# Area Enclosed by First Turn of Archimedean Spiral

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## Contents

## Theorem

Let $S$ be the Archimedean spiral defined by the equation:

- $r = a \theta$

The area $\mathcal A$ enclosed by the first turn of $S$ and the polar axis is given by:

- $\mathcal A = \dfrac {4 \pi^3 a^2} 3$

## Proof

\(\displaystyle \mathcal A\) | \(=\) | \(\displaystyle \int_0^{2 \pi} \frac {\left({a \theta}\right)^2} 2 \, \mathrm d \theta\) | Area between Radii and Curve in Polar Coordinates | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left[{\frac {a^2 \theta^3} 6}\right]_0^{2 \pi}\) | Primitive of Power | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\left({2 \pi}\right)^3 a^2 \theta^3} 6\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {4 \pi^3 a^2} 3\) | after simplification |

$\blacksquare$

## Historical Note

The Area Enclosed by First Turn of Archimedean Spiral was first determined by Archimedes in his book *On Spirals*.

His proof appears as Proposition $24$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)