The area (in sq. units) of the region \(\{(x, y): |x| \le 2,\ 0 \le y \le x+3\}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 12
The condition \(|x| \le 2\) means \(-2 \le x \le 2\). On this interval \(x+3 \ge 1\), so the line stays above the x-axis and the region lies between the x-axis and the line y = x + 3:
\( \displaystyle A = \int_{-2}^{2}(x+3)\,dx \)
\( \displaystyle = \left[\frac{x^{2}}{2}+3x\right]_{-2}^{2} \)
\(= (2+6)-(2-6)\)
\(= 12\)
The region is a trapezium with parallel sides 1 (at x = −2) and 5 (at x = 2) and width 4, and \(\frac{1}{2}(1+5)\times4 = 12\) agrees.