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Find the area of the region bounded by t...

Find the area of the region bounded by the curve `y^(2)=2x" and "x^(2)+y^(2)=4x`.

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The correct Answer is:
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We have, `y^(2)=2x" and "x^(2)+y^(2)=4x`

`rArr x^(2)+2x-4x`
`rArr x^(2)-2x=0`
`rArr x(x-2)=0`
`rArr x=0, 2`
Also, `x^(2)+y^(2)=-y^(2)`
`rArr x^(2)-4x=-y^(2)`
`rArr x^(2)-4x+4=-y^(2)+4`
`rArr (x-2)^(2)-2^92)=-y^(2)`
`:." Required area " =2. int_(0)^(2)[sqrt(2^(2)-(x-2)^(2))-sqrt(2x)]dx`
`=2[[(x-2)/2.sqrt(2^(2)-(x-2)^(2))+(2^(2))/2sin^(-1)((x-2)/2)]_(0)^(2)-[sqrt2 .x^(3//2)/(3//2)]_(0)^(2)]`
`=2[(0+0-1.0+3.pi/2)-(2sqrt2)/3(2^(3//2-0))]`
`=(4pi)/2 -8.2/3=2pi-16/3=2(pi-8/3)" sq units"`
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