Finding the area inside the plot $x^4+y^4=x^2+y^2$ Find the area inside the plot $x^4+y^4=x^2+y^2$.


After proceeding with the polar coordinate, it suffices to evaluate the following: $$I = 4\int_{0}^{\frac{\pi}{2}}\dfrac{d\theta}{2-\sin^2(2\theta)}= 2\int_0^{\pi}\dfrac{dx}{2-\sin^2x}$$. 
Now do a tangent substitution: $$\tan x = t\Rightarrow \sin^2x = \dfrac{t^2}{1+t^2},\,\, dx = \dfrac{dt}{1+t^2}$$
Finally, $$I = 4\int_0^{\infty}\dfrac{dt}{2+t^2} =2\sqrt{2}\cdot(\arctan(\infty) - \arctan(0)) = \sqrt{2}\pi = 4.4429...$$
Or this is also a routine exercise on complex contour integration using the Euler identity $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$.
