Area of Lobe of Lemniscate of Bernoulli

Theorem
Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively.

Let $O$ denote the origin.

The area of one lobe of $M$ is $a^2$.

Proof
By the definition of the lemniscate of Bernoulli, we have that the polar equation of $M$ is:


 * $r^2 = 2 a^2 \cos 2 \theta$

Let $\mathcal A$ denote the area of one lobe of $M$.

The boundary of the right hand lobe of $M$ is traced out where $-\dfrac \pi 2 \le 2 \theta \le \dfrac \pi 2$.

Thus from Area between Radii and Curve in Polar Coordinates: