13. Find the area enclosed by the graph $x^{2}+y^{2}=|x|+|y|$ on the $x y$-plane.
(2 marks)
求 $x y$ 坐標平面上由圖像 $x^{2}+y^{2}=|x|+|y|$ 所圍出的面積。
(2 分)
13. $\pi+2$
13. Observe that the graph is symmetric about the $x$-axis and the $y$-axis. Hence we need only consider the first quadrant. In the first quadrant, the equation of the graph can be written as $x^{2}+y^{2}=x+y$, or $\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=\frac{1}{2}$, which is a circle passing through $(0,0),(1,0)$ and $(0,1)$. By symmetry, the whole graph can be constructed as shown.
Now the area bounded by the curve can be thought of as a square of side length $\sqrt{2}$ plus four semi-circles of diameter $\sqrt{2}$. Its area is
$$
(\sqrt{2})^{2}+4 \cdot \frac{1}{2} \cdot \pi\left(\frac{\sqrt{2}}{2}\right)^{2}=\pi+2
$$