Question

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

Answer 1

On solving the equation

$x^2+y^2=1$ and $y^2=1-x$

we get the point of induction as $(1,0)$ and $(0,1)(0,-1)$

Hence the required area is

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

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

$=2\times [\frac{\pi }{2}\times \frac{1}{2}]+2\times \frac{2}{3}$

$=\frac{\pi }{2}+\frac{4}{3}$ sq. units