Question

Find the area of the minor segment of the circle x² + y² = 4 cut off by the line x = 1, using integration.

Answer 1

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`