Surface integral and what it computes

Can anyone tell me what the surface integral of $f(x,y,z)$ actually calculates in a physical sense in three dimensions?

I know that the surface integral computes the surface area of some given surface:

$$\iint_D ||r_u\times r_v||dA$$

However what does the surface integral below calculate ($f(x,y,z)$ is a continuous function defined on the surface $S$): $$\iint_S f(x,y,z)dS=\iint_D f(r(u,v))||r_u\times r_v||dA$$

Does it calculate volume? Can anyone explain to me what it computes?

Edit: If a smooth parametric surface is given by the parametrization: $$r(u, v) = <x(u, v), y(u, v), z(u, v)>, (u,v)\in D$$ and $r$ is possibly injective on $D$ except possibly on the boundary of $D$, then the surface area of $S$ over $D$ is given by:

$$\iint_D ||r_u\times r_v||dA$$

Let the domains of integration $B,S,L,C$ and $SW$ be a body, surface, lamina, curve, and a straight wire, respectively.

• The following integrals give the integration domain's length/area/volume: \begin{align}\int_{SW} &\:\mathrm dx_1 \\\int_C &\:\mathrm ds \\\iint_L &\:\mathrm dx_1\,\mathrm dx_2 \\\int_S &\:\mathrm dS \\\iiint_B &\:\mathrm dx_1\,\mathrm dx_2\,\mathrm dx_3\end{align}
• If the integration domain's (mass) density function is $f,$ then the following integrals give its mass: \begin{align}\int_{SW} f(x_1) &\:\mathrm dx_1 \\\int_C f(x_1,x_2,x_3) &\:\mathrm ds \\\iint_L f(x_1,x_2) &\:\mathrm dx_1\,\mathrm dx_2 \\\int_S f(x_1,x_2,x_3) &\:\mathrm dS \\\iiint_B f(x_1,x_2,x_3) &\:\mathrm dx_1\,\mathrm dx_2\,\mathrm dx_3\end{align}
• The following integrals give the signed area/volume between the integration domain and the curve $x_2=f(x_1)$ or surface $x_3=f(x_1,x_2)$: \begin{align}\int_{SW} f(x_1) &\:\mathrm dx_1 \\\int_C f(x_1,x_2) &\:\mathrm ds \\\iint_L f(x_1,x_2) &\:\mathrm dx_1\,\mathrm dx_2\end{align}