Question

The area of the region described by $A=\{(x,y)x^2+y^2\le 1and{y}^2\le 1-x\}$ is

• A. $\frac{\pi }{2}+\frac{4}{3}$
• B. $\frac{\pi }{2}-\frac{4}{3}$
• C. $\frac{\pi }{2}-\frac{2}{3}$
• D. $\frac{\pi }{2}+\frac{2}{3}$

Answer 1

The correct option is A $\frac{\pi }{2}+\frac{4}{3}$

So area is $\frac{\pi }{2}+(\frac{\pi }{2}$ - unshaded region or inside circle Region II )

${\int }_0^1{\int }_{-\sqrt{1-x}}^{\sqrt{1-x}}dydx$

$={\int }_0^{1}2\sqrt{1-x}dx$

$={(-\frac{2(1-x)3/2}{3})}_0^1\times 2$

$=4/3$

$\therefore \frac{\pi }{2}+\frac{4}{3}$

A part