<s>[INST] Find the area bounded by the curve $x^4+y^4=x^2+y^2$ I am stuck with this problem which deals with evaluating an Area
The problem reads :

Find the area bounded by the curve $x^4+y^4=x^2+y^2$.

I tried factorizing the expression and expressing $y$ in terms of $x$, not able to proceed with that idea. Someone please help me out.
 [/INST] Use polar co-ordinates $x=r \cos t, y=r \sin t$, then this curve is $$r^2=\frac{1}{\sin^4 t+ \cos^4 t}=\ $$, the area the area is
$$A=4 ~ \frac{1}{2}\int_{0}^{\pi/2} r^2 dt=2 \int_{0}^{\pi/2} \frac{1}{\cos^4 t+\sin^4 t}dt =2 \int_{0}^{\pi/2}\frac{\sec^4 t}{1+\tan^4 t} dt $$
Let $z=\tan t \implies dz=\sec^2 t dt$
$$=2\int_{0}^{\infty} \frac{1+z^2}{1+z^4} dz=2 \int_{0}^{\infty}\frac{1+1/z^2}{z^2+1/z^2}=2\int_{0}^{\infty} \frac{1+1/z^2}{(z-1/z)^2+2}$$ $$\implies A=2 \int_{-\infty}^{\infty} \frac{du}{u^2+2}=\sqrt{2} \pi.$$
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