# Area between curves

Integrate the top curve minus the bottom curve over the interval where they enclose a region. Find the crossing points first.

Maths (university) · Part 05: Integration · Card 8 of 10 · Last updated 2026-10-19

## Analogy

Cutting a strip of pastry between two lines: stack up all the thin vertical slices.

## Example

**Between y = x and y = x^2**

They meet at x = 0 and x = 1.
On (0, 1), x is above x^2.
Integral of (x - x^2) = 1/2 - 1/3.
Area = 1/6.

## In practice

Used for areas, and later for volumes and probabilities.

## Practice

### Flashcards

- **What is the area between y = x and y = x^2 on [0, 1]?** 1/6.

### Interview questions

**Find the area enclosed between y = x and y = x^2.**

- Solve x = x^2 for crossing points x = 0 and 1
- Line is above parabola on (0, 1)
- Integrate x - x^2 from 0 to 1
- Result 1/2 - 1/3 = 1/6

## Questions people ask

### How do I know which curve is on top?

Pick a test value between the crossing points and compare the two functions there. At x = 0.5, y = x gives 0.5 while y = x squared gives 0.25, so the line is on top. Swapping them would give a negative area.

### What if the curves cross more than once?

Split the integral at every crossing point. On each piece, subtract the lower function from the upper one. Then add the pieces. If you use one formula across a crossing, areas above and below cancel and you get the wrong total.

## Related concepts

- [Average value](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-9)
- [Volume of revolution](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-6)
- [Work as an integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-9)
- [Antiderivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1)

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