Intuitively, a surface is a deformed plane domain lying in space. As a model you can think of a piece of paper (or a flat stretchable material) which you pick up from the table and deform it, possibly gluing together some edges. In this section we want to learn how to integrate functions defined on surfaces. This yields a generalisation of double integrals over plane domains to double integrals over curved domains, that is, surfaces. Similarly, in Chapter 7, line integrals were a generalisation of integrals over an interval to integrals along a “bent” interval, that is, a curve.
