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

`y^(2)=4aximpliesy=2sqrt(a)sqrt(x) " " `...(1)
` "and " x^(2)=4ayimpliesy=(x^(2))/(4a) " " `...(2)
On solving eqs. (1) and (2), we get the point of intersection are O(0, 0) and P(4a, 4a)

Now, `PQ bot OX`
Area of bounded region
`=` Area of OQPAP - Area of OQPBO
`=int_(0)^(4a)2sqrt(a)sqrt(x)dx-int_(0)^(4a)(x^(2))/(4a)dx`
`=2sqrt(a)[(2)/(3)x^(3//2)]_(0)^(4a)-(1)/(4a)[(1)/(3)x^(3)]_(0)^(4a)`
`=(4)/(3)sqrt(a).(4a)^(3//2)-(1)/(12a)(4a)^(3)`
`=(32)/(3)a^(2)-(16)/(3)a^(2)=(16)/(3)a^(2)sq.` units.