# What is a surface integral?

Sep 8, 2015

A surface integral is a generalisation of integration from linear intervals to two dimensions.

#### Explanation:

Limit of a sum

If a function is defined on a linear interval, then we can approximate the integral - the 'area under the curve' - by splitting our interval into several subintervals, evaluating the function at some point in each of the subintervals, multiplying that value by the length of the subinterval and summing. The actual integral is the limit as we make our subintervals smaller and smaller.

Analogously, if a function is defined on a surface $S$, then we can approximate the integral of the function over the surface by splitting the surface into sub-surfaces - "patches", evaluate the function at some point in each patch, multiply by the area of the patch and sum. The surface integral is the limit as we make the patches smaller and smaller.

Integral over bounded surface

Suppose $S$ is a finite surface (e.g. the surface of a sphere) and $f$ is a function from $S$ to $\mathbb{R}$ that is defined on almost all points of $S$.

Then ${\int}_{S} f$ is essentially the average value of $f$ multiplied by the area of $S$.

For example, ${\int}_{S} 1 = \text{area of } S$

Generalisation to unbounded surface

If $S$ is unbounded (e.g. the whole of the XY plane), and you define bounded subsets ${S}_{1} \subseteq {S}_{2} \subseteq \ldots \subseteq S$ such that ${\lim}_{n \to \infty} {S}_{n} = S$, then

${\int}_{S} f = {\lim}_{n \to \infty} {\int}_{{S}_{n}} f$