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Question
Find the area of the minor segment of the circle x2 + y2 = 4 cut off by the line x = 1, using integration.
Solution
Given equation of circle x2 + y2 = 4
Required Area = `int_1^2 y dx`
= `2int_1^2 sqrt(4 - x^2)dx`
= `2[x/2 sqrt(4 - x^2) + 4/2 sin^-1 x/2]_1^2`
= `2[0 + 2 sin^-1 (1) - (1/2 sqrt(3) + 2 sin^-1 (1/2))]`
= `4(π/2) - sqrt(3) - 4(π/6)`
= `(2π - (2π)/3) - sqrt(3)`
= `(4π - 3sqrt(3))/3`
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