Question

Find the area of the region bounded by the curves $y^2=4x,x^2=4y$.

Answer 1

Given equation of curves are
$y^2=4x....\left(i\right)$
and $x^2=4y....\left(ii\right)$
$\Rightarrow \left(\frac{x^2}{4}\right)=4x$
$\Rightarrow \frac{x^4}{16}=4x$
$\Rightarrow x^4=64x$
$\Rightarrow x^4-64x=0$
$\Rightarrow x(x^3-4^3)=0$
$\Rightarrow x=4,0$
$\therefore$ Area of shaded region, $A={\int }_0^4(\sqrt{4x}-\frac{x^2}{4})dx$
$={\int }_0^4(2\sqrt{x}-\frac{x^2}{4})dx$

$={[\frac{2x^{3/2}\times 2}{3}-\frac{1}{4}\times \frac{x^3}{3}]}_0^4$

$=\frac{2.2}{3}\times 8-\frac{1}{4}\times \frac{64}{3}-0=\frac{32}{3}-\frac{16}{3}=\frac{16}{3}squnits$.