I need to prove the size of a hyperbolic angle is twice the area of its hyperbolic sector but with the Leibniz Sector Formula which is defined as:
(Sector Formula by Leibniz)}
Let $\gamma: [a, b] \to \mathbb{R}^2$ be a piecewise continuously differentiable curve. Then the curve sweeps out the oriented area given by the formula: $ F(\gamma) = \frac{1}{2} \int_a^b (x \dot{y} - y \dot{x}) \, dt, $ where $x$ and $y$ denote the coordinate components of $\gamma$, and $\dot{x}$ and $\dot{y}$ are their derivatives with respect to $t$.
My starting Problem is given as:
The term "area" for "surface" is motivated as follows:
Let $a \in [0, 1[$ and let $P = (x_a, y_a)$ and $Q = (x_a, -y_a)$ be the intersection points of the right branch of the unit hyperbola with the lines passing through the origin with slopes $a$ and $-a$, respectively. Then, the points $O$, $P$, and $Q$ define a region with area $A$.
It can be shown that:
$
x_a = \cosh A, \quad y_a = \sinh A,
$
so that:
$
A = \operatorname{arcosh} x_a = \operatorname{arsinh} y_a.
$
What I tried: In our case, the hyperbola $( x^2 - y^2 = 1)$ is parameterized, and we choose the hyperbolic angle ( t ) (analogous to the polar angle for circles). The hyperbola can be parameterized as: $ x(t) = \cosh(t), \quad y(t) = \sinh(t), $ where ( t ) is the hyperbolic angle.
Derivatives of $ x(t)$ and $y(t) $ The derivatives are: $ \dot{x}(t) = \sinh(t), \quad \dot{y}(t) = \cosh(t). $
Applying the Sector Formula: Substituting the parameterized forms into the sector formula: $ F = \frac{1}{2} \int \left( x(t) \dot{y}(t) - y(t) \dot{x}(t) \right) \, dt. $ Substitution of Functions: Expanding the integrand: $ F = \frac{1}{2} \int \left( \cosh(t) \cosh(t) - \sinh(t) \sinh(t) \right) \, dt. $
Using the identity: $\cosh^2(t) - \sinh^2(t) = 1,$ the integrand simplifies to: $ F = \frac{1}{2} \int 1 \, dt.$
Integration: $F = \frac{1}{2} \left[ t \right] $
Question
Now I am not sure if this way is even correct and I do not know how to find out my limits for the Integral... Furthermore I would be happy if someone could explain thw leibniz formula to me - why is it good to use it, what are advantages, how does it work here etc.
