The parabola y^2=x divides the circle x^2+y^2=2 into two parts whose areas are in the ratio

Step-by-step solution

aligned & Area of circle = ( 2 )^2=2 & Area of O C A D O=2 Area of O C A O & =2 area of O C B+ area of B C A & =2 ₀^1 y_p d x+2 ₁^ 2 y_c d x & where y_p= x and y_c= 2-x^2 & Required Area & =2 1 x d x+2 ₀^ 2 2-x^2 d x & =2 [ 2 3 1-0 ]+2 [ x 2-x^2 2 + ⁻¹ x 2 ]₁^ 2 aligned aligned & = 4 3 +2 2 - 4 - 1 2 = 4 3 +2 4 - 1 2 = 3 +2 6 & Bigger area =2 - 3 +2 6 = 9 -2 6 & Required Ratio = 9 -2 3 +2 i.e., 9 -2: 3 +2 aligned