Question

Find the area of the region included by the curves $x^2=5y$ and $y^2=5x$

Answer 1

Intersection point -

$\Rightarrow x^2=5y$

$y=\frac{x^2}{5}$

$\Rightarrow y^2=5x$

$\Rightarrow {\left(\frac{x^2}{5}\right)}^2=5x$

$\frac{x^4}{5^2}=5x$

$x^3=5^3$

$\Rightarrow x=5$

$\Rightarrow$ Area bounded $=\left|{\int }_0^5\frac{x^2}{5}-\sqrt{5x}\right|$

$=\left|{[\frac{x^3}{15}-\sqrt{5}\times \frac{2}{3}{x}^{3/2}]}_0^5\right|$

$=\mathrm{8.3333.}$

Hence, the answer is $\mathrm{8.3333.}$