
We compute surface area with double integrals.

In the past, we’ve used definite integrals to compute the arc length of curves. The natural extension of the concept of arc length over an interval is surface area over a region. Consider the surface $$ over some region in the $$-plane. To compute the surface area, first consider the surface area of a ‘‘patch’’ of the surface, determined by $$ and $$ below:
In essence, we zoom in on this portion of the surface to the extent that the tangent plane approximates the function so well that in this figure, it is virtually indistinguishable from the surface itself. Therefore we can approximate the surface area $$ of a ‘‘patch’’ of this region of the surface with the area of the parallelogram spanned by $$ and $$. Here

hence

Summing these ‘‘patches’’ together leads to a double integral.

A table of gradient vectors for a function $$ is given below:
Let $$ be the shaded region above. Estimate the surface area of $$ over $$.
Use the fact that

Let’s train ourselves to use our new tools by computing the surface areas of known surfaces. We start with a triangle.

It’s common knowledge that the surface area of a sphere of radius $$ is $$. While there are several ways to confirm this formula, we will use a double integral. Our computation will involves using our formula for surface area, polar coordinates, and improper integrals!

Let’s find the surface area of a general region now.

In practice, technology helps greatly in the evaluation of such integrals. High powered computer algebra systems can compute integrals that are difficult, or at least time consuming, by hand, and can at the least produce very accurate approximations with numerical methods. In general, just knowing how to set up the proper integrals brings one very close to being able to compute the needed value. Most of the work is actually done in just describing the region $$ in terms of polar or rectangular coordinates. Once this is done, technology can usually provide a good answer.