We are familiar with single-variable integrals of the form [latex]\displaystyle\int_a^b f(x)dx[/latex], where the domain of integration is an interval [latex][a,b][/latex]. Such an interval can be thought of as a curve in the [latex]xy[/latex]-plane, since the interval defines a line segment with endpoints [latex](a,0)[/latex] and [latex](b,0)[/latex] — in other words, a line segment located on the [latex]x[/latex]-axis. Suppose we want to integrate over any curve in the plane, not just over a line segment on the [latex]x[/latex]-axis. Such a task requires a new kind of integral, called a line integral.

Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.