Find the area of curve enclosed by : |x+y|+|x-y| ≤4,|x| ≤1,...

Key Concepts

Absolute Value Inequalities

This concept involves understanding how inequalities containing absolute value expressions define regions in the plane. Expressions such as |x+y| + |x-y| ? 4 can be analyzed to reveal standard geometric shapes (in this case, a rotated square or diamond) by leveraging properties of absolute values, symmetry, and sometimes by applying a linear transformation. Recognizing these shapes helps simplify the analysis of regions defined by such inequalities.

Transformation of Quadratic Expressions

When dealing with expressions like ?(x² - 2x + 1), it is often beneficial to simplify them through algebraic manipulation such as completing the square. In this example, the quadratic expression factors neatly into (x - 1)², leading to the form y ? |x - 1|. This transformation reveals that the inequality describes a V-shaped region, facilitating further geometric interpretation and integration.

Intersection of Planar Regions

In problems where the area enclosed by a combination of inequalities is sought, it is crucial to consider the intersection of multiple regions defined by each constraint (such as |x+y|+|x-y| ? 4, |x| ? 1, and y ? |x-1|). Understanding how these regions overlap and determining their boundaries allows one to set up the proper integrals or to apply geometric reasoning in order to compute the area of the complex region.

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