Find the integral of the second kind for Bernoulli leminscate

Question

I have tried to solve this integral of the second kind( with respect to $x,y$) but I stumbled on finding a correct path over which to integrate. I know from the definition that I should get at $ \int\limits_{C}{{Pdx\, + Q\,dy}} = \int\limits_{C}{{P\left( {x,y} \right)dx}} + \int\limits_{C}{{Q\left( {x,y} \right)\,dy}}$, but I lack examples and I don`t know how to apply it. I am new to this type of problems and I do not have many examples, could you provide a full proof, or at least in the form of an answer, such that it would serve as a model for similar problems I encounter? Thank you very much!!!

Answer 1

$$(x^2+y^2)^2=a^2(x^2-y^2)$$ is a quadratic equation in both $x^2$ and $y^2$. For example you can draw

$$y=\pm\sqrt{\frac{\sqrt{8ax^2+a^2}-2x^2-a}2}.$$

But integration will be arduous in Cartesian coordinates, it is better to switch to polar.

Answer 2

Green's theorem says that $$\int_{\Gamma} x\,dy-y\,dx = 2\int_{\operatorname{Int} \Gamma}dx\,dy = 2\operatorname{Area}(\operatorname{Int} \Gamma)$$ where $\operatorname{Int} \Gamma$ is the area enclosed by $\Gamma$. The equation of $\Gamma$ an be rewritten in polar coordinates as $r^2 = a^2\cos(2\phi)$ so the area is simply $$\operatorname{Area}(\operatorname{Int} \Gamma) = 2\int_{-\pi/4}^{\pi/4}\int_0^{a\sqrt{\cos(2\phi)}}r\,dr\,d\phi = \frac{a^2}2.$$