surface integral of a scalar function

A surface integral $\iint_S f\,dS$ that weights each patch of a curved surface by a varying scalar density before summing.

The surface integral of a scalar function is $\iint_S f\,dS = \iint_D f(\mathbf{r}(u,v))\,|\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv$ : $f(\mathbf{r}(u,v))$ evaluates the density at the point the parametrization currently sits at, and $|\mathbf{r}_u\times\mathbf{r}_v|$ supplies the same area-stretch factor from the surface-area formula. Whenever $f\equiv1$ , the surface integral collapses back to plain surface area.

Examples

For the cone $\mathbf{r}(u,v)=\langle u\cos v,\,u\sin v,\,u\rangle$ with density $f(x,y,z)=z$ : mass $= \iint_S z\,dS = \dfrac{16\sqrt2\,\pi}{3} \approx 23.71$ .