2. Plot on the plane $(x ; y)$ the set of points satisfying the equation $\left|9+8 y-x^{2}-y^{2}\right|+|8 y|=16 y+9-x^{2}-y^{2}$, and find the area of the resulting figure.
Answer: $25 \pi-25 \arcsin 0.6+12$.

Solution. Note that the equality $|a|+|b|=a+b$ holds if and only if the numbers $a$ and $b$ are non-negative (since if at least one of them is negative, the left side is greater than the right side). Therefore, the first equation is equivalent to the system of inequalities

$$
\left\{\begin{array} { l } 
{ 9 + 8 y - x ^ { 2 } - y ^ { 2 } \geq 0 , } \\
{ 8 y \geq 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
(y-4)^{2}+x^{2} \leq 25, \\
y \geq 0 .
\end{array}\right.\right.
$$

The first inequality defines a circle of radius 5 centered at $(0 ; 4)$, and the entire system defines the part of this circle lying in the half-plane $y \geq 0$. The area of this segment is equal to the difference between the area of the circle and the area of the segment of this circle lying in the half-plane $y \leq 0$. Since the central angle of this segment is $2 \arcsin 0.6$, we obtain that its area $S$ is $25 \pi-25 \arcsin 0.6+12$.