Area between Radii and Curve in Polar Coordinates

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 $\AA$ between $\theta_a$, $\theta_b$ and $C$ is given by:
 * $\ds \AA = \int \limits_{\theta \mathop = \theta_a}^{\theta \mathop = \theta_b} \frac {\paren {\map g \theta}^2 \rd \theta} 2$

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

Proof

 * AreaPolarIntegral.png

Consider the area of the brown triangle.

This would be:
 * $a_\triangle = \dfrac 1 2 r^2 \map \sin {\delta \theta}$

We will be using non-standard analysis, so let $\delta \theta = \varepsilon > 0$, an infinitesimal.

Thus:
 * $a_\triangle = \dfrac 1 2 r^2 \sin \varepsilon$

Using the Power Series Expansion for Sine Function: