Area between Radii and Whorls of Archimedean Spiral
Jump to navigation
Jump to search
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:
- $\AA = a^2 \pi \paren {\theta_2 - \theta_1} \paren {\theta_2 + \theta_1 + 2 \pi \paren {2 n + 1} }$
![]() | This article needs proofreading. In particular: My algebra may have failed me If you believe all issues are dealt with, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2} 6 \paren {\paren {\theta_2 + 2 \paren {n + 1} \pi}^3 - \paren {\theta_1 + 2 \paren {n + 1} \pi}^3 - \paren {\theta_2 + 2 n \pi}^3 + \paren {\theta_1 + 2 n \pi}^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a^2} 6 \paren {\theta_2^3 + 3 \paren {2 \paren {n + 1} \pi} \theta_2^2 + 3 \paren {2 \paren {n + 1} \pi}^2 \theta_2 + \paren {2 \paren {n + 1} \pi}^3}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {a^2} 6 \paren {\theta_1^3 + 3 \paren {2 \paren {n + 1} \pi} \theta_1^2 + 3 \paren {2 \paren {n + 1} \pi}^2 \theta_1 + \paren {2 \paren {n + 1} \pi}^3}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \frac {a^2} 6 \paren {\theta_2^3 + 3 \paren {2 n \pi} \theta_2^2 + 3 \paren {2 n \pi}^2 \theta_2 + \paren {2 n \pi}^3}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \frac {a^2} 6 \paren {\theta_1^3 + 3 \paren {2 n \pi} \theta_1^2 + 3 \paren {2 n \pi}^2 \theta_1 + \paren {2 n \pi}^3}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \pi \paren {\paren {n + 1} \theta_2^2 + 2 \pi \paren {n + 1}^2 \theta_2 - \paren {n \theta_2^2 + 2 \pi n^2 \theta_2} }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds a^2 \pi \paren {\paren {n + 1} \theta_1^2 + 2 \pi \paren {n + 1}^2 \theta_1 - \paren {n \theta_1^2 + 2 \pi n^2 \theta_1} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \pi \paren {\theta_2^2 + 2 \pi \paren {2 n + 1} \theta_2}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds a^2 \pi \paren {\theta_1^2 + 2 \pi \paren {2 n + 1} \theta_1}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \pi \paren {\theta_2 - \theta_1} \paren {\theta_2 + \theta_1 + 2 \pi \paren {2 n + 1} }\) |
![]() | This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Historical Note
The Area between Radii and Whorls of Archimedean Spiral was first determined by Archimedes.