The area of the region \(\{(x, y): y \ge x^2 \text{ and } y \le |x|\}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{3}\)
The region lies above the parabola \(y=x^2\) and below \(y=|x|\). Both curves are unchanged when x is replaced by −x, so the region is symmetric about the y-axis.
For x ≥ 0, \(|x|=x\). The curves y = x and \(y=x^2\) meet where
\(x=x^2\)
\(x=0 \text{ or } x=1\)
and for 0 < x < 1, \(x \gt x^2\). Area of the right half:
\( \displaystyle \int_0^1\left(x-x^2\right)dx=\left[\frac{x^2}{2}-\frac{x^3}{3}\right]_0^1 \)
\( \displaystyle =\frac{1}{2}-\frac{1}{3}=\frac{1}{6} \)
By symmetry, the left half has the same area:
\( \displaystyle \text{Area}=2\times\frac{1}{6}=\frac{1}{3} \)