Question

Area of the region bounded by the curve |x|+|y|=1 and x-axis is :

• A. 1
• B. 2
• C. \(\frac{1}{2}\)
• D. \(\frac{3}{2}\)

Answer 1

The Correct Option is A

The given curve is |x| + |y| = 1. This represents a square centered at the origin (0, 0) with vertices (1,0), (0,1), (-1,0), and (0,-1). The area of the region bounded by this curve can be found by considering the geometry of the figure.

The equation |x| + |y| = 1 defines a square in the coordinate plane. It's necessary to analyze the equations for each segment:

• In the first quadrant where x ≥ 0 and y ≥ 0, the equation is x + y = 1.
• In the second quadrant where x ≤ 0 and y ≥ 0, the equation is -x + y = 1.
• In the third quadrant where x ≤ 0 and y ≤ 0, the equation is -x - y = 1.
• In the fourth quadrant where x ≥ 0 and y ≤ 0, the equation is x - y = 1.

The entire square is symmetric about the x-axis and y-axis. To find the area below the x-axis, integrate over the relevant area below y = 0:

The area in one quadrant of the square is (1/2)*1*1, because it’s a right triangle with legs of length 1.

Since the figure is symmetric about both axes, the total area under the x-axis is \(2 \times \frac{1}{2} \times 1 = 1\).

Thus, the area of the region bounded by the curve |x| + |y| = 1 and the x-axis is 1.

This confirms that the correct answer is: 1