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The area of the region enclosed by ...

The area of the region enclosed by ` x ^ 2 + y ^ 2 = 2 , y ^ 2 = x ` and y - axis is

A

` ( pi ) /(4 ) + (1)/(3) `

B

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

C

` ( pi ) /(4 ) - (1)/(3) `

D

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

Text Solution

Verified by Experts

The correct Answer is:
D
`x^(2) + y^(2) = 2 , " "y^(2) = x`
Area = 2 ` int_(0)^(1) y^(2) dy + 2 int_(1)^(sqrt(2)) sqrt(2-y^(2)) dy`
`= [(2)/(3) y^(3)]_(0)^(1) + [ y sqrt(2-y^(2))+sin^(-1) .(y)/(sqrt(2))]_(1)^(sqrt(2))`
` = (2)/(3) + [pi - ( 1 + (pi)/(2)) ] = (pi)/(2) - (1)/(3) " ""Area" = (pi)/(2) - ( 1)/(3)`
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