The area of the region bounded by the parabola \(y^2 = x\) and the straight line \(2y = x\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4/3 sq. units
Putting \(x = 2y\) in \(y^2 = x\) gives the points of intersection:
\(y^2 = 2y\)
\(y(y - 2) = 0,\quad y = 0 \text{ or } y = 2\)
The points are (0, 0) and (4, 2). For \(0 \lt y \lt 2\) the line \(x = 2y\) lies to the right of the parabola \(x = y^2\) (at y = 1: 2 > 1). Integrating with respect to y:
\( \displaystyle \text{Area} = \int_{0}^{2} (2y - y^2)\,dy \)
\( \displaystyle = \left[y^2 - \frac{y^3}{3}\right]_{0}^{2} \)
\( \displaystyle = 4 - \frac{8}{3} = \frac{4}{3} \)
The area is 4/3 sq. units.
NCERT Exemplar:
Application of Integrals
