Proving Principle Binormal of Arc Length Parametrized Curve is Contained in Hemisphere Let $\alpha:[a,b]\rightarrow \mathbb{R}^3$ be naturally parametrized with $$ \int_a^b|\tau(s)|ds<2\pi. $$
Then, $B(s)$, the principle binormal, lies in a hemisphere of the unit sphere for all $s\in[a,b]$.
I'm meant to use Crofton's formula for this, but I have no idea how it is applicable.

You don't need Crofton. This is just a classic fact (from spherical geometry) that any curve of length $\le 2\pi$ in the unit sphere is contained in a closed hemisphere. It usually appears in the proof of Fenchel's Theorem.
See, for example, the last paragraph of the proof on pp. 25-26 in my differential geometry text or the lemma on pp. 113-114 of Chern's "Curves and Surfaces in Euclidean Space" in Global Differential Geometry, MAA Studies in Mathematics, volume 27.
