Brain Teaser - Area enclosed by a parabola and a line

[fhub]

(09-15-2015 01:30 PM)Gerson W. Barbosa Wrote:  Franz, I'm trying to extend the problem to the biquadratic parabola (y = x^4). The expression for lines normal to it at a point P(u, u^4) was obtained from Thomas Klemm's method above.

Aaah ok, but that's in fact a very complicated calculation!

You get the left intersection point from the cubic equation x^3+u*x^2+u^2*x+1/(4*u^3)+u^3=0 (which is the quartic polynomial divided by (x-u)), and then you have to calculate the definite integral from this complicated left point to u.
And finally solve the terrible equation A'(u)=0 ...

Well, I get a different result:
u=0.743514881896 with A=1.22954355289

Franz