How do I calculate the area of Bernoulli's Lemniscate?

Question

can anyone help me calculate this area? I have to use double integrals, and the question sounds like this: " Calculate the area bounded by the curve $(x^2+y^2)^2=a^2(x^2-y^2)$, where $a$ is a real constant. I have searched online and found that this type of curve is a lemniscate, but I do not know how to convert $x,y$ to polar coordinates.

Could you please point me in the right direction ?

Answer 1

Since the lemniscate encloses 4 equal subregions, one in each quadrant, you could use

$\displaystyle A=4\int_0^{\frac{\pi}{4}}\int_0^{a\sqrt{\cos2\theta}}r\; dr d\theta$.

Answer 2

$$A=2(\dfrac{1}{2}\int r^2d\theta)=a^2\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\cos(2\theta)d {\theta}=a^2$$