An Area Worth Exploring

Geometry Level 2

It's common knowledge that the graph of \(x^2 + y^2 \leq 1\) is a unit disk, and the area of the region is \(\pi.\)

However, it's not common for someone to know the shape of the graph \[ \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1,\] where \(\lfloor \cdot \rfloor\) is the floor function. To three decimal places, what is the area of the region on the coordinate plane that satisfies the equation \( \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1?\)

Note: Try to draw a hand sketch of this curve and provide analysis instead of using any software!


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