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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.

The figure here shows triangle AOC inscribed in the region cut from the parabola yx2 by the line ya2 Find the limit of the ratio of the area of the triangle to class=

Respuesta :

Edufirst
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

 



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