Hyperbolic sector
A hyperbolic sector is a region of the Cartesian plane {(x,y)} bounded by rays from the origin to two points (a, 1/a) and (b, 1/b) and by the hyperbola xy = 1.
A hyperbolic sector in standard position has a = 1 and b > 1 .
The area of a hyperbolic sector in standard position is loge b .
Proof: Integrate under 1/x from 1 to b, add triangle {(0, 0), (1, 0), (1, 1)}, and subtract triangle {(0, 0), (b, 0), (b, 1/b)}.
When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle.
[edit] Hyperbolic logarithm
Students of integral calculus know that f(x) = xp has an algebraic antiderivative except in the case p = −1 corresponding to the quadrature of the hyperbola. The other cases are given by Cavalieri's quadrature formula. Whereas quadrature of the parabola had been accomplished by Archimedes in the 3rd century BC (The Quadrature of the Parabola), the hyperbolic quadrature required the invention of a new function: Gregoire de Saint-Vincent addressed the problem of computing the area of a hyperbolic sector. His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola.
The natural logarithm is a transcendental function, an entity beyond the class of algebraic functions. Evidently transcendental functions are necessary in integral calculus.
