The surface defined by the equation

$x^2+y^2-z^2=1$

is known as a one$-$sheeted hyperboloid. It can be described parametrically by the parametrization

$r(u,v)=(coshucosv)i+(coshusinv)j+(sinhu)k$

One$-$sheeted hyperboloids are used in the construction of cooling towers for nuclear power stations and other industrial applications, as shown in the figure below, and discussed in this Wikipedia article

Compute the surface area of the section of a one$-$sheeted hyperboloid described by the parametrization above, where the parameters $u,v$ are restricted by the limits

$-1u1,0v2,(correspondingto$ the surface $in$ the figure$).$

Note. The calculation may involve a tricky integral evaluation; feel free to use Wolfram Alpha to get the value of the integral once the integral is set up$.$