Consider again the example in § {Flux}, which involved the part of the plane $x+y+z=1$ which lies in the first quadrant. Suppose you want to find the average height of this triangular region above the $xy$-plane. To do this, chop the surface into small pieces, each at height $z=1-x-y$. In order to compute the average height, we need to find \begin{equation} \hbox{avg height} = \frac{1}{\hbox{area}} \Sint z \,\dS \end{equation} where the total area of the surface can be found either as \begin{equation} \hbox{area} = \Sint \dS \end{equation} or from simple geometry. So we need to determine $\dS$. But we already know $d\SS$ for this surface from (5) of § {Flux}! It is therefore straightforward to compute \begin{equation} \dS = |d\SS| = |\xhat+\yhat+\zhat|\,dx\,dy = \sqrt{3} \,dx\,dy \end{equation} and therefore \begin{equation} \hbox{avg height} = \frac{1}{\sqrt{3}/2} \int_0^1 \int_0^{1-x} (1-x-y) \sqrt{3}\,dy \,dx = \frac13 \end{equation}