Area between Radii and Curve in Polar Coordinates

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Theorem

Let $C$ be a curve expressed in polar coordinates $\polar {r, \theta}$ as:

$r = \map g \theta$

where $g$ is a real function.


Let $\theta = \theta_a$ and $\theta = \theta_b$ be the two rays from the pole at angles $\theta_a$ and $\theta_b$ to the polar axis respectively.


Then the area $\mathcal A$ between $\theta_a$, $\theta_b$ and $C$ is given by:

$\displaystyle \mathcal A = \int \limits_{\theta \mathop = \theta_a}^{\theta \mathop = \theta_b} \frac {\paren {\map g \theta}^2 \rd \theta} 2$

as long as $\paren {\map g \theta}^2$ is integrable.


Proof

AreaPolarIntegral.png