A surface integral accumulates a scalar quantity over a curved surface $S$. If the surface is parametrized by

$$\mathbf r(u,v),$$

with parameter region $D$, then

$$\iint_S f,dS

\iint_D f(\mathbf r(u,v)) |\mathbf r_u\times\mathbf r_v|,du,dv.$$

Surface area

Taking $f=1$ gives the area of the surface:

$$\operatorname{Area}(S)

\iint_D|\mathbf r_u\times\mathbf r_v|,du,dv.$$

The factor

$$|\mathbf r_u\times\mathbf r_v|$$

converts a small parameter rectangle into the corresponding area on the actual surface.

Accumulating a surface density

If $\sigma$ is a mass per unit area distributed over a thin surface, its total mass is

$$m=\iint_S\sigma,dS.$$

The same construction applies to any scalar density distributed across a surface.

Parametrization independence

Different regular parametrizations of the same surface give the same scalar surface integral. Reversing orientation also leaves $dS$ unchanged because the magnitude of the normal cross product is used.

When a vector field is integrated through an oriented surface, the direction of the normal matters. That oriented surface integral is the idea of flux.