# The area (in sq. units) of the region

Question:

The area (in sq. units) of the region $\mathrm{A}=\left\{(x, y):|x|+|y| \leq 1,2 y^{2} \geq|x|\right\}$ is:

1. (1) $\frac{1}{3}$

2. (2) $\frac{7}{6}$

3. (3) $\frac{1}{6}$

4. (4) $\frac{5}{6}$

Correct Option: , 4

Solution:

Required area $=4\left[\int_{0}^{\frac{1}{2}} 2 y^{2} d y+\frac{1}{2} \operatorname{area}(\Delta P A B)\right]$

$=4\left[\frac{2}{3}\left[y^{3}\right]_{0}^{\frac{1}{2}}+\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\right]=4\left[\frac{2}{3} \times \frac{1}{8}+\frac{1}{8}\right]$

$=4 \times \frac{5}{24}=\frac{5}{6}$