Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 3 / a / iii

Parametrize the finite cylindrical surface patch byThe surface area element is $|{\mathbf{R}}_t\times {\mathbf{R}}_z|dtdz=dtdz$. The seam $t=\pm \pi$ has zero surface area, so it causes no overcounting. ThereforeUnrolling the unit circular cylinder preserves surface area, which explains the equality with the area under the hyperbolic cosine in the figure. The patch consists of the cylindrical surface alone; no flat base is included.