Question

The figure here shows triangle AOC inscribed in the region cut from the parabola y=x^2 by the line y=a^2. Find the limit of the ratio of the area of the triangle to the area of the parabolic region as a approaches zero.

Answer 1

Area of the parabolic region = Integral of [a^2 - x^2 ]dx | from - a to a =

(a^2)x - (x^3)/3 | from - a to a = (a^2)(a) - (a^3)/3 - (a^2)(-a) + (-a^3)/3 =

= 2a^3 - 2(a^3)/3 = [4/3](a^3)

Area of the triangle = [1/2]base*height = [1/2](2a)(a)^2 = a^3

ratio area of the triangle / area of the parabolic region = a^3 / {[4/3](a^3)} =

Limit of a^3 / {[4/3](a^3)} as a -> 0 = 1 /(4/3) = 4/3