Area Enclosed by First Turn of Archimedean Spiral

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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:

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


ArchimedeanSpiralArea.png


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