Area common to the circle `x^(2)+y^(2)=9` an the parbola `y^(2)=8x` is

Options

A. 0
B. `(2sqrt(2))/(3)+(9pi)/(2)-9 "sin"^(-1) ((1)/(3))`
C. `16 pi`
D. None of these

Answer

B

Explanation

The points of intersection of two curves are A `(1,2sqrt2)andB(1,-2sqrt2)`.

`:.` Required area
`=2[int_(0)^(1)ydx+int_(1)^(3)ydx]`
`=2[int_(0)^(1)2sqrt2(x)^(1\//2)dx+int_(1)^(3)sqrt(9-x^(2)dx)]`
`=2[2sqrt2[(x^(3//2))/(3//2)]_(0)^(1) +[(x)/(2)sqrt(9-x^(2))+(9)/(2)sin^(-1)((x)/(3))]_(1)^(3)]`
`=2[(4sqrt2)/(3)+(9)/(2)xx(pi)/(2)-(sqrt(8))/(2)-(9)/(2)sin^(-1)((1)/(3))]`
`=(2sqrt2)/(3)+(9pi)/(2)-9sin^(-1)((1)/(3))`