Let A(k) be the total area of the regions bounded by the curve y = |x^2 - 1| and the line y = k , where 0 < k < 1 . If the value of k for which A(k) is minimum is p q where p and q are coprime positive integers, find the value of p+q .

Step-by-step solution

The line y = k intersects the curve y = |x^2 - 1| where |x^2 - 1| = k . This gives x^2 - 1 = k x = 1+k and 1 - x^2 = k x = 1-k . The bounded regions consist of one central region above y=k and two symmetric regions below y=k . Area of the central region A₁ is bounded above by y = 1 - x^2 and below by y = k for x [- 1-k , 1-k ] : A₁ = 2 ₀^ 1-k (1 - x^2 - k) dx = 2 [ (1-k)x - x^3 3 ]₀^ 1-k = 4 3 (1-k)^ 3/2 Area of the two symmetric regions A₂ bounded above by y = k and below by y = |x^2 - 1| for x [ 1-k , 1+k ] and i