Length of Perimeter of Cardioid/Proof 1
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Theorem
Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
- $r = 2 a \paren {1 + \cos \theta}$
where $a > 0$.
The length of the perimeter of $C$ is $16 a$.
Proof
Let $\LL$ denote the length of the perimeter of $C$.
The boundary of the $C$ is traced out where $-\pi \le \theta \le \pi$.
From Arc Length for Parametric Equations:
- $\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
where, from Equation of Cardioid:
- $\begin {cases} x & = 2 a \cos \theta \paren {1 + \cos \theta} \\ y & = 2 a \sin \theta \paren {1 + \cos \theta} \end {cases}$
We have:
\(\ds \frac {\d x} {\d \theta}\) | \(=\) | \(\ds 2 a \map {\frac \d {\d \theta} } {\cos \theta + \cos^2 \theta}\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds -2 a \paren {\sin \theta + 2 \cos \theta \sin \theta}\) | Derivative of Cosine Function, Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -2 a \paren {\sin \theta + \sin 2 \theta}\) | Double Angle Formula for Sine | |||||||||||
\(\ds \frac {\d y} {\d \theta}\) | \(=\) | \(\ds 2 a \map {\frac \d {\d \theta} } {\sin \theta + \sin \theta \cos \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \paren {\cos \theta + \cos^2 \theta - \sin^2 \theta}\) | Derivative of Sine Function, Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \paren {\cos \theta + \cos 2 \theta}\) | Double Angle Formula for Cosine |
Thus:
\(\ds \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2}\) | \(=\) | \(\ds \sqrt {4 a^2 \paren {\paren {\sin \theta + \sin 2 \theta}^2 + \paren {\cos \theta + \cos 2 \theta}^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \sqrt {\sin^2 \theta + 2 \sin \theta \sin 2 \theta + \sin^2 2 \theta + \cos^2 \theta + 2 \cos \theta \cos 2 \theta + \cos^2 2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \sqrt {2 + 2 \sin \theta \sin 2 \theta + 2 \cos \theta \cos 2 \theta}\) | Sum of Squares of Sine and Cosine in $2$ instances | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \sqrt {2 + 2 \sin \theta \paren {2 \sin \theta \cos \theta} + 2 \cos \theta \paren {\cos^2 \theta - \sin^2 \theta} }\) | Double Angle Formulas | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \sqrt {2 + 4 \sin^2 \theta \cos \theta + 2 \cos^3 \theta - 2 \sin^2 \theta \cos \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \sqrt {2 + 2 \sin^2 \theta \cos \theta + 2 \cos^3 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \sqrt {2 + 2 \cos \theta \paren {\sin^2 \theta + \cos^2 \theta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 a \sqrt {\dfrac {1 + \cos \theta} 2}\) | Sum of Squares of Sine and Cosine and extracting factor | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds 4 a \cos \dfrac \theta 2\) | Half Angle Formula for Cosine |
\(\ds \LL\) | \(=\) | \(\ds \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta\) | Area between Radii and Curve in Polar Coordinates | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\pi}^\pi 4 a \cos \dfrac \theta 2 \rd \theta\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 a \intlimits {2 \sin \dfrac \theta 2} {-\pi} \pi\) | Primitive of $\cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 8 a \paren {\sin \dfrac \pi 2 - \sin \dfrac {-\pi} 2}\) | evaluation between $-\pi$ and $\pi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 8 a \paren {1 - \paren {-1} }\) | Sine of Right Angle, Sine Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds 16 a\) |
$\blacksquare$