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The area of the region bounded by the cu...

The area of the region bounded by the curves `x^2 + y^2 = 8 and y^2=2x` (in sq. unit) is

A

`2pi+(1)/(3)`

B

`pi+(1)/(3)`

C

`2pi+(4)/(3)`

D

`pi+(4)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given curves `x^(2)+y^(2)=8" ... (i)"`
and `y^(2)=2x" ... (ii)"`

On solving Eqs. (i) and (ii), we get
`x^(2)+2x-8=0`
`x^(2)+4x-2x-8=0`
`x(x+4)-2(x+4)=0`
`(x-2)(x+4)=0`
`therefore` x = 2 and `y = pm 2`
`therefore` Required area = 2 [Area of OAP + Area of PAB]
`2[int_(0)^(2)sqrt(2x)dx+int_(2)^(2//sqrt(2))sqrt(8-x^(2))dx]`
`=2[sqrt(2)(x^(3//2).(2)/(3))_(0)^(2)+((x)/(2)sqrt(8-x^(2))+(8)/(2)sin^(-1).(x)/(2sqrt(2)))_(2)^(2sqrt(2))]`
`=2[(2sqrt(2))/(3).2sqrt(2)+2pi-2-pi]`
`=2[(8)/(3)-2+pi]=2((2)/(3)+pi)=2pi+(4)/(3)`
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