Question

The area bounded by the curves x = ay² and y = ax² is 1 unit, then find the value of a, a>0

Answer 1

See diagram.

Given curves :  a > 0,

       x = a y²    and  y =  a x²

The points of intersection of these two curves are:
       y = a x² = a (ay²)² = a³ y⁴
       y = 0   => x = 0.
  and     y = 1/a    =>  x = 1/a

Area between the two graphs:
[tex] = \int \limits_{0}^{1/a} {- ax^2 + \sqrt{\frac{x}{a}} } \, dx\\\\= - a[x^3 /3]_0^{1/a} + \frac{2x^{\frac{3}{2}}}{3\sqrt{a}}]_0^{1/a}\\\\=-\frac{1}{3a^2}+\frac{2}{3a^2}=\frac{1}{3a^2}\\\\=1 \: \: Given.\\\\a=\frac{1}{\sqrt{3}}[/tex]

So a = 1 / √3   Answer

Answer 2

a=1/√3 is my answer....