The area included between the parabolas $y^{2}=5 x$ and $x^{2}=5 y$ is
The area included between the parabolas $y^{2}=5 x$ and $x^{2}=5 y$ is
$\frac{25}{7}$ sq. units
$\frac{25}{3}$ sq. units
$\frac{25}{4}$ sq. units
$25 \mathrm{sq}$. units
Solution
Prabolas$y^{2}=5 x$ and $x^{2}=5 y$ intersect at
$\frac{x^{2}}{5}=5 x \Rightarrow x^{*}=5^{3} x \Rightarrow x\left(x^{3}-125\right)=0 \Rightarrow x=0.5 \Rightarrow y=0,5$
Let points of intersection be $Q(0,0)$ and $A(5.5)$. Required area is shaded. Ares included becween the parabolas
$=\int \sqrt{5 x}-\frac{x}{5} d x$
$\left.=\frac{\sqrt{5} x}{3}-\frac{x^{3}}{15}=\frac{2}{3}\right)(\sqrt{5})(5 \sqrt{5})-\frac{125}{15}$
$=\frac{50}{3}-\frac{125}{15}=\frac{25}{3}$
