1. Let $b$ be a real number with $\frac{\pi}{2} < b < \pi$, and let $C_b$ be the curve parametrized by
$f: [\frac{\pi}{2}, b] \to \mathbb{R}^2$
$t \mapsto (t, \ln(\csc(t))).$
(a) Find the arc-length function $\sigma_f$, i.e., find $\sigma_f(t)$ in terms of $t$ for each $t \in [\frac{\pi}{2}, b]$. You may use the fact that
$\frac{d}{dx}(\ln(\csc(x) + \cot(x))) = -\csc(x)$.
(b) Express the total length of $C_b$ in terms of $b$.
