I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems:
In the upper half-plane model, show that the $H_2$-length of the segment of the $y$-axis from $i$ to $iB$ is $|\log B|$. This is what I'm given as distance: $$ ds = \frac{\sqrt{dx^2 + dy^2}}{y} = \frac{|dz|}{\operatorname{Im}z}. $$
In the Poincaré disk model, show that a circle of Euclidean radius $r < 1$ and center $0$ has hyperbolic radius twice the integral from $0$ to $r$ of $dx/(1 - x^2) = 2\tanh^{-1}r$. Deduce that the hyperbolic circumference of a circle of hyperbolic radius $\rho$ is $2\pi\sinh \rho$. This is what I'm given as distance: $$ ds = \frac{2\sqrt{dx^2 + dy^2}}{1 - (x^{2} + y^{2})} = \frac{2|dz|}{1 - |z|^{2}}. $$
Any help explaining how to work with this, an example, or any hints, would be most welcome.