Computing the area and length of a curve 
Using the Riemannian hyperbolic metric $$g = \frac{4}{(1-(u^2+v^2))^2}\pmatrix{ 1 & 0 \\ 0 & 1 \\}$$ on the disk $D_p = \{(u,v)\ | \ u^2 +v^2 \le p^2\}$ compute the area of $D_p$ and the length of the curve $\partial D_p$.


This is just the hyperbolic metric
$$ds={2|dw|\over 1-|w|^2}, \qquad w:=u+iv,\tag{1}$$
on the unit disk in the $(u+iv)$-plane. Therefore the length of $\partial D_p$ computes to
$$L(\partial D_p)=\int\nolimits_{\partial D_p} ds={2\over 1-p^2}\int\nolimits_{\partial D_p} |dw|={4\pi p\over1-p^2}\ .$$
For any conformal metric, i.e., a metric of the form $ds=g(w)|dw|$, one has
$$d{\rm area}_g(w)=g^2(w)\>d{\rm area}(w)\ ,$$
whereby $d{\rm area}$ denotes the euclidean area in the $w$-plane. Therefore the $g$-area of $D_p$ for the metric $(1)$ computes to
$${\rm area}_g(D_p)=4\int\nolimits_{D_p}{1\over(1-|w|^2)^2}\ d{\rm area}(w)=
8\pi\int_0^p{r\over(1-r^2)^2}\ dr={4\pi p^2\over 1-p^2}\ .$$
