Question

The area bounded by the curves $x=a{y}^2$ and $y=a{x}^2$is $1$ unit, then fins the value of $a,a>0$.

Answer 1

First we solve for point P.

$y=a{x}^2$ and $x=a{y}^2$

$⟹x=a(a{x}^2{)}^2$

$⟹x=a^3{x}^4$

$⟹x=0,\frac{1}{a}$

Now, $y_1=\sqrt{\frac{x}{a}}$ and $y_2=a{x}^2$

Area$=A={\int }_0^{\frac{1}{a}}(y_1-y_2)dx$

$⟹A={\int }_0^{\frac{1}{a}}(\sqrt{\frac{x}{a}}-a{x}^2)dx$

$⟹A=\frac{2}{3\sqrt{a}}{\left[x^{\frac{3}{2}}\right]}_0^{\frac{1}{a}}-\frac{a}{3}{\left[x^3\right]}_0^{\frac{1}{a}}$

$⟹1=\frac{2}{3a^2}-\frac{1}{3a^2}$

$⟹a=\frac{1}{\sqrt{3}}$