Question

Find area of common region of circle $x^2+y^2=2$ and parabola $y^2=x$

Answer 1

$y^2=x{x}^2+y^2=2x^2+x-2=0x^2+2x-x-2=0x(x+2)-1(x+2)=0$

$x=1,-2$

$A=2[{\int }_0^1\sqrt{x}dx+{\int }_1^{\sqrt{2}}\sqrt{2-x^2}dx]A=2\{{\left[\frac{2x^{3/2}}{3}\right]}_0^1+{[\frac{x}{2}\sqrt{2-x^2}+{\sin }^{-1}\left(\frac{x}{\sqrt{2}}\right)]}_1^{\sqrt{2}}\}A=\frac{4}{3}+2[0+{\sin }^{-1}\left(1\right)-(\frac{1}{2}+{\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)]A=\frac{4}{3}+2[\frac{\pi }{2}-\frac{1}{2}-\frac{\pi }{4}]A=\frac{4}{3}+\pi -1-\frac{\pi }{2}A=\frac{2+3\pi }{6}$