Question

The area of the region described by $A=\left(\right(x,y):x^2+y^2\le 1)$ and $B=\left(\right(x,y):y^2\le 1-x)$

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

Answer 1

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

Area $ABCA$ $\frac{\pi }{2}{r}^2=\frac{\pi }{2}$

Due to symmetry

$ADCA=2AODA$

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

$=\frac{-2\left[\right(1-x{)}^{3/2}{]}_0^1}{\frac{3}{2}}$

$=\frac{-4}{2}-1=\frac{4}{3}$

Hence the total area = $\frac{\pi }{2}$+ $\frac{4}{3}$