Question

Find the area of the region common to the circle x² + y² =9 and the parabola y² =8x

Answer 1

`x^2+y^2=9   and y^2=8x`

`x^2+8x=9`

`x^2+8x-9=0`

`therefore (x+9)(x-1)=0`

`therefore x=1 or x=-9 `

`therefore y=+-2sqrt2`

 ∴ The points of intersections are

`P(1,2sqrt2) and Q(1,-2sqrt2)`

`y^2=8x`

`therefore y=sqrt8sqrtx=2sqrt2 x^(1/2)-> f_1(x)`

`and x^2+y^2=9 therefore y^2=9-x^2`

`therefore y=sqrt(9-x^2)-> f_2(x)`

Required area,
= Area OPAQO = 2 Area OPAMO
= 2(Area OPMO + Area APMA)

`=2[int_0^1f_1(x)dx+int_1^3f_2(x)dx]`

`=2[int_0^12sqrt2 x^(1/2)dx+int_1^3sqrt(9-x^2)dx]`

`=2[2sqrt2(x^(3/2)/(3/2))_0^1+(x/2sqrt(9-x^2)+9/2sin^-1(x/3)_1^3)]`

`=2[(4sqrt2)/3+(3/2(0)+9/2sin^-1 (1)-1/2sqrt8-9/2sin^-1 (1/3))]`

`=2[(4sqrt2)/3+9/2.x/2-sqrt2-9/2sin^-1 (1/3)]`

`=2[(sqrt2/3+(9pi)/4-9/2 sin^-1 (1/3)) sq. units`