> [!NOTE] Lemma > Let $S$ be a [[Parametrized Surface|surface parametrized]] by $\underline{r}(u,v)=(x(u,v),y(u,v),z(u,v))$ where $(u,v)\in \Omega.$ Then its 'surface area' is given by $\int \int_{\Omega} |\underline{r}_{u} \times \underline{r}_{v} | \, du \, dv $where $\underline{r}_{u}=(x_{u},y_{u},z_{u})$ where $x_{u}$ denotes the first [[Fréchet Differentiation|partial derivative]] of $x$; $\times$ denotes [[Cross Product in Real 3-Space|cross product]] and $|\dots|$ denotes [[Euclidean Norm|length]]. **Proof**: Follows directly from [[Change of variables formula]].