# Area between Radii and Whorls of Archimedean Spiral

## 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 $\AA$ of $R$ is given by:

$\AA = a^2 \pi \paren {\theta_2 - \theta_1} \paren {\theta_2 + \theta_1 + 2 \pi \paren {2 n + 1} }$

## Proof

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

The corners of $R$ are located where:

$\theta = \theta_1 + 2 n \pi$
$\theta = \theta_2 + 2 n \pi$
$\theta = \theta_1 + 2 \paren {n + 1} \pi$
$\theta = \theta_2 + 2 \paren {n + 1} \pi$

 $\ds \AA$ $=$ $\ds \int_{\theta_1 + 2 \paren {n + 1} \pi}^{\theta_2 + 2 \paren {n + 1} \pi} \frac {\paren {a \theta}^2 \rd \theta} 2 - \int_{\theta_1 + 2 n \pi}^{\theta_2 + 2 n \pi} \frac {\paren {a \theta}^2 \rd \theta} 2$ Area between Radii and Curve in Polar Coordinates $\ds$ $=$ $\ds \intlimits {\frac {a^2 \theta^3} 6} {\theta_1 + 2 \paren {n + 1} \pi} {\theta_2 + 2 \paren {n + 1} \pi} - \intlimits {\frac {a^2 \theta^3} 6} {\theta_1 + 2 n \pi} {\theta_2 + 2 n \pi}$ Primitive of Power $\ds$ $=$ $\ds \frac {a^2} 6 \paren {\paren {\theta_2 + 2 \paren {n + 1} \pi}^3 - \paren {\theta_1 + 2 \paren {n + 1} \pi}^3 - \paren {\theta_2 + 2 n \pi}^3 + \paren {\theta_1 + 2 n \pi}^3}$ $\ds$ $=$ $\ds \frac {a^2} 6 \paren {\theta_2^3 + 3 \paren {2 \paren {n + 1} \pi} \theta_2^2 + 3 \paren {2 \paren {n + 1} \pi}^2 \theta_2 + \paren {2 \paren {n + 1} \pi}^3}$ $\ds$  $\, \ds - \,$ $\ds \frac {a^2} 6 \paren {\theta_1^3 + 3 \paren {2 \paren {n + 1} \pi} \theta_1^2 + 3 \paren {2 \paren {n + 1} \pi}^2 \theta_1 + \paren {2 \paren {n + 1} \pi}^3}$ $\ds$  $\, \ds - \,$ $\ds \frac {a^2} 6 \paren {\theta_2^3 + 3 \paren {2 n \pi} \theta_2^2 + 3 \paren {2 n \pi}^2 \theta_2 + \paren {2 n \pi}^3}$ $\ds$  $\, \ds + \,$ $\ds \frac {a^2} 6 \paren {\theta_1^3 + 3 \paren {2 n \pi} \theta_1^2 + 3 \paren {2 n \pi}^2 \theta_1 + \paren {2 n \pi}^3}$ $\ds$ $=$ $\ds a^2 \pi \paren {\paren {n + 1} \theta_2^2 + 2 \pi \paren {n + 1}^2 \theta_2 - \paren {n \theta_2^2 + 2 \pi n^2 \theta_2} }$ $\ds$  $\, \ds - \,$ $\ds a^2 \pi \paren {\paren {n + 1} \theta_1^2 + 2 \pi \paren {n + 1}^2 \theta_1 - \paren {n \theta_1^2 + 2 \pi n^2 \theta_1} }$ $\ds$ $=$ $\ds a^2 \pi \paren {\theta_2^2 + 2 \pi \paren {2 n + 1} \theta_2}$ $\ds$  $\, \ds - \,$ $\ds a^2 \pi \paren {\theta_1^2 + 2 \pi \paren {2 n + 1} \theta_1}$ $\ds$ $=$ $\ds a^2 \pi \paren {\theta_2 - \theta_1} \paren {\theta_2 + \theta_1 + 2 \pi \paren {2 n + 1} }$

## Historical Note

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