---
tag: 0GXA8Z205E
kind: concept
name: Area under a curve
msc: [26A42]
uses: [0QM4TXT3DR, 0HBY48G8G1, 0NHGY0N0JB]
sources:
  - url: https://en.wikipedia.org/wiki/Integral
    locator: Interpretations
---

## Summary

For a nonnegative continuous function $f$ on $[a, b]$, the area under its graph is the [[integral|0QM4TXT3DR]] $\int_a^b f(x) \, dx$. The integral approximates the region by thin rectangles, through [[Riemann sums|0HBY48G8G1]]. For a function of any sign, the integral counts area below the $x$-axis as negative.

## Remarks

The area between the graphs of $f \le g$ over $[a, b]$ is $\int_a^b (g(x) - f(x)) \, dx$.
For example, the area under $x^2$ over $[0, 1]$ is $1/3$. In the 3rd century BC, Archimedes found the area under a parabola by the method of exhaustion.
The [[fundamental theorem of calculus|0NHGY0N0JB]] computes areas from antiderivatives.

## Explanations

### L1 human

The integral measures the area of the region under a graph.
We fill the region with thin rectangles and add up their areas.