The area (in sq. units) of the region `{(x,y) : y^(2) ge 2x` and `x^(2) + y^(2) le 4x, x ge 0, y ge 0}` is

Options

A. `pi-(4)/(3)`
B. `pi-(8)/(3)`
C. `pi-(4sqrt(2))/(3)`
D. `(pi)/(2)-(2sqrt(2))/(3)`

Answer

b

Explanation

The required area A is given by
`A=underset(0)overset(2)|y_(1)-y_(2)|dx`

`implies A=underset(0)overset(2)int(sqrt(4x-x^(2))-sqrt(2x))dx`
`implies A = underset(0)overset(2)int(sqrt(2^(2)-(x-2)^(2))-sqrt(2x))dx`
`implies A =[(1)/(2)(x-2)sqrt(4x-x^(2))+2 sin^(-1)(x-2)/(2)(2sqrt(2))/(3)x^(3//2)]_(0)^(2)`
`implies A=-(8)/(3)-2(-(pi)/(2))=pi-(8)/(3)`