Trying to solve for the area enclosed by $x^4+y^4=1$. A friend posed this question to me today, but I have no clue what to do to solve this. Keep in mind, we don't even know if there is a straightforward solution. I think he just likes thinking up problems out of thin air. 


Anyway, the question becomes more general, since we *think* that 


$lim\_{n\to\infty}\int\_0^1{(1-x^n)^{1/n}} = {1\over4}$    (it approaches a square / becomes linear)


can anyone confirm that this is true or not?



Area enclosed by x^4 + y^4 = 1
I always prefer not to skip $dx$:
$$
I\_n=\int\_0^1(1-x^n)^{1/n}dx.
$$
After the change of variable $t=x^n$, the integral becomes the beta integral,
$$
I\_n=\frac1n\int\_0^1(1-t)^{1/n}t^{1/n-1}dt
=\frac1n\frac{\Gamma(1+1/n)\Gamma(1/n)}{\Gamma(1+2/n)}
=\frac1n\frac{\Gamma(1/n)^2\cdot 1/n}{\Gamma(2/n)\cdot 2/n}
\to1 \quad\text{as $n\to\infty$},
$$
as $1/\Gamma(z)\sim z$ as $z\to 0$.


