The area (in sq. units) of the region bounded by the curve \(y = x^{2}+1\), the x-axis and the lines x = −1 and x = 2 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6
Since \(x^{2}+1 \ge 1 \gt 0\), the curve lies above the x-axis throughout, so the area is the integral of y itself:
\( \displaystyle A = \int_{-1}^{2}(x^{2}+1)\,dx \)
\( \displaystyle = \left[\frac{x^{3}}{3}+x\right]_{-1}^{2} \)
\( \displaystyle = \left(\frac{8}{3}+2\right)-\left(-\frac{1}{3}-1\right) \)
\( \displaystyle = \frac{14}{3}+\frac{4}{3} = 6 \)