Length of Perimeter of Cardioid/Proof 2
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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 Polar Curve:
- $\ds \LL = \int_{\theta \mathop = -\pi}^{\theta \mathop = \pi} \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$
where:
- $r = 2 a \paren {1 + \cos \theta}$
Note that we have:
\(\ds \frac {\d r} {\d \theta}\) | \(=\) | \(\ds 2 a \frac \d {\d \theta} \paren {1 + \cos \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -2 a \sin \theta\) | Sum Rule for Derivatives, Derivative of Cosine Function |
We therefore have:
\(\ds \LL\) | \(=\) | \(\ds \int_{-\pi}^\pi \sqrt {4 a^2 \paren {1 + \cos \theta}^2 + 4 a^2 \sin^2 \theta} \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \int_{-\pi}^\pi \sqrt {1 + 2 \cos \theta + \cos^2 \theta + \sin^2 \theta} \rd \theta\) | extracting a factor of $\sqrt {4 a^2} = 2 a$, noting that $a \ge 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \int_{-\pi}^\pi \sqrt {2 + 2 \cos \theta} \rd \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 a \int_0^\pi \sqrt {4 \cos^2 \frac \theta 2} \rd \theta\) | Definite Integral of Even Function, Double Angle Formula for Cosine: Corollary $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 8 a \int_0^\pi \size {\cos \frac \theta 2} \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 a \int_0^\pi \cos \frac \theta 2 \rd \theta\) | Definition of Absolute Value, noting that $\cos \theta \ge 0$ for $0 \le \theta \le \dfrac \pi 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 16 a \intlimits {\sin \frac \theta 2} 0 \pi\) | Primitive of $\cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 16 a \paren {\sin \frac \pi 2 - \sin 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 16 a\) | Sine of Right Angle, Sine of Zero is Zero |
$\blacksquare$