Surface Integrals

A surface integral of a function is simply a way to add up something that lives on a curved surface: the mass of a dome, the heat escaping through a curved wall, the average temperature over a wing. In $\iint_S f\,dS$, $S$ is the surface, $dS$ is a tiny patch of its area and $f$ is how much stuff there is per unit area on that patch. Multiply and add. The formula with $\|\vec r_u\times\vec r_v\|$ looks scary, but it's only telling you how big each patch really is.

Take a look at the line integral $\int_C f\,ds$ from the last unit. You parametrized the curve, and the speed $\|\vec r\,'(t)\|$ turned $dt$ into real length $ds$. This is the same move one dimension up: you parametrize the surface, and the stretch factor $\|\vec r_u\times\vec r_v\|$ from the last lesson turns $du\,dv$ into real area $dS$.

Recipe: parametrize, substitute into $f$, multiply by the area factor, integrate over the parameter region.

For a graph $z = g(x, y)$, the area factor is $\sqrt{1 + g_x^2 + g_y^2}$, as in the surface area lesson. For a sphere of radius $a$, it's $a^2\sin\varphi$.

With $f = 1$ you get the surface area; with a density, the mass; with a temperature, divided by the area, the average temperature over the surface.

A common mistake. It's tempting to integrate $f$ over the shadow region with plain $dA$, forgetting the area factor. That treats every patch as if it were lying flat. On a steep part of the surface the real patch is much bigger than its shadow, so you'd undercount exactly where the surface is steepest. There's an animation further down showing a floor square lifting onto a bowl and growing as the bowl gets steeper.

The total heat flowing out through a curved wall, the mass of a dome, and the average pressure over an airplane wing are surface integrals.