# Definite integral

The integral of f from a to b is the signed area between the curve and the x-axis. Areas above the axis count positive, below count negative.

Maths (university) · Part 05: Integration · Card 4 of 10 · Last updated 2026-10-19

## Analogy

A bank balance: money in is positive, money out is negative, and the total is the net change.

## Example

**Integral of x^2 from 0 to 3**

Antiderivative: x^3/3.
At 3: 27/3 = 9.
At 0: 0.
Area = 9 - 0 = 9.

## In practice

Gives total distance from speed, total water from a flow rate, or total cost from a rate.

## Practice

### Flashcards

- **What is the integral of x^2 from 0 to 3?** 9.

## Questions people ask

### What does a negative definite integral mean?

It means the signed area below the x-axis is larger than that above. The integral of sine from pi to 2 pi is -2: the curve lies below the axis, so the area counts as negative. Take the absolute value when you need actual area.

### What are the limits a and b?

They are the start and end of the interval you are adding up over. Swapping them changes the sign of the answer. If a equals b, the interval has no width, so the integral is zero. The bounds go in after the antiderivative is found.

## Related concepts

- [Fundamental theorem of calculus](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-5)
- [Antiderivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1)
- [Power rule for integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-2)
- [Riemann sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-3)

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