Line Integrals: Learn It 1

  • Calculate scalar line integrals
  • Calculate vector line integrals
  • Apply line integrals to compute work
  • Describe flux and circulation

Line Integrals

Single-variable integrals of the form [latex]\int_a^b f(x) dx[/latex] integrate over an interval [latex][a,b][/latex] on the [latex]x[/latex]-axis. We can think of this interval as a curve—specifically, a line segment with endpoints [latex](a,0)[/latex] and [latex](b,0)[/latex].

But what if we want to integrate over any curve in the plane or in space, not just a line segment on the [latex]x[/latex]-axis? This requires a new kind of integral called a line integral.

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

There are two types of line integrals:

  • Scalar line integrals integrate a scalar function over a curve in a plane or in space.
  • Vector line integrals integrate a vector field over a curve in a plane or in space.

We’ll begin by studying scalar line integrals.