Area between Radii and Whorls of Archimedean Spiral

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Theorem

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

$r = a \theta$


Let $\theta = \theta_1$ and $\theta = \theta_2$ be the two rays from the pole at angles $\theta_1$ and $\theta_b$ to the polar axis respectively.

Let $R$ be the figure enclosed by:

$\theta_1$ and $\theta_2$
the $n$th turn of $S$ and the $n+1$th turn of $S$


The area $\mathcal A$ of $R$ is given by:

$\mathcal A = a^2 \pi \left({\theta_2 - \theta_1}\right) \left({\theta_2 + \theta_1 + 2 \pi \left({2 n + 1}\right)}\right)$


Proof

The straight line boundaries of $R$ are given as $\theta_1$ and $\theta_2$.

The corners of $R$ are located where:

$\theta = \theta_1 + 2 n \pi$
$\theta = \theta_2 + 2 n \pi$
$\theta = \theta_1 + 2 \left({n + 1}\right) \pi$
$\theta = \theta_2 + 2 \left({n + 1}\right) \pi$


ArchimedeanSpiralAreaBetweenRadii.png


\(\displaystyle \mathcal A\) \(=\) \(\displaystyle \int_{\theta_1 + 2 \left({n + 1}\right) \pi}^{\theta_2 + 2 \left({n + 1}\right) \pi} \frac {\left({a \theta}\right)^2 \, \mathrm d \theta} 2 - \int_{\theta_1 + 2 n \pi}^{\theta_2 + 2 n \pi} \frac {\left({a \theta}\right)^2 \, \mathrm d \theta} 2\) $\quad$ Area between Radii and Curve in Polar Coordinates $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left[{\frac {a^2 \theta^3} 6}\right]_{\theta_1 + 2 \left({n + 1}\right) \pi}^{\theta_2 + 2 \left({n + 1}\right) \pi} - \left[{\frac {a^2 \theta^3} 6}\right]_{\theta_1 + 2 n \pi}^{\theta_2 + 2 n \pi}\) $\quad$ Primitive of Power $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2} 6 \left({\left({\theta_2 + 2 \left({n + 1}\right) \pi}\right)^3 - \left({\theta_1 + 2 \left({n + 1}\right) \pi}\right)^3 - \left({\theta_2 + 2 n \pi}\right)^3 + \left({\theta_1 + 2 n \pi}\right)^3}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2} 6 \left({\theta_2^3 + 3 \left({2 \left({n + 1}\right) \pi}\right)\theta_2^2 + 3 \left({2 \left({n + 1}\right) \pi}\right)^2 \theta_2 + \left({2 \left({n + 1}\right) \pi}\right)^3}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \frac {a^2} 6 \left({\theta_1^3 + 3 \left({2 \left({n + 1}\right) \pi}\right)\theta_1^2 + 3 \left({2 \left({n + 1}\right) \pi}\right)^2 \theta_1 + \left({2 \left({n + 1}\right) \pi}\right)^3}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \frac {a^2} 6 \left({\theta_2^3 + 3 \left({2 n \pi}\right)\theta_2^2 + 3 \left({2 n \pi}\right)^2 \theta_2 + \left({2 n \pi}\right)^3}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac {a^2} 6 \left({\theta_1^3 + 3 \left({2 n \pi}\right)\theta_1^2 + 3 \left({2 n \pi}\right)^2 \theta_1 + \left({2 n \pi}\right)^3}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle a^2 \pi \left({\left({n + 1}\right) \theta_2^2 + 2 \pi \left({n + 1}\right)^2 \theta_2 - \left({n \theta_2^2 + 2 \pi n^2 \theta_2}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle a^2 \pi \left({\left({n + 1}\right)\theta_1^2 + 2 \pi \left({n + 1}\right)^2 \theta_1 - \left({n \theta_1^2 + 2 \pi n^2 \theta_1}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle a^2 \pi \left({\theta_2^2 + 2 \pi \left({2 n + 1}\right) \theta_2}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle a^2 \pi \left({\theta_1^2 + 2 \pi \left({2 n + 1}\right) \theta_1}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle a^2 \pi \left({\theta_2 - \theta_1}\right) \left({\theta_2 + \theta_1 + 2 \pi \left({2 n + 1}\right)}\right)\) $\quad$ $\quad$


Historical Note

The Area between Radii and Whorls of Archimedean Spiral was first determined by Archimedes.