<s>[INST] Finding the area inside the plot $x^4+y^4=x^2+y^2$ Find the area inside the plot $x^4+y^4=x^2+y^2$.

 [/INST] Let's look at
the area below the part of the curve
from 
$(0, 1)$ to $1, 1)$
and the line $y = x$.
The total area is
8 times this.
The area below
$y = x$
is obviously
$1/2$.
If
$x^4+y^4=x^2+y^2
$,
$y^4-y^2
=x^2-x^4
$
or,
trying to be clever,
$y^4-y^2+1/4
=x^2-x^4-1/4+1/2
$
or
$(y^2-1/2)^2
=1/2-(x^2-1/2)^2
$
or
$y^2
=\frac12+\sqrt{\frac12-(x^2-\frac12)^2}
$
so that
$y
=\sqrt{\frac12+\sqrt{\frac12-(x^2-\frac12)^2}}
$.
At the moment,
I don't know how to integrate this.
But,
Wolfy numerically says that
$\int_0^1 y dx
\approx 1.05536
$,
so the area between this
ans $y=x$
is about
$0.55536
$
so that
the total area is about
$8*0.55536
=4.44288
$
</s>