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

The final solution should be $\sqrt{2}\pi$. I tried to change it into polar coordinates, but I couldn't calculate the integral.


Here is the solution:

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Integrate to get the area. By symmetry, it is eight times the integral $\theta = \pi/4$ and $\pi/2$. In (simpler) rectilinear coordinates:

$$\int\limits_{x=0}^1 \frac{\sqrt{\sqrt{-4 x^4+4 x^2+1}+1}}{\sqrt{2}} - x\ dx = 0.55536$$


$$y(x) = \frac{\sqrt{\sqrt{-4 x^4+4 x^2+1}+1}}{\sqrt{2}}$$

is the solution to the original stated equation in the relevant region.

So the total area is: $8 \times 0.55536 = 4.44288 = \sqrt{2} \pi$.




Solve for $r^2$


$r$ is non-zero, so:



As you stated the area is eight times a single octant of the region so:

$$A=4\int_0^{\frac{\pi}{4}} \frac{2\sec^2{2\theta}\text{d}\theta}{2+\tan^2{2\theta}}=4\int_0^{\frac{\pi}{4}}\frac{\text{d}(\tan{2\theta})}{2+\tan^2{2\theta}}=4\int_0^\infty \frac{\text{d}t}{2+t^2}=4\frac{\pi}{2\sqrt{2}}=\sqrt{2}\,\pi$$

  • $\begingroup$ Nice... keeping it in radial coordinates (+1). $\endgroup$ – David G. Stork Mar 26 at 0:46

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