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 $\AA$ enclosed by the first turn of $S$ and the polar axis is given by:

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


ArchimedeanSpiralArea.png


Proof

\(\ds \AA\) \(=\) \(\ds \int_0^{2 \pi} \frac {\paren {a \theta}^2} 2 \rd \theta\) Area between Radii and Curve in Polar Coordinates‎
\(\ds \) \(=\) \(\ds \intlimits {\frac {a^2 \theta^3} 6} 0 {2 \pi}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {\paren {2 \pi}^3 a^2} 6\)
\(\ds \) \(=\) \(\ds \frac {4 \pi^3 a^2} 3\) 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

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