give new proofs of the Greek results as well as trying to prove results going beyond what the Greeks had achieved.
Article by: J J O'Connor and E F Robertson