# Work as an integral

When a force changes with position, work is the integral of force over distance. For a spring, force is kx, where k is the spring constant.

Maths (university) · Part 06: Integration techniques and uses · Card 9 of 9 · Last updated 2026-10-19

## Analogy

Pushing a heavier and heavier shopping trolley: you add up the effort over each small step.

## Example

**Spring k = 200 N/m, stretch 0.1 m**

Force F = 200x newtons.
W = integral of 200x dx, x from 0 to 0.1.
= 100 x^2 from 0 to 0.1.
= 100(0.01) = 1 joule.

## In practice

Used to find energy needed to stretch springs or pump liquids.

## Practice

### Flashcards

- **How much work stretches a spring (k = 200 N/m) by 0.1 m?** 1 joule.

## Questions people ask

### Why isn't work just force times distance here?

That formula needs a constant force. A spring pushes back harder the further you stretch it, so the force changes along the way. Integrating adds up force times tiny distance for each small step, giving the correct total energy. Here the answer is 1 joule, half of 20 newtons times 0.1 metres.

### What does Hooke's law say?

It says a spring's force is proportional to how far it is stretched or compressed, F equals k times x. The constant k measures stiffness. A spring with k = 200 newtons per metre pulls with 20 newtons when stretched 0.1 metres. Doubling the stretch doubles the pull.

## Related concepts

- [Volume of revolution](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-6)
- [Area between curves](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-8)
- [Average value](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-9)
- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)

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