Note that this curve is symmetric in all the four quadrants. Therefore, it is sufficient to just compute the area in the first quadrant, i.e. $x > 0$ and $y > 0$. It is easy to find that the $x$-intercept is $1$. Then because

$$\int_0^1 (1-x^{2/5})dx = \left.(x-\frac{5}{7}x^{7/5}\right|_0^1=\frac{2}{7} $$

therefore the final answer is $\boxed{\frac{8}{7}}$.