Let the area of the region bounded by the curves y = 5-x^2 and y = |x-1| be A . If 4A = - , where and are positive integers, then the value of + is equal to

Step-by-step solution

The required area is bounded above by the semicircle y = 5-x^2 and below by the V-shaped curve y = |x-1| . First, find the points of intersection of the two curves: For x 1 , y = x-1 . x-1 = 5-x^2 x^2 - 2x + 1 = 5 - x^2 2x^2 - 2x - 4 = 0 x^2 - x - 2 = 0 (x-2)(x+1) = 0 x = 2 (since x 1 ). For x 1-x = 5-x^2 1 - 2x + x^2 = 5 - x^2 2x^2 - 2x - 4 = 0 x^2 - x - 2 = 0 (x-2)(x+1) = 0 x = -1 (since x The area A is given by: A = _ -1 ^2 5-x^2 dx - _ -1 ^2 |x-1| dx The second integral represents the area of two triangles form