Question

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

• A. $\frac{1}{6}$
• B. $\frac{5}{6}$
• C. $\frac{1}{3}$
• D. $\frac{7}{6}$

Answer 1

The correct option is B $\frac{5}{6}$


Total area =$4{\int }_0^{\frac{1}{2}}\left[\right(1-x)-\left(\sqrt{\frac{x}{2}}\right)]dx$
$=4[x-\frac{x^2}{2}-\frac{1}{\sqrt{2}}\frac{x^{\frac{3}{2}}}{\frac{3}{2}}{|}_0^{1/2}]$
$=4[\frac{1}{2}-\frac{1}{8}-\frac{\sqrt{2}}{3}{\left(\frac{1}{2}\right)}^{\frac{3}{2}}]$
$=4\times \frac{5}{24}=\frac{5}{6}$