Answer: 800

Solution:

For $x \ge 0$ rewrite the equation as $2x(y + 20)$ $= 400 - y^2$ $= (20 - y)(20 + y),$ so either $y = -20$ or $y = 20 - 2x.$ For $x \le 0$ it becomes $2x(y - 20) = -(y - 20)(y + 20),$ so either $y = 20$ or $y = -20 - 2x.$ The graph therefore consists of two horizontal rays and two rays of slope $-2.$

These rays bound a parallelogram: the top edge runs from $(-20, 20)$ to $(0, 20)$ along $y = 20,$ the bottom edge from $(0, -20)$ to $(20, -20)$ along $y = -20,$ and the two slanted edges of slope $-2$ connect them.

The parallelogram has horizontal base $20$ and height $40$ between the lines $y = 20$ and $y = -20,$ so its area is $20 \cdot 40 = 800.$