The parabola y2=x divides the circle x2+y2=2 into two parts whose areas are in the ratio:
Solutions · 1
Official solution
The circle has area 2π. The parabola meets the circle at (1,±1).
\[\text{Area}(\text{parabolic region}) = 2\int_0^1\sqrt x\,dx + 2\int_1^{\sqrt{2}}\sqrt{2-x^2}\,dx\]
\[= 2\cdot\frac23 + 2\left[\frac x2\sqrt{2-x^2}+\sin^{-1}\frac{x}{\sqrt{2}}\right]_1^{\sqrt{2}} = \frac43 + 2\left(\frac\pi2-\frac\pi4-\frac12\right) = \frac{3\pi+2}{6}\]
Larger part =2π−63π+2=69π−2.
\[\text{Ratio} = (9\pi-2):(3\pi+2)\]