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Area of a Hyperbolic Segment

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Calculates the area and arc length of a hyperbolic segment.
\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Coordinate a
b
Cut position x
≧a
Area of a Hyperbolic Segment S 
Eccentricity e 
Length of a hyperbolic arc L 
\(\normalsize Hyperbolic\ Segment\\ (1)\ area: \hspace{20px} y={\large\frac{b}{a}}\sqrt{x^2-a^2}\\ \hspace{50px} S=xy-ab\cosh^{\tiny -1}{\large\frac{x}{a}}\\\\ (2)\ eccentricity : \hspace{10px} e={\large\frac{\sqrt{a^2+b^2}}{a}}\\ (3)\ hyperbolic\ arc\ length:\\ \hspace{50px} k={\large\frac{1}{e}},\hspace{5px} z=\sqrt{\large\frac{x^2-a^2}{x^2-(ka)^2}}\\ \hspace{50px} L={\large\frac{1}{k}}\left\{a(1-k^2)F(z,k)-aE(z,k)+xz\right\}\\\\ \)

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