Find the area of the region included between the parabolas `y^2=4a xa n dx^2=4a y ,w h e r ea > 0.`

Explanation

`Rightarrow frac{x^{4}}{16 a^{2}}=4 a x quad Rightarrow quad x^{4}=64 a^{3} x`


`Rightarrow x(x^{3}-64 a^{3})=0 quad Rightarrow quad x=0` or `x=4 a`


We plot the curves on same system of axes to get the required region.



`text { The enclosed area }=int_{0}^{4 a}(sqrt{4 a x}-frac{x^{2}}{4 a}) d x `


`=2 sqrt{a} frac{2}{3} x^{frac{3}{2}}-frac{x^{3}}{12 a}]_{0}^{4 a} `


`=frac{4}{3} sqrt{a}(4 a)^{frac{3}{2}}-frac{(4 a)^{3}}{12 a}-0=frac{32 a^{2}}{3}-frac{16 a^{2}}{3}=frac{16 a^{2}}{3} text { square units. }`