# Riemann sum

An estimate of area under a curve made by adding up thin rectangles. Using more rectangles gives a better answer, and the limit is the definite integral.

Maths (university) · Part 05: Integration · Card 3 of 10 · Last updated 2026-10-19

## Analogy

Filling an uneven puddle with square tiles: smaller tiles fit the edge better.

## Example

**Area under x^2 on [0, 3], 3 strips**

Width 1. Right-hand heights: 1, 4, 9.
Sum = 1 + 4 + 9 = 14.
This overestimates.
True area is 9.

## In practice

It is the definition of the integral, and how computers estimate areas.

## Practice

### Flashcards

- **Why does a right-hand Riemann sum overestimate an increasing function?** Each rectangle's height is taken from its taller right edge.

## Questions people ask

### Why does a Riemann sum overestimate here?

On [0, 3] the curve x squared is increasing, so the right-hand corner of each rectangle sits above the curve. That adds extra area. Using left-hand heights would underestimate, giving 0 + 1 + 4 = 5. The true area, 9, lies between them. More strips squeeze both estimates towards 9.

### How do I get the exact area from Riemann sums?

Let the number of rectangles grow without bound, so each width shrinks to zero. The sums then approach one number, which is the definite integral. In practice we avoid this limit and use the fundamental theorem of calculus, which gives the same exact value far faster.

## Related concepts

- [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)
- [Definite integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-4)
- [Fundamental theorem of calculus](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-5)

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