<s>[INST] Area enclosed by the curve $x^2+y^2=|x|+|y|$. 
Find the area enclosed by the curve $x^2+y^2=|x|+|y|$ on the coordinate plane.

I have no idea how this curve looks like, therefore can't find the area. Please help. Thank you.
 [/INST] Wolfram|Alpha shows you how the curve looks like. To obtain the area, first note that by symmetry we can concentrate on the first quadrant. There the equation is equivalent to $x^2 + y^2 = x+y\iff y^2 - y + (x^2-x) = 0$, so
\[
  y_{1,2} = \frac 12 \pm \sqrt{\frac 14 - (x^2-x)}
\]
The randicand is nonnegative for $x^2 - x \le \frac 14$, that is $0 \le x \le \frac 12(1 + \sqrt 2)$ (note that we are in the first quadrant, hence $x \ge 0$). $y_2$ is positive if moreover $x^2-x \ge 0$, that is $x \ge 1$. So the area is 
\begin{align*}
  A &= 4 \left(\int_0^1 \left(\frac 12 + \sqrt{\frac 14 - (x^2-x)}\right)\,dx 
       + \int_1^{\frac 12(1 +\sqrt 2)} 2\sqrt{\frac 14 - (x^2 - x)}\, dx\right)
\end{align*} 
</s>