3.5: Integrals on a Line

Math 4A: Multivariable Calculus · LibreTexts

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    Intro

    Learning Objectives
    • Calculate a scalar line integral along a curve.
    • Calculate a vector line integral along an oriented curve in space.
    • Use a line integral to compute the work done in moving an object along a curve in a vector field.
    • Describe the flux and circulation of a vector field.

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

     

    This page titled 3.5: Integrals on a Line was last modified on Wed, 20 Aug 2025 01:16:50 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Kenn Huber.