The area of the region bounded by the curves \( x^2=4y \) and \( x=4y \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{24} \) Sq.unit
The curves meet where \( x^2=x \), i.e. at x = 0 and x = 1. For 0 < x < 1, \( x\gt x^2 \), so the line \( y=\frac{x}{4} \) is above the parabola \( y=\frac{x^2}{4} \).
\( \displaystyle \text{Area}=\int_0^1\left(\frac{x}{4}-\frac{x^2}{4}\right)dx \)
\( \displaystyle =\frac{1}{4}\left[\frac{x^2}{2}-\frac{x^3}{3}\right]_0^1 \)
\( \displaystyle =\frac{1}{4}\cdot\frac{1}{6} \)
\( \displaystyle =\frac{1}{24}\ \text{sq. unit} \)