Area/Arc Length of a Hyperbolic Segment I'm given a hyperbolic segment, similar to the parabolic segment shown here: http://mathworld.wolfram.com/ParabolicSegment.html
I know the height of the segment ("h" in the wolfram article), and the length of the line segment joining the endpoints of the hyperbola ("2a" in the wolfram article).
Is it possible to find the area of the segment? Also, does there exist an approximation formula, or rapidly converging method to determine the approximate arc length of the given hyperbola?
 A: It is not possible to find an answer just from the information supplied.
Consider the hyperbola with equation
$$\frac{(y-h)^2}{d^2} -\frac{x^2}{c^2}=1.$$
One branch of this has shape roughly similar to the parabola illustrated in your picture. In particular, it has "height" $h$.
In order for the $x$-intercepts to be at $\pm a$ as in the picture, the relevant condition is $c\sqrt{h^2-d^2}=da$.  There are infinitely many hyperbolas for specified $h$ and $a$. The areas are not all the same for these hyperbolas, and neither are the arclengths.
Once the hyperbola is completely specified, arclength, though somewhat unpleasant, can be handled by setting up the usual integral.  It is one of the relatively rare cases where the integration can be carried out explicitly in terms of elementary functions.
A: translate everything so that the vertices of the hyperbola are on the x axis.
find the partial integral of the positive part of the hyperbola from vertex to the positive intersection, and subtract the integral of the positive part of the line (from the x-intercept to the same intersection point).
do the same thing with the negative parts, and add the absolute value of both parts.  
