Question

The area of the region $R=\{(x,y):\left|x\right|\le \left|y\right|\}$ 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

The correct option is C $\frac{\pi }{2}$ sq. units

Required area$=4$ (Area of the shaded region in first quadrant)
$=4{\int }_0^{1/\sqrt{2}}(y_1-y_2)dx=4{\int }_0^{1/\sqrt{2}}(\sqrt{1-x^2}-x)dx$
$=4{[\frac{x}{2}\sqrt{1-x^2}+\frac{1}{2}{\sin }^{-1}x-\frac{x^2}{2}]}_0^{1/\sqrt{2}}$
$=4[\frac{1}{2\sqrt{2}}\times \frac{1}{\sqrt{2}}+\frac{1}{2}\times \frac{\pi }{4}-\frac{1}{4}]=\frac{\pi }{2}$ sq. units