What is the area of the largest circle that can be enclosed by the regions |y| = 2 and |x| = 1?

The area of the largest circle that can be enclosed by the regions |y| = 2 and |x| = 1 is determined by the smallest distance between the lines.

The lines |y| = 2 form a rectangle with the lines |x| = 1. The smallest distance between these lines is the shorter side of the rectangle, which is 2 units (from x = -1 to x = 1).

The radius of the largest circle that can be enclosed in this rectangle is half of this distance, which is 1 unit.

The area of a circle is given by the formula πr², where r is the radius of the circle.

Substituting r = 1 into the formula gives:

Area = π(1)² = π square units.