---
title: "Work as a Line Integral Along a Path"
description: "When force varies along the route, work is the integral of the force component along the displacement rather than a product. The work-energy theorem then follows from the equation of motion instead of"
canonical: https://lightmysky.com/learn/science/work-as-a-line-integral-along-a-path-mt_UFuQUPGAt7
source: https://lightmysky.com/learn/science/work-as-a-line-integral-along-a-path-mt_UFuQUPGAt7.md
retrieved: 2026-09-12
---

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# Work as a Line Integral Along a Path

When force varies along the route, work is the integral of the force component along the displacement rather than a product. The work-energy theorem then follows from the equation of motion instead of being quoted.

Subject: Science · Area: Energy · Ages 18 to 19
Page: https://lightmysky.com/learn/science/work-as-a-line-integral-along-a-path-mt_UFuQUPGAt7

## Ready when they can

- Computes the work done by a varying force as an integral along a stated path
- Derives the work-energy theorem from Newton's second law
- Explains why only the component of force along the motion contributes

## Lesson: Work adds up along the path

Last stop drag pushed against every metre you moved and drained energy of motion. In physics, work means energy handed over by a force during a displacement. When the force varies, you chop the path into tiny steps, multiply force by each tiny step, and add them all. That sum is the integral of force along the path. For a steady push along the motion the sum collapses to force times distance.

Only the part of the force pointing along the motion counts. A push with the motion does positive work, a push against it does negative work, and a sideways push at right angles does zero work. The angle between force and step decides everything.

**Example.** Stretch a spring and the pull grows with the stretch, so force times distance no longer works. Integrating the growing force over the stretch gives the stored work. That is exactly how you find the work of pulling a spring from its natural length.

Add up the work of every force and you get the net work, which equals the change in kinetic energy. Speeding pushes add energy of motion, and resisting pushes remove it. That link is the work energy theorem, derived straight from Newton, not quoted.

**Recap.** Work is the integral of force along the path, and net work equals the change in kinetic energy.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Drag and Terminal Speed from the Equation of Motion](https://lightmysky.com/learn/science/drag-and-terminal-speed-from-the-equation-of-motion-mt_In6VeoADzZ)
- [Work Done by a Force at an Angle](https://lightmysky.com/learn/science/work-done-by-a-force-at-an-angle-mt_OvBtur6dW5)
- [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd)

## Opens up

- [Conservative Forces and the Potential Energy Function](https://lightmysky.com/learn/science/conservative-forces-and-the-potential-energy-function-mt__053R9wrQy)
- [Continuity and Bernoulli's Equation for Ideal Flow](https://lightmysky.com/learn/science/continuity-and-bernoullis-equation-for-ideal-flow-mt_pKxlVYTy0Y)