Question

The area of the region $R=\{(x,y):|x|\le |y|$ and $x^2+y^2\le 1\}$ is

• A. $\frac{3\pi }{8}$ sq units
• B. $\frac{5\pi }{8}$ sq units
• C. $\frac{\pi }{2}$ sq units
• D. $\frac{\pi }{8}$ sq units

Answer 1

Answer: (C)

Required area $=4$ (Area of the shaded region in first quadrant)
$=4\underset{0}{\overset{1\sqrt{2}}{\int }}\left(y_1-y_2\right)dx=4\underset{0}{\overset{1\sqrt{2}}{\int }}\left(\sqrt{1-x^2}-x\right)dx$
$=4{\left[\frac{x}{2}\sqrt{1-x^2}+\frac{1}{2}{\sin }^{-1}x-\frac{x^2}{2}\right]}^{1/\sqrt{2}}$
$=4\left[\frac{1}{2\sqrt{2}}\times \frac{1}{\sqrt{2}}+\frac{1}{2}\times \frac{\pi }{4}-\frac{1}{4}\right]$
$=4\left[\frac{1}{4}+\frac{\pi }{8}-\frac{1}{4}\right]$
$=\frac{4\pi }{8}=\frac{\pi }{2}$ sq units