# Area under Curve

It has been suggested that this page or section be merged into Definite Integral is Area.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## Theorem

Let $f: \R \to \R$ be a real function which is defined and (Darboux) integrable on the closed interval $\closedint a b$.

Consider the graph of $f$ embedded in a Cartesian plane.

Let $\AA$ denote the area between the curve $\map f x$, the straight lines $x = a$ and $x = b$, and the $x$-axis.

Then:

- $\AA = \ds \int_a^b \map f x \rd x$

where $\ds \int_a^b \map f x \rd x$ denotes the (Darboux) definite integral of $f$ from $a$ to $b$.

## Proof

Let $x \in \closedint a b$.

Let $\delta x$ be an arbitrarily small positive real number such that $x + \delta x \in \closedint a b$.

Consider the small strip between:

- the $x$-axis
- the vertical straight lines through $\tuple {x, 0}$ and $\tuple {x + \delta x, 0}$
- the curve $\map f x$.

The area of this strip is approximated by $y \delta x$.

Hence we create a model of the geometric interpretation of the definite integral.

This theorem requires a proof.In particular: A rigorous proof based on the above half-baked waffleYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Examples

### Area under $y = \paren {x - 1} \paren {x - 2}$

The area between the $x$-axis and the curve $y = \paren {x - 1} \paren {x - 2}$ is $\dfrac 1 6$.

### Area under $y = \sin x$ between $0$ and $\pi$

The area bounded by the curve $y = \sin x$ and the $x$-axis between $x = 0$ and $x = \pi$ is $2$.

### Area between $x = y^2$ and $x^2 = 8 y$

Area under Curve/Examples/Between x=y^2 and x^2=8y

## Sources

- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text {II}$. Calculus: Area, Volume and Centre of Gravity