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

i find the ratio to be a constant 3/4

Answer 2

Area of triangle = (1/2)*2a*a^2 = a^3

Area of parabolic region =
\[2\int\limits_{0}^{a}(a^{2}-x^{2}) dx = \frac{4a^{3}}{3}\]

Ratio: \[\frac{a^{3}}{\frac{4a^{3}}{3}} = \frac{3}{4}\]

Answer 3

Right, thank you very much.

Answer 4

your welcome