Brain Teaser - Area enclosed by a parabola and a line

[Gerson W. Barbosa]

Just for the record, here are some numerical results for the cases y=x^4, y=6 and y=8. Notice the minimum areas occur for normal lines passing near y(3/4), y(5/6) and y(7/8), respectively. The values of u for minimum areas should be accurate to 7 or 8 digits.

y=x^4

u: 0.730000000000 Area: 1.232203807010
u: 0.740000000000 Area: 1.229721793524
u: 0.741869418802 Area: 1.229582549556
u: 0.743490000000 Area: 1.229543561785
u: 0.743514882370 Area: 1.229543552881
u: 0.743515349953 Area: 1.229543552884
u: 0.743543197140 Area: 1.229543564411
u: 0.744136760955 Area: 1.229549111764
u: 0.75000000000o Area: 1.230145031369


y=x^6

u: 0.810000000000 Area: 1.120523733964
u: 0.814713174965 Area: 1.119930602299
u: 0.814723174965 Area: 1.119930598819
u: 0.814724747205 Area: 1.119930598753
u: 0.814733174965 Area: 1.119930600634
u: 0.815000000000 Area: 1.119932604732
u: 0.820000000000 Area: 1.120664993006


y=x^8

u: 0.86790000000 Area: 1.24466886334
u: 0.86800000000 Area: 1.24466861630
u: 0.86796579708 Area: 1.24466859303
u: 0.86797579708 Area: 1.24466858824
u: 0.86798579708 Area: 1.24466859303
u: 0.86804996475 Area: 1.24466885164
u: 0.86810000000 Area: 1.24466932686

Gerson.