Question

The area of the region described by $A=\left\{\right(x,y):x^2+y^2\le 1\text{and}{y}^2\le 1-x\}$ 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 $\frac{\pi }{2}+\frac{4}{3}$

Required area $=P+Q$

Now, area of the region $P=\frac{1}{2}\times \pi \times (1{)}^2=\frac{\pi }{2}$
Now, area of the region $Q=\left|2\underset{0}{\overset{1}{\int }}\sqrt{1-x}dx\right|$
$={|2\frac{(1-x{)}^{3/2}}{\frac{3}{2}}\cdot (-1\left)\right|}_0^1$
$=|-\frac{4}{3}|$
$=\frac{4}{3}$
Hence, Required area $=\frac{\pi }{2}+\frac{4}{3}$