# Thread: Calculating area using Integrals

1. ## Calculating area using Integrals

calculate the surface of the area determined with curve 81y^2=4x^2(9-x^2)

2. Originally Posted by the prince
calculate the surface of the area determined with curve 81y^2=4x^2(9-x^2)
your problem statement is incomplete. Is this enclosed region rotated about an axis? ... or are you simply wanting the area enclosed by the lemniscate?

3. Hello, the prince!

I assume you want the area of the region . . .

Calculate the area of the region determined by: . $81y^2\:=\:4x^2(9-x^2)$

We have: . $y \;=\;\pm\frac{2}{9}x\sqrt{9-x^2}$

The graph is symmetric to the $x$-axis, $y$-axis and the origin
. . and has intercepts: . $(0,0),\;(\pm3,0)$

The graph looks like a infinty-sign.
Code:
                  |
* *        |        * *
*         *    |    *         *
* | *
- * - - - - - - - * - - - - - - - * -
* | *
*         *    |    *         *
* *        |        * *
|

As skeeter pointed out, it is called a lemniscate.

The total area is: . $A \;=\;4 \times \tfrac{2}{9}\int^3_0 x\left(9-x^2\right)^{\frac{1}{2}}dx$