The area (in sq. units) bounded between the curves \( y=x^2 \) and \( y=1 \) in the first quadrant is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{2}{3} \)
The curves meet where \( x^2=1 \), i.e. at x = 1 in the first quadrant. Between x = 0 and x = 1 the line y = 1 lies above the parabola.
\( \displaystyle \text{Area}=\int_0^1(1-x^2)\,dx \)
\( \displaystyle =\left[x-\frac{x^3}{3}\right]_0^1 \)
\( \displaystyle =1-\frac{1}{3}=\frac{2}{3} \)
Why the other options are incorrect
Option (2) \( \displaystyle\frac{4}{3} \) — Incorrect. This is the area on both sides of the y-axis; only the first quadrant is required.
Option (3) \( \displaystyle\frac{1}{3} \) — Incorrect. This is the area under the parabola from 0 to 1, not the area between the two curves.
Option (4) \( \displaystyle\frac{8}{3} \) — Incorrect. The area is \( \frac{2}{3} \), not \( \frac{8}{3} \).