Question

Find the area of a minor segment of the circle x² + y² = a² cut off by the line x = `"a"/2`

Answer 1

Solving the equation x² + y² = a² and x = `"a"/2`

We obtain their points of intersection which are `("a"/2, sqrt(3) "a"/2)` and `("a"/2, - (sqrt(3)"a")/2)`.

Hence, From the figure in the question, we get

Required Area = 2 Area of OAB

= `2 int_("a"/2)^"a" sqrt("a"^2 - x^2)  "d"x`

= `2[x/2 sqrt("a"^2 - x^2) + "a"^2/2 sin^-1  x/"a"]_("a"/2)^"a"`

= `2["a"^2/2 * pi/2 - "a"/4 * "a" sqrt(3)/2 - "a"^2/2 * pi/6]`

= `"a"^2/12 (6pi - 3sqrt(3) - 2pi)`

= `"a"^2/12 (4pi - 3sqrt(3))` sq.units