Surface Integrals

A curved roof has a surface area that is generally larger than its flat projection. To estimate the cost of painting it, the curved surface area must first be calculated. If the roof is represented parametrically by a vector-valued function r(u,v), then each point in a parameter domain D corresponds to a point on the surface S. This connection allows the curved surface to be studied through a two-dimensional parameter region.

The parameter domain D is divided into many small rectangles. A reference point in each rectangle is mapped to a point on the surface using the vector-valued function. At that point, two tangent vectors are formed from the partial derivatives of the parametrization. These vectors describe how the surface changes in the two parameter directions.

When the tangent vectors are scaled by the dimensions of a small rectangle in the parameter domain, they form the sides of a small parallelogram on the tangent plane. This parallelogram gives a linear approximation of the corresponding curved surface patch. Its area is found using the magnitude of the cross product of the tangent vectors.

Adding the areas of all such parallelograms produces a Riemann sum approximation for the total surface area. As the rectangles become smaller and more numerous, this sum approaches a double integral over the parameter domain:

\begin{equation*}A=\iint_D \left\|\mathbf{r}_u \times \mathbf{r}_v\right\|\,dA\end{equation*}

For a general surface integral, a function f is evaluated at a representative point on each patch and multiplied by the corresponding patch area:

\begin{equation*}\iint_S f\,dS=\iint_D f(\mathbf{r}(s,t))\left\|\mathbf{r}_s \times \mathbf{r}_t\right\|\,dA\end{equation*}

When f = 1, this formula gives surface area. Multiplying the resulting area by the paint cost per unit area gives the total painting cost.