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Find the area bounded by the curves x^(2...

Find the area bounded by the curves `x^(2)+y^(2)=4,x^(2)=-sqrt(2)y, and x=y`.

Text Solution

Verified by Experts

The correct Answer is:
`((1)/(3)-pi)` sq units
Given curves are `x^(2) +y^(2)=4,x^(2)= -sqrt(2)y and x=y`.

Thus, the required area
`=|int_(sqrt(2))^(sqrt(2)) sqrt(4-x^(2)) dx|-|int_(-sqrt(2))^(0)x dx|-|int_(0)^(sqrt(2))(-x^(2))/(sqrt(2)) dx| `
`=2int_(0)^(sqrt(2)) sqrt(4-x^(2))dx - |((x^(2))/(2))_(-sqrt(2))^(0)|-|(x^(3))/(3sqrt(2))|_(0)^(sqrt(2))`
`=2{(x)/(2) sqrt(4-x^(2))-(4)/(2) "sin"^(-1)(x)/(2)}_(0)^(sqrt(2)) -1-(2)/(3)`
`=(2-pi)-(5)/(3)`
`=((1)/(3)-pi)` sq units
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