Area of the region bounded by the lines \( |x|+|y|=1,\ x\ge 0,\ y\ge 0 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{1}{2} \) square units
For \( x\ge 0,\ y\ge 0 \), \( |x|=x \) and \( |y|=y \), so the line is \( x+y=1 \).
With the axes it bounds the triangle with vertices (0, 0), (1, 0) and (0, 1).
\( \displaystyle \text{Area}=\int_0^1(1-x)\,dx \)
\( \displaystyle =\left[x-\frac{x^2}{2}\right]_0^1 \)
\( \displaystyle =1-\frac{1}{2} \)
\( \displaystyle =\frac{1}{2}\ \text{square units} \)
Shortcut: The region is a right triangle with both legs of length 1: area \( =\frac{1}{2}\times 1\times 1=\frac{1}{2} \).
Tip: \( |x|+|y|=a \) is a square with vertices (±a, 0), (0, ±a) and total area \( 2a^2 \); each quadrant holds one quarter, \( \frac{a^2}{2} \).