How to measure the length of a curve in hyperbolic space?

Question

Let $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)>0\}$. If we are considering this as hyperbolic space, then we have that the line element $ds^2=\frac{dx^2+dy^2}{y^2}$ is how we measure the length of a curve in hyperbolic space. The square is what scares me, so I wanted to check to see if my intuition is correct. Thus, say that we have that $z(t):[0,1]\rightarrow\mathbb{H}$ is a curve in the upper half plane (not necessarily a geodesic, and let us assume that it has continuous partial derivatives or is sufficiently nice so I don't have to worry about taking derivatives), and say we have that $z(t)=x(t)+iy(t)$. Then I am curious as to measure the arc length of this curve in hyperbolic space. Would the formula be given by $$ \int_0^1\sqrt{\frac{x'(t)^2+y'(t)^2}{y(t)^2}}dt $$ If not then how do I measure the length of the curve?