The area bounded by the curve y^2=2x,y- axis and the line y=1 , is

The correct answer is Option (4) → 1/6 Sq. units

The parabola \( y^2=2x \) passes through the origin and meets the line y = 1 at \( x=\frac{1}{2} \). The region lies between the y-axis and the curve for 0 ≤ y ≤ 1, so it is covered by horizontal strips with \( x=\frac{y^2}{2} \):

    \( \displaystyle \text{Area}=\int_0^1 x\,dy \)

    \( \displaystyle =\int_0^1\frac{y^2}{2}\,dy \)

    \( \displaystyle =\left[\frac{y^3}{6}\right]_0^1 \)

    \( \displaystyle =\frac{1}{6} \)