The area of the region bounded by |x| + |y| = 1, x ≥ 0 , y ≥ 0 is:

The correct answer is Option (2) → \(\displaystyle\frac{1}{2}\)

For \(x \ge 0\) and \(y \ge 0\), |x| = x and |y| = y, so the curve is the straight line

    \(x + y = 1,\quad y = 1 - x\)

It meets the axes at (1, 0) and (0, 1). The region is the triangle under this line from x = 0 to x = 1:

    \( \displaystyle \text{Area} = \int_{0}^{1}(1 - x)\,dx = \left[x - \frac{x^2}{2}\right]_{0}^{1} \)

    \( \displaystyle = 1 - \frac{1}{2} = \frac{1}{2} \)

This is also \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1\) for the right triangle with legs 1 and 1.