Area of the region bounded by the curves \( y=x^{2} \), \( x=0 \), \( x=2 \) and x-axis is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{8}{3} \) sq. units
Since \( y=x^{2}\ge 0 \), the region lies above the x-axis, between the ordinates x = 0 and x = 2:
\( \displaystyle \text{Area}=\int_{0}^{2}x^{2}\,dx \)
\( \displaystyle =\left[\frac{x^{3}}{3}\right]_{0}^{2} \)
\( \displaystyle =\frac{8}{3}\text{ sq. units} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{16}{3} \) sq. units — Incorrect. \( \frac{16}{3} \) is the area on the other side of the curve, between \( y=x^{2} \), the y-axis and the line y = 4: the rectangle 2 × 4 = 8 minus \( \frac{8}{3} \).
Option (3) \( \displaystyle\frac{13}{6} \) sq. units — Incorrect. The area is \( \frac{8}{3} \), not \( \frac{13}{6} \).
Option (4) \( \displaystyle\frac{13}{3} \) sq. units — Incorrect. The area is \( \frac{8}{3} \), not \( \frac{13}{3} \).