Hello, the prince!
I assume you want the area of the region . . .
Calculate the area of the region determined by: .$\displaystyle 81y^2\:=\:4x^2(9x^2) $
We have: .$\displaystyle y \;=\;\pm\frac{2}{9}x\sqrt{9x^2}$
The graph is symmetric to the $\displaystyle x$axis, $\displaystyle y$axis and the origin
. . and has intercepts: .$\displaystyle (0,0),\;(\pm3,0)$
The graph looks like a infintysign. Code:

* *  * *
* *  * *
*  *
 *        *        * 
*  *
* *  * *
* *  * *

As skeeter pointed out, it is called a lemniscate.
The total area is: .$\displaystyle A \;=\;4 \times \tfrac{2}{9}\int^3_0 x\left(9x^2\right)^{\frac{1}{2}}dx$