Find the parametric equation for $x^2-y^2=1$, $x>0$ $y \in [-1, 1]$ and $z \in [0,1]$. Write the integral to find the area without calculating it.
The parametric equation I found (using the identity $\cosh^2t - \sinh^2 t = 1)$ for the given equation is $\phi(t,z)=(\cosh{t}, \sinh{t}, z)$, but now I'm having problems when setting up the integral:
$\int _{0} ^{1} \int _{?} ^{?} ||\phi_t(t,z) \times \phi_z(t,z)|| = \int _{0} ^{1} \int _{?} ^{?} \sqrt{\cosh^2t+\sinh^2t} dtdz$
I really don't know how to find the domain for $t$...
The domain is determined by the substitutions you have made, namely, $x = cosh(t), y = sinh(t)$. Thus, you have to determine the domain's endpoints by solving $-1 = sinh(t)$ and $1 = sinh(t)$ for $t$.