A point P moves in X Y -plane in such a way that...

A point $P$ moves in $X Y$ -plane in such a way that $[|x|]+[|y|]=1$, where [ ] denotes the greatest integer function. Area of the region representing all possible of the point $P$ is equal to ............

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Key Concepts

Area Calculation in the Plane

Calculating the area of regions defined by algebraic conditions involves breaking the region into simpler parts whose areas are easier to compute, often using rectilinear geometry. This technique is fundamental when dealing with problems that require summation of areas from multiple disjoint regions defined by piecewise functions.

Piecewise Analysis in Coordinate Geometry

Many geometric conditions defined with floor and absolute value functions naturally break the coordinate plane into different regions or intervals. Recognizing these disjoint subregions based on the integer parts of coordinates is key in setting up a correct analysis of the overall area or shape.

Absolute Value Function

The absolute value function measures the distance of a number from zero, disregarding sign. In multidimensional problems, it is used to create symmetric conditions around the origin, allowing the problem to be interpreted in all quadrants of the coordinate plane.

Greatest Integer Function

This function, often called the floor function, returns the largest integer less than or equal to a given real number. It is crucial in problems where continuous variables are converted into discrete parts, and understanding it helps one analyze regions or conditions that involve integer step changes.

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