The area (in sq. units) bounded by the curves \( x=y^2 \) and \( x=y \) in the first quadrant is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{1}{6} \)
The curves meet where \( y^2=y \), i.e. at y = 0 and y = 1. For 0 < y < 1, \( y\gt y^2 \), so the line x = y is to the right of the parabola.
\( \displaystyle \text{Area}=\int_0^1(y-y^2)\,dy \)
\( \displaystyle =\left[\frac{y^2}{2}-\frac{y^3}{3}\right]_0^1 \)
\( \displaystyle =\frac{1}{2}-\frac{1}{3}=\frac{1}{6} \)