Area inside Cardioid
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Theorem
Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
- $r = 2 a \paren {1 + \cos \theta}$
The area inside $C$ is $6 \pi a^2$.
Proof
Let $\AA$ denote the area inside $C$.
The boundary of $C$ is traced out where $-\pi \le \theta \le \pi$.
Thus:
\(\ds \AA\) | \(=\) | \(\ds \int_{-\pi}^\pi \dfrac {\map {r^2} \theta} 2 \rd \theta\) | Area between Radii and Curve in Polar Coordinates | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\pi}^\pi \dfrac {\paren {2 a \paren {1 + \cos \theta} }^2} 2 \rd \theta\) | Equation of Cardioid | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a^2 \int_{-\pi}^\pi \paren {1 + 2 \cos \theta + \cos^2 \theta} \rd \theta\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a^2 \intlimits {\theta + 2 \sin \theta + \frac \theta 2 + \frac {\sin 2 \theta} 4} {-\pi} \pi\) | Primitive of $\cos a x$, Primitive of Square of Cosine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a^2 \paren {\paren {\pi + 2 \sin \pi + \dfrac \pi 2 + \frac {\sin 2 \pi} 4} - \paren {-\pi + 2 \, \map \sin {-\pi} + \dfrac {-\pi} 2 + \frac {\map \sin {-2 \pi} } 4} }\) | evaluation between $-\pi$ and $\pi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a^2 \paren {\pi + \dfrac \pi 2 - \paren {-\pi} - \paren {\dfrac {-\pi} 2} }\) | Sine of Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds 6 \pi a^2\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Cardioid: $11.13$