We have so far integrated "over'' intervals, areas, and volumes with single, double, and triple integrals. We now investigate integration over or "along'' a curve—"line integrals'' are really "curve integrals''.

As with other integrals, a geometric example may be easiest to understand. Consider the function $f=x+y$ and the parabola $y=x^2$ in the $x$-$y$ plane, for $0\le x\le 2$. Imagine that we extend the parabola up to the surface $f$, to form a curved wall or curtain, as in figure 16.2.1. What is the area of the surface thus formed? We already know one way to compute surface area, but here we take a different approach that is more useful for the problems to come.