Find the area, lying above the x=axis and included between the circle `x^2+y^2=8x` and the parabola `y^2=4xdot`

Explanation

We have, given equations `x^2+y^2=8x` .....(i)
`y^2=4x` .....(ii)
equation can be written as `(x-4)^2 +y^2 = 4^2`
so equation (i) represents a circle with center (4,0) and radius 4 .
we have points of intersection `0(0,0)` and `A(4,4),C(4,-4)` , now we have the area of region bounded
Required area of OBAO =` int_0^4 ( sqrt(8x-x^2) - sqrt(4x))dx`
= ` int_0^4 ( sqrt(4^2- (x-4)^2) - 2sqrt(x))dx`
{ using this formula `int sqrt(a^2-x^2).dx= (x/2) sqrt(a^2-x^2) + a^2/(2) sin^-1(x/a) +c `}
by solve the equation we get A= `(4pi - (32)/3 )`sq. units