What is the limit of a bounded region with a specific boundary?

[Blandongstein]

Let $A_r(r \in \mathbb{N})$ be the area of the bounded region whose boundary is defined by $(6y^2r-x)(6\pi^2 y-x)=0$ then find the value of

$$ \lim_{n \to \infty}(\sqrt{A_1 A_2 A_3}+\sqrt{A_2 A_3 A_4}+\cdots \text{n terms})$$

 

[Evgeny.Makarov]

Start by finding \(A_r\).