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

Equations of the paravbolas are
`y^(2)=4ax and x^(2)=4ay`

Solving the equations
`((x^(2))/(4a))=4ax`
`:. X^(4)=64a^(3)x`
`:. x^(4)-64a^(3)x=0`
`:.x(x^(3)-64a^(2))=0`
`:.x=0 or x^(3)=64a^(3)`
`:.x^(3)=(4a)^(3)`
`:. x=4a`
when `x=0,y^(2)=4ax=0 :. y=0`
when `x=4a,y^(2)=4ax=4axx4a :. y=4a`
`:.` The points of intersecton of the parabolas are `O(0,0,0),P(4a,4a)`
`:.` The required area is bounded by `y^(2)=4ax and x^(2)=4ax` between x=0 and `x=4a`
`:. A= overset(4a)underset(0) int sqrt(4ax*)dx-overset(4a)underset(0) int (x^(2))/(4a)*dx`
`:. =sqrt(4a) overset(4a)underset(0) int sqrt(x*)dx(1)/(4a)-overset(4a)underset(0) int x^(2)*dx`
`=sqrt(4a)*(2)/(3)[x^(3//2)]_(0)^(4a)-(1)/(4a)xx(1)/(3)[x^(3)]_(0)^(4a)`
`=(4sqrt(a))/(3)[4a]^(3//2)-(1)/(12a)(4a)^(3)`
`=(4sqrt(a))/(3)xx4asqrt(4a)-(1)/(12a)xx64a^(3)`
`=(32)/(3)a^(2)-(16)/(3)a^(2)=(16)/(3)a^(2)sq`. units.