The area bounded by the line \( y=x \), the x-axis and the ordinates \( x=-1 \) and \( x=2 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{5}{2} \) Sq.units
The line y = x is below the x-axis for −1 ≤ x < 0 and above it for 0 < x ≤ 2, so the area is split at x = 0 and the part below is taken as positive:
\( \displaystyle \text{Area}=\left|\int_{-1}^{0}x\,dx\right|+\int_{0}^{2}x\,dx \)
\( \displaystyle =\left|\left[\frac{x^{2}}{2}\right]_{-1}^{0}\right|+\left[\frac{x^{2}}{2}\right]_{0}^{2} \)
\( \displaystyle =\left|0-\frac{1}{2}\right|+(2-0) \)
\( \displaystyle =\frac{1}{2}+2 \)
\( \displaystyle =\frac{5}{2}\text{ sq. units} \)
Shortcut: The region is two triangles: legs 1 and 1 on the left (area \( \frac{1}{2} \)) and legs 2 and 2 on the right (area 2), total \( \frac{5}{2} \).
Why the other options are incorrect
Option (1) \( \displaystyle\frac{3}{2} \) Sq.units — Incorrect. \( \frac{3}{2}=\int_{-1}^{2}x\,dx \), the signed integral, in which the part below the x-axis has been subtracted instead of added.
Option (3) \( \displaystyle\frac{7}{2} \) Sq.units — Incorrect. The two triangles have areas \( \frac{1}{2} \) and 2, total \( \frac{5}{2} \), not \( \frac{7}{2} \).
Option (4) \( \displaystyle\frac{9}{2} \) Sq.units — Incorrect. The total is \( \frac{5}{2} \), not \( \frac{9}{2} \).