So we have a cylinder surface $Q$ given by the following parametric equation,
$$ \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} u \\ \cosh u \\ v\sinh u \\ \end{pmatrix} $$
where, $0\le u\le 1$ and $0\le v\le 1$.
Question
I have to find the length of the rim of $Q$.
I am aware that I have to use the line integral, but I do not know how to convert this function into t parametric equation so that I can use the formula.
Any help or hint will be highly appreciated. Thank you :)
The parametric equation of the rim can be obtained by setting $v=1$ as
$$\begin{align} x &= u \\ y &= \cosh u \\ z &= \sinh u \end{align}$$
which is showed by the $\color{blue}{blue}$ curve below. So the integral for the length of this curve will be
$$\begin{align} L &= \int_{0}^{1} \sqrt{\left(\frac{dx}{du}\right)^2+\left(\frac{dy}{du}\right)^2+\left(\frac{dz}{du}\right)^2}du \\ &= \int_{0}^{1} \sqrt{1+\sinh^2 u +\cosh^2 u} \,\, du \end{align}$$