Surface Integrals

Consider painting a curved roof. The cost of the paint needed depends on the curved surface area, which is calculated by partitioning the parameter domain D into small rectangles.

Each rectangle’s reference point maps to a specific coordinate on the surface S through a vector-valued function.

At this mapped point, a tangent plane is constructed using two tangent vectors, which are the partial derivatives of the vector-valued function.

Scaling these by the grid dimensions produces edge vectors that form a small parallelogram. This shape linearly approximates the area of the surface patch. This area is calculated as the magnitude of the cross product of the vectors.

The areas of all the parallelograms are then added together to form a Riemann sum that approximates the surface area.

As the number of rectangles approaches infinity, this sum becomes a double integral over the domain D.

For a general surface integral, a function f is evaluated at a point in each patch and multiplied by the patch area. This process yields the final area formula.

Finally, multiply the total area by the paint price per unit area to get the total cost.