Multiplicity of a parametrized surface integral (${\int }_{U}f(\mathbf{r}(u,v))|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv$)

For a regular parametric surface $\mathbf{r}:U\to X$, the displayed surface integral counts each image point as many times as its preimages in $U$, apart from sets of zero surface area. It equals the ordinary integral over the image when the parametrization is injective except on such sets. A periodic parametrization integrated over more than one period can give a different answer from integration over its image set. For example, $(u,v)\mapsto (\cos u,\sin u,v)$ with $0\le u\le 4\pi$ and $0\le v\le 1$ gives parameter area $4\pi$, whereas the image circular cylinder has surface area $2\pi$.